vVj t Wilis CORNELL UNIVERSITY LIBRARY MATHEMATICS CORNELL UNIVERSITY LIBRARY 924 067 762 280 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/cletails/cu31924067762280 A TREATISE SOLID GEOMETEY. A 1:eeatise SOLID GEOMETRY BY THE REV. PEECIVAL FEOST, M.A. LATE FELLOW OP ST JOHN'S COLLEOE, MATHEMATICAL LECTUBEB OF EINS'S COLLEGE, AND THE BEV. JOSEPH WOLSTENHOLME. M.A. FELLOW AND ASSISTANT TUTOK OF CHEIST'S OOLLEQE. etambrttrge anJj Utonljott: MACMILLAN AND CO. 1863 [The Right of Trar^laiion is reserved.] PREFACE. surfaces of the second degree, and, in doing so, their object has been to direct the student to the selection of the methods which are best adapted to the exigencies of each problem. In the more diflScult portions of the subject, they have considered themselves to be address- ing a higher class of students, and here they have tried to lay a good foundation on which to build, if any of their readers should wish to pursue their stu- dies in any department of the science, beyond the limits to which the work extends. The authors would willingly have given references to all the writers from whom they have derived in- formation in the course of their work, but they have found this to be impossible, and they regret it the less, because it will not be supposed that they lay claim to every thing in "VKbich they have made no reference. They have, however, in a very large number of cases mentioned the names of eminent men, who have ad- vanced the boundaries of the subject, and they hope it will be apparent, that they have appreciated the labours of such men as Cayley, Salmon, M^CuUagh, Roberts and Townsend; at all events they are sen- sible that, in many departments,- the treatise lately published by Salmon on the same subject proves how far their own work is from being perfect. They cannot conclude this work without making PEEFACE. VU acknowledgments to Mr Ferrers of Caius College, and Mr Home of St John's College, for their kindness in examining and commenting upon the proof sheets of the earlier parts of their work, and at the same time without expressing their regret that they have not escaped a large number of errors, which it will be punishment enough to them to see tabulated in an adjoining page. ERBATA. Page ^3, for minimum, read maximum. „ 15, Problem !£ is wrong. „ 42, line 17, for (63), read (61). ,, pa, „ II, for in, reati is. „ 119, „ 30, /or | = ooa(9 + o), reod | = Bin (9+a). „ tJ. „ ^4, /or sin 9, read coa ff. z . „ ti. „ 27, /or - Bin 6 coa o, read - sin 9 am a. „ 126, „ 14, for >', read »'. „ I2g, „ 10 and 13, omit t throughout. „ 143, „ S3, /or suppose, ♦■cad surface. „ 167, „ 23, for the othera, reaci another, and for the three, read two of the three. „ 204, „ 51, imert +ii,vm. ,, t6. „ 35, /or TO - rZ, read »-«2- »J;. „ Hi, „ 29, coffer cylinder, write or cone. „ 243, „ 23, remove I from ^„_a to D'ip.. ,, 246, ,, 8, for theae, read three. „ 269, „ 5, for pole, read polar. j> 306, „ 8, after cosines, write multiplied by .4, £, C, D respectively. Lithograph to face jaage 393. CONTENTS. CHAPTER I. CO-ORDINATE SYSTEMS. PAQB System of three planes » Four-plane system 4 Fundamental relation of co-ordmatea 5 Tetrahedral co-ordinates ,, Polar system 6 CHAPTER II. GENERAL DESCRIPTION OF LOCI OF EQUATIONS. Nature of the locus of an equation in the three-plane system 9 Locus of the polar equation , 13 Curves in space 14 Defect in the representation of curves by the intersection of surfaces 1 5 CHAPTER III PROJECTIONS OF LINES AND AREAS. DIRECTIONS OF LINES. Definitions of projection 16 Projection of a straight line upon a straight line „ Direction-cosines of a line 17 Fundamental relation between direction-cosines 18 Angle between two lines in terms of their direction-cosines , „ Direction-ratios 19 Fundamental relation between directioUiratios , , Angle between two lines in terms of their direction -ratios 20 Projection of a line on a plane ,, Projection of a plane area on a plane. 21 Area of a plane figure in terms of its prcgections on co-ordinate planes 22 h CONTENTS. CHAPTER IV. DISTANCES OP POINTS. EQUATIONS OF A STRAIGHT LINE. PAGE Distance between two points. Bectangular axes ^" Oblique axes ^7 Polar system '^S Equations of a straight line. Syijmetrical equations ^9 Non-symmetrical equations 3° Number of independent constants >> Angle between two straight lines 3^ Conditions of parallelism 33 „ perpendicularity 34 „ intersection ,, Equations of a straight line satisfying given conditions. Passing through a given point 3S „ two given points „ ,, a given point and parallel to a given straight line ... ,, Passing through a given point, perpendicular to and intersecting a given straight line 56 Passing through a given point, parallel to a given plane, and inter- secting a given straight line ,, Distance from a given point to a given line 3jr Shortest distance between two lines 38 Equations of a straight line in the four-plane system 39 CHAPTER V. GENERAL EQUATION OP THE FIRST DEGREE. Locus of general equation of first degree is a plane 43 Equation of a plane 44. Angle between two planes 46 Angle between a straight line and plane ^^ Length of the perpendicular from a point on a plane. Kectangular system ,j Oblique system 4y Distance of apoint from a, plane measured in any direction 48 Various forms of equation of a plane. Transversal form ...-. 40 Non-symmetrical form ; 50 Polar system 51 CONTENTS. XI PAGE Pour-plane system. General equation of the first degree represents a plane 51 Interpretation of constants 52 Condition of parallelism of two planes 53 Equation of a plane under particular conditions. ; through a given point 53 : through two given points 55 Passing through three given points „ Passing through a given point, and parallel to a given plane 56 Passing through two given points, and parallel to a given line „ Passing through a given point, and parallel to two given straight lines 57 Equidistant from two given straight lines not in the same plane 58 CHAPTEE VI. FOUR-POINT CO-OEDINATE SYSTEM. General explanation of the four- point co-ordinate system 61 Distance of a point from a plane ,, Equation of a point 62 Interpretation of constants 63 Equation of a point dividing a straight line in a given ratio „ Point at an infinite distance 65 Position of a point relative to the fundamental tetrahedron 66 Distance between two points 67 Fundamental relation between the co-ordinates of a plane _ 69 Perpendicular from a point upon a plane referred to tetrahedral co-ordinatea 7 1 Condition of a parallelism of two planes Ji General equation of a straight line ,> CHAPTEK VII. TRANSFORMATION OF CO-ORDINATES. Change of the origin of co-ordinates • > 75 Transformation from one system to another of rectangular co-ordinates 76 Euler's formulae 78 Transformation from a system of rectangular to one of oblique co-ordinates... 83 Transformation from one system to another of oblique co-ordinates „ The degree of an equation cannot be changed by transformation of co-ordi- nates 85 52 xii CONTENTS. FAOIE Transformation from rectangle to polar co-ordinates °5 Tranaformation from a four-plane to a three-plane co-ordinate system , , Transformation from one four-point system to another °o CHAPTER VIII. ON CERTAIN SURFACES OF THE SECOND ORDER. Equation of a, sphere °° Passing through a given point °9 Passing through two given points u Touching the three axes 9° Touching a plane in a given point » Kadical plane of two spheres 9^ Poles of similitude 9^ Equation of cylindrical surface 94 Equation of a conical surface on elliptic base 95 Equation of an oblique circular cone > Circular sections of oblique circular cone 9^ Equation of a spheroid 97 Equation of an ellipsoid 9^ Construction of the locus of the equation 99. Equation of an hyperboloid of one sheet „ Construction of the locus of the equation 100 Asymptotic surface loi Equation of an hyperboloid of two sheets 103 Construction of the locus of the equation 104 Asymptotic surface '. 105 Equation of the elliptic paraboloid 106 Construction of the locus of the equation 107 Equation of th* hyperbolic paraboloid 108 Construction of the locus of the equation 109 Asymptotic surface „ Principal sections 112 CHAPTER IX. ON GENERATION BY LINES AND CIRCLES. Generating lines of an hyperboloid of one sheet 115 Two lines of the same system do not intersect 117 Lines of opposite systems intersect '. 118 Generation of an hyperboloid by the motion of a straight line intersecting three fixed straight lines ^^ CONTENTS. xiii PAGE Figure representing the generating linea of an hyperboloid 121 Generating lines of a hyperbolic paraboloid 122 Lines of the same system do not, those of opposite systems do intersect 122 Figure representing the generating lines of a hyperbolic paraboloid 124 Plane section of a central surface, when circular ; locus of centers of circular seetioDS 12s Two circular sections of opposite systems lie in one sphere 128 Circular sections of non-central surfaces Circular sections of an eUiptio paraboloid of opposite systems Ke on one , sphere 131 • CHAPTEE X. MODULAR ASD UMBILICAL GENERATION OF SURFACES OF THE SECOND DEGREE. General account of modular generation 135 Surfaces capable of modular generation 136 Modular, focal, and dirigent conies for central surfaces 137 Focal conies for conical surfaces 139 Nature of the two moduli and modular conies in the principal central sur- faces 140 General relation between the moduli in central surfaces 1 ,, Modular conies for non-central surfaces + . 141 Tracing of the changes of surfaces and real focal conies corresponding t* changes of the modulus from o to 00 143 General account of umbilical generation ,, Surfaces capable of umbilical generation 144 Umbilical, focal, and dirigent conies for central surfaces 145 Umbilioalj focal, and dirigent conies for non-central surfaces 146 Coniooid 147 Properties of coniooids deduced by the modular and umbilical methods of generation 148 Properties of cones of the second degree 150 Reciprocal cones 151 CHAPTER XI. GENERAL EQUATION OF THE SECOND DEGREE. General account of the reduction by transformation of co-ordinates to a simpler form IS5 Transformation so that the products of the variables vanish. Discriminating cubic 156 Discriminating cubic has always real roots 160 xiv CONTENTS. PAGE SurtaceB wUch are loci of the general equation of the second degree:..... i6i Conditions that the general equation may represent particular species of surfaces. Central surfaces ^ Non-central surfaces ' Elliptic or hyperbolic cylinder '"7 Parabolic cylinder " Two planes '^^ CHAPTER XII. DIAMETRAL SUKFACES. DIAMETKAL PLANES. CONJUGATE DIAMETERS.* Diametral surfaces. Degree of diametral surface '7^ Diametral plane for a system of chords in a central conicoid I73 Conditions that, of three central planes of a central conicoid, each may be diametral to the intersection of the other two i74 Relations between the co-ordinates of the extremities of three conjugate diameters of a conicoid ft Relations between the lengths of three conjugate diameters and the angles between them '75 Diametral plane for a system of chords of non-central conicoids 178 Diametral plane for a system of chords of any conicoid 179 Principal planes of any conicoid. Discriminating cubic 180 The three principal planes of any conicoid are at right angles 181 Maxima and minima central radii ,, Nature of the section of a conicoid made by a given plane 182 Similarity of sections of similar and similarly situated conicoids, made by parallel planes 187 Area of a plane section of a central conicoid ,, Area of a plane section of an elliptic paraboloid 190 CHAPTER XIII. DEGREES OP SURFACES AND CURVES. INTERSECTIONS OF SURFACES. NUMBER OF CONDITIONS WHICH SURFACES CAN SATISFY. CLASSES OF SURFACES. Degrees of surfaces and curves 194 Number of points of intersection of three surfaces 195 Degree of the complete intersection of two surfaces ,, Number of conditions which a surface of the »* degree can satisfy „ JjJONTENTS, XV FAQE Number of points through which, if surfaces of the n*^ degree pass, they will intersect in a common curve ig5 Number of points on a surface of a degree lower than n, through which, if surfaces of the n*^ degree pass, they will intersect the former surface in one and the same curve in^ Curves which are partial intersections of surfaces log Number of points through which, if surfaces of the rat'i degree pass, they will also pass through points which will be fixed in space ioi When three surfaces intersect in a multiple line, to find the number of points of intersection which correspond to this multiple line 203 Co-ordinates of a plane which passes through the line of intersection of planes whose co-ordinates are given in the four-point system 205 Angle between two given planes Class of a surface 206 CHAPTER XIV. TANGENT LINES PLANES AND CONES. POLES AND POIAB PLANES. NORMALS. Tangent line to a surface 208 Singular or multiple points 209 Tangent plane to a surface ,, Tangent cone at a multiple point 211 Nature of the intersection of a surface with its tangent plane at any point ... 212 Illustration by an anchor-ring 213 Tangent line to the curve of intersection of a surface with its tangent plane . 2 14 Number of triple tangent planes onasurfaceof the third degree 2i5 Euled surfaces 218 Developable surfaces ,, Edge of regression 220 Intersection of a developable surface with a tangent plane „ The shortest line between two points on a developable surface 221 Skew surfaces „ Nature of contact of a tangent plane to a skew surface 223 The equation of the tangent plane to a developable surface contains only one parameter „ Form of the curve of intersection of a developable surface with a tangent plane 223 Theorem by Bouquet 225 Tangent planes touching along a curved line ,, Curve of greatest inclination to a given plane 228 Tangent plane referred to tetrahedral co-ordinates 229 XVI CONTENTS. PAOE Tangent cone at a singular point referred to tetrahedral co-ordinates 230 Claaa of a surface of the jj* degree 231 Equation of the point of contact of a tangent plane in four point co-ordinates 132 Tangential equation of the cui-ve of contact of a singolar tangent plane 233 Degree of a tangential surface of a given class ^34 Polar equation of a tangent plane >, Perpendicular from the pole upon the tangent plane 23S Polar plane corresponding to a given pole 237 Equation of polar plane with respect to a given oouicoid ,, Center of a given conicoid co-ordinates 238 Four-point co-ordinates of the polar plane 239 Equation of the center of a given conicoid in four point co-ordinates „ Two conicoids which intersect in one plane curve intersect also in another ... 240 Equation of the enveloping cone of a given conicoid having a given vertex ... 24 1 Asymptotic lines, planes, and surfaces, to a given surface 243 Asymptotes of a central conicoid 244 Lines of closest contact in an asymptotic plane 245 Asymptotic surfaces having contact of a higher degree than the second 247 Normal at any point of a surface 249 Normal cone at a singular point 250 Tangential equation of a normal 251 Number of normals from a given point to a surface of the m* degree 252 Helicoid ijj Condition for a conjugate curved line 256 Cone, whose vertex is any point of a conicoid, and base a focal conic, has for its axis the normal to the conicoid at the vertex 258 CHAPTER XV. METHOD OF RECIPROCAL POLARS. Nature of reciprocity ^g Reciprocal points and planes 26" Auxiliary conicoids and polar reciprocals 266 Species of polar reciprocals 268 Eeciprooal polar of a conicoid with respect to a given point Equation of the reciprocal polar of a conicoid with respect to any point on a focal conic g Reciprocal of a conicoid, with respect to a point on a focal conic, is a conicoid of revolution capable of generation by the method to which the conic corresponds _ Cones and cylinders which are reciprocal polars of circles with respect to »P°'°'- 272 Examples of the method of reciprocal polars 273 Degree of the polar reciprocal of a surface 2»£ CONTENTS. XVll CHAPTER XYI. GENERAL THEORY OF POLAES A2JD TANGENT LINES. PAGE Positiona of the points of interaeotion of a given surface with a straight line passing through two given points 280 Polara of a surface with respect to a point 281 Geometrical properties of polara 282 Connexion between diametral and polar surfaces 283 Every polar of a surface, with respect to a given pole, is a polar of eveiy polar of a higher degree than its own 284 Every line, drawn through a pole to a point in the curve of intersection of the first polar with the surface, meets the surface in two coincident points „ A multiple point on a surface will he also a multiple point of one degree less on the first polar, with respect to any point not on the surface 284 If a tangent cone at a double point of a surface becomes two non-coincident tangent planes, the first polar touchea the line of intersection at the double point 285 If the two tangent planes coincide, they also coincide with the tangent plane to the first polar „ If r generating lines of a conical tangent coincide, r - i of that of the first polar will-coincide „ If a surface have a multiple line of the m"' degree, the first polar contains the same Une of the m— 1 1 degree of multiplicity 286 If the pole be on the surface, the polar plane will be a tangent plane to the surface and also to the polars .- „ Locus of poles whose polar planes pass through a given point 287 Number of poles corresponding to a given plane ,, Polar curve corresponding to a straight line ,, Effect of a double point upon the class of a surface 288 Reduction of class when the conical tangent reduces to two planes 289 Reduction corresponding to a double straight line 290 Reduction corresponding to a multiple straight line of any degree of multi- plicity ^91 Reduction corresponding to a multiple curve line ipi Relation of straight lines to surfaces. Condition that a straight line may touch a surface at a given point 293 Locus of points of contact of all tangents from a given point to a surface 294 Tangent lines at an ordinary point meeting a surface in three coincident points 3) Conical envelope '^95 Number of tangents from a given point which jneet a surface in three con- secutive points • !• Number of tangents at a given point which touch at another point as well... „ Locus of tangents from a singular point ^9'^ Number of double sides of a conical envelope having a given vertex 297 XVIU CONTENTS. CHAPTER XVII. FtTlfCTIONAL AND DIFFERENTIAL EQUATIONS OP FAMILIES OP SURFACES. FAGB Parameters and their elimination ^99 Functional equations of families of ruled surfaces 301 Cylindrical surfaces ,, Conical surfaces 3°* Conoidal surfaces ,, Surfaces generated by a straight line moving parallel to a fixed plane. . 303 Functional equation of surfaces of revolution 304 Differential equations of families of surfaces 305 Cylindrical surfaces 306 Conical surfaces .,. „ Conoidal surfaces 307 Surfaces generated by a straight line moving parallel to a fixed plane. ,, Developable surfaces 309 Surfaces of revolution 310 Application to coniooids '. 311 CHAETER XVIII. PHOPEETIES OF CONIOOIDS SATISFYING GIVEN CONDITIONS. FOEMS OF THE EQUATION OF A SPHERE. Coniooids through eight given points. General form of the equation 318 Polar plane of any other given point passes through a fixed straight Wne „ Pole of any given plane lies on the curve of intersection of two coni- coids 319 Eeciprocal propositions Cones through eight points ,, Conicoids also touching a given plane 321 Conicoids through seven given points. General form of the equation 322 Polar plane of any other given point passes through a fixed point „ Locus of the pole of a given plane 323 Number touching two given planes 324 Form of equation when six of the points lie by threes on non-intersect- ing straight lines Form when the points lie in four straight lines passing through four given points 32g CONTENTS. XIX PAGE Coniooids through six giTen points and touching three givqn planes..., 337 Conicoida through five given points and touching four given planes „ Conicoid containing two non-intersecting straight lines. General form of the equation ,. 328 Condition of being paraboloids „ Pole of a given plane lies on a fixed plane „ Conicoids having a common plane section or a common enveloping cone 329 Conicoids having two common plane sections 330 Tangential equation of a surface is the same equation as the equation, in tetrahedral co-ordinates, of its reciprocal with respect to the surface 0^-1-^^7^32=0 331 Equation of a sphere circumscribing the fundamental tetrahedral 332 General equation of a sphere in tetrahedral co-ordinates 333 Kadius of the circumscribing sphere 334 General tangential equation of a sphere 335 Tangential equation of spheres touching the faces of the fundamental tetra- hedron ,, Centers of similarity of the inscribed and four escribed spheres 336 Distance between the centers of the inscribed and circumscribed spheres 337 CHAPTEE XIX. cruvEs. Equations of a tangent line to a curve 54*"^ Direction of branches of a curve at a multiple point 342 Differential coefficient of the arc 344 Equation of normal plane ; 345 Edge of regression of the developable surface enveloped by the normal planes „ Equation of the osculating plane.; 346 Direction-cosines of the osculating plane 347 Equations of the principal normal 348 Limiting position of the line joining the extremities of equal distances mea- sured along a curve and its tangent 350 Four-point system. Equation of a curve of double curvature „ Equation of the tangent |^ , Equation of the osculating plane ) Singularities of curves and developables 350 Connexion between singularities in curves of double curvature and those in plane curves 355 XX CONTENTS. CHAPTER XX. ENVELOPES. PAGE Nature of tlie locus of ultimate interaeotiou of surfaces corresponding to the number of parameters 3^1 Locus of ultimate intersection when the general equation of the surfaces in- volves one arbitrary parameter 362 Equations of the characteristic and the edge of the locus of ultimate inter- section }j The locus of ultimate intersections in general touches eaoh of the surfaces along a curve 363 Geometrical proof that the locus of ultimate intersections is an envelope 364 Envelope of spheres having for diametral planes one series of ciroiJar sec- tions of an ellipsoid „ Locus of ultimate intersection of surfaces, the general equation of which in- volves two arbitrary parameters 366 The locus of ultimate intersections is the envelope of the surfaces 367 Envelope of surfaces involving m parameters connected by » - i, or n- 2 equations 368 Envelope of diametral planes of an ellipsoid whose sections are of constant area 369 Envelope of spheres of which parallel chords of an ellipsoid are diameters ... 370 Differential equation of envelope of surfaces containing two parameters, one of which is an arbitrary function of the other 371 Differential equation of envelope of surfaces involving three parameters, two being arbitrary functions of the third „ When surfaces depend on two parameters, one of which is an arbitrary func- tion of the other, condition that the envelope may contain a given curve, or touch a given surface 372 Edge of a tubular surface 373 Equation of the Wave suiface 374 CHAPTER XXI. VOLUMES, AREAS OF SUBFACES, &C. Differential coefiBcient of Volume, rectangular co-ordinates -ijg "Wedge 3jg Surface, rectangular co-ordinates 380 Surface of wedge Volume, polar co-ordinates Surface, polar co-ordinates 381 CONTENTS. xxi PAas Volume or surface of a wedge cut off by a given surface of any form, and a cylindrical surface 382 Volume of a solid, rectangular co-ordinates 383 Volume contained between an ellipsoid and cylinder 384 Volume cut off by a plane from an elliptic paraboloid 386 Volume of a solid, polar co-ordinates ....■ 387 Volume of a solid, tetrahedral co-ordinates 389 CHAPTEE XXII. CUHV ATTIRE OF CtTEVES. Polar developable 3po Osculating circle 303 Angle of contingenoe -. 394 Sphericai curvature „ Polar line ,, Curvature of torsion ,, Angle of torsion ,, Osculating cone 30^ Evolutes „ Evolutes are geodesic lines of polar developable „ Locus of centers of circular curvature „ Kectifying developable ,, Eectifying line ,, Binormal 390 Angle between consecutive radii of curvature 397 Kadius of complex curvature 398 Angle of osculating coiie ,, Eectifying line is the axis of osculating cone 399 Eectifying surface is the locus of centers of pi'incipal curvature of the develop- able of a curve '. ,, Angle of contingence of locus of centers of circular curvatures 400 Element of arc of locus of center of curvature Eadius of spherical curvature 401 Eadius and co-ordinates of center of circular curvature ,, Equations of principal normal 402 Eadius of torsion 404 Singular points 40S Limits of distances, Two consecutive tangents 4^1 Osculating plane from adjacent point 1, Consecutive principal normals 4oS XXll CONTENTS. CHAPTER XXIII. CURVATUHE OP SURFACES AND LINES OP CUKVATUBE. PAGE Conditions of complete contact of the »*•> order 4'" Number of constants in order to insure complete contact of any order with a given surface 4" Euler's theorem 413 Indicatrix >, Meunier's theorem 415 Radius of curvature of normal section 416 Sections and radii of principal curvature 417 Conditions for an umbilicus 418 Line of spherical curvature 419 Number of umbilici 420 Lines of curvature 421 Differential equations of lines of curvature ,, Normals at consecutive points intersect 422 Non-symmetrical form of equations .' 425 Lines of curvature on a conicoid are its intersections with confocal surfaces. . 427 In a line of curvature of a central conicoid pd is constant 431 Umbilici of a central conicoid 432 Osculating plane of a line of curvature 435 Gauss' measure of curvature 436 Measure of curvature not altered by deformation of an inextensible surface . . 437 Dupin's theorem ^j CHAPTER XXIV. GEODESIC LINES. Osculating plane contains normal to surface ^.n Geodesic on right circular cylinder .50 Geodesic on right cone .ej Number of geodesies on cone ^c-. Geodesic on surface of revolution ^54 p(2 constant throughout geodesic on central conicoid 4Se pd the same for all geodesies through an umbilicus >£§ Locus of point the sum of difference of whose geodesic distance from adjacent umbilici is constant ..g Geodesic distances of opposite umbilici are all equal , ^eg pd the same for all geodesies touching the same line of curvature 459 Geodesic tangents to a line of oiu^ature make equal angles with the line of curvature at their intersection Locus of intersection of perpendicular geodesic tangents to two lines of cur- vature CONTENTS. XXUl • PAGE Properties of projeotiona of geodesies and lines of curvature upon the planes of circular section 461 Intersection of a geodesic tlirough' an umbilicus with a geodesic tangent to a line of curvature which it meets at a constant angle lies on one of two spheres 463 Tangent lines to a geodesic on a oonicoid touch a fixed confocal conicoid 464 Developable enveloped by tangent planes along a geodesic on a conicoid has its" edge of regression on another conicoid 465 Geodesic lines of a paraboloid 466 Rectilinear geodesies on surfaces 467 Kadius of torsion at any point on a geodesic line 467 Radius of geodesic curvature , 468 GEOMETRY OF THREE DIMENSIONS. CHAPTER I. ON CO-ORDINATE SYSTEMS. 1. Befoee entering upon the application of Algebra to the investigation of Theorems, and to the solution of Problems, in Solid Geometry, we shall premise on the part of the student a complete knowledge of all the ordinary processes adopted in the case of Plane Geometry. By this means we shall avoid the necessity of entering upon the subject of the interpretation of the affection denoted by the sign (— ) prefixed to a symbol ; since it is known that, if + a de- note a line of length a measured in any direction from a point in a line straight or curved, — a may be interpreted to denote a line of length a measured in the opposite direction from any other point in the line, without this hypothesis involving any infringe- ment of the laws of combination of these signs, assumed as the fundamental laws of Symbolical Algebra. 2. Our first object will be to explain how the position of a point in space can be represented by algebraical symbols, and with this view we shall describe the different co-ordinate systems which it has been found convenient to adopt ; each of which has its peculiar advantage, according to the nature of the theorem or problem which is the subject of examination. B ON CO-ORDINATE SYSTEMS. Co-ordinate System of Three Planes. 3. In the co-ordinate system of three planes, three planes xOy, yOz, zOx are fixed upon as planes of reference, which may be either perpendicular to one another, or inclined at angles which are known. The three lines in which they intersect are called co-ordinate axes, and the point in which they meet the origin of co-ordinates. The position of a point in space is then completely deter- mined, when its distance from each of the planes, estimated pa- rallel to the co-ordinate axes, and the direction in which those distances are measured, are given. The absolute distance, and the direction of measurement are included in the term algebraical distance. Thus +a and —a are the algebraical distances of two points whose absolute distances from the plane yOz are each a, and which are measured, the first in the direction Ox, the second in the direction x from that plane. These algebraical distances are called the co-ordinates of a point in this system, and aire usually denoted by the letters x, y, and s. The point, of which these are co-ordinates, is described as the point {x, y, z). ON CO-ORDINATE SYSTEMS. 3 Produce xO, yO, zO backwards to x, y', z' ; then, if a, b, c are absolute lengths, (a, b, c) denotes a point in the compart- ment xysO, {—a,b, c) in x'yzO, {a, —b, c) in xy'sO, {a, b, — c) in xys 0, {a, —b,—c) in xy'z 0, {-a, b, — c) in x'yz' 0, (- a, — b, c) in x'y'z 0, (- a, —b, - c) in x'y'z 0. 4. If a parallelopiped be constructed, whose faces are pa- rallel to the co-ordinate planes, the point P (a, b, c) being the other extremity of the diagonal drawn from the origin, the edges LP, MP, NP will be the co-ordinates of the point P, supposed in the compartment xyz 0. Also, it is obvious that x = a for every point in the plane face PNIM, or that a; = a is the equation of that plane, &s y = b and is = c are the equations of the planes PLmN and PMnL indefinitely extended in every direction. Thus, the point P may be considered as the intersection of the three planes, whose equations are x = a, y = b, z=c. The points I, may be denoted by (a, 0, 0) and (0, 0, 0) and the points L and If by (0, b, c) and {a, 0, c). I. (1) Construct the positions of points -which are represented by the equations X + y = 4a, as — y =a. (2) ■ x' + f=2!?, X +y =2z, xy = o^ (3) Shew that for every point in OP, P being (a, b, c), X _y _z (4) Shew that for every point in the plane LMlm a h ^ + ^=1. b2 4 ON CO-ORDINATE SYSTEMS. Four-Plane Co-ordinate System. 5. In the co-ordinate system of foiu- planes, four planes are fixed upon as planes of reference, which form by their intersec- tions a pyramid or tetrahedron ABCD. The position of a point is determined in this system by the algebraical distances a, ^, 7, S from the four planes respectively opposite to the vertices A, B, G, D, these distances being all absolute distances when the point is within the tetrahedron. Hence, for a point in the compartment between the plane AGD and the other three produced, yS will be negative and a, 7, S positive; between BAG, GAD, and DAB, produced through A, a. will be positive, and ^, 7, 8 all negative. If a be positive, a = a is the equation of a plane parallel to BGD, at a distance a from it, on the side towards A; a.== — a that of a plane on the opposite side at the same distance. 6. In this system of co-ordinates the following peculiarity must be observed, viz. that any three of the co-ordinates a, ^, 7, S are sufficient to determine the position of the point, since, when a, 0, 7 are given, three planes are determined parallel to the faces opposite to A, B, G which intersect in the point, and so determine its position completely. Hence, when o, /3, 7 are given, 8 ought to be known from the geometry of the figure, and we proceed to determine the relation between the co-ordinates in this system. ON CO-OEDINATK SYSTEMS. 5 Belation of Co-ordinates in the Four-Plane System. 7. Let V be the volume of the tetrahedron contained by the four fixed planes, A, B, G, D the areas of the triangular faces. If the point P whose co-ordinates are a, /3, 7, S be joined by- straight lines to the angular points of the tetrahedron, four pyra- mids are formed, whose vertices are at P, and whose bases are the faces of the tetrahedron. The algebraical sum of these four pyramids make up the volume of the tetrahedron; therefore, remembering that the volume of a pyramid is three times the base x the altitude, Aa.+ B^+ (77 + i:»S=3F, whence, when any three of the co-ordinates of a point are given, the fourth may be found. The object of the introduction of a fourth co-ordinate, in this system, is the same as that for which trilinear co-ordinate's are employed in Plane Geometry, viz. to obtain equations homo- geneous with reference to the co-ordinates, and thus to arrive at symmetrical results. By means of the equation given above, any equation which does not appear in a homogeneous form can be reduced to such a form immediately. Thus the equation a = a of a plane may be reduced to the homogeneous form 3Fa = a (^« +-B^+ C7 + Dh), Tetrahedral Co-ordinates. 8. The expressions Involving these co-ordinates are fre- quently simplified by the employment, in their stead, of the tetrahedrons, which are proportional to them, namely, ^Aa, J-B/8, JC7, ^Dh, which we shall call "Tetrahedral Co-ordinates." If a, /3, 7, S denote these co-ordinates, the relation always subsisting between them is 6 ON CO-ORDINATE SYSTEMS. Any expressions involving the former (four-plane) co-ordi- nates, may be at once transformed so as to involve the latter (tetrahedral) by the substitution of 3a 3/3 37 3S , _ jj ^. , Z ' 'F' ~D' ^ ^°^ °^' ^> y' ^ respectively. The equation of condition is farther simplified, if we take the volimie of the fiindamental tetrahedron as the unit of volume, or, which amounts to the same supposition, take as the co-ordinates of any point the ratios of its algebraical distances from the faces to the distances of the angular points respectively opposite to them. In this case the equation of condition becomes a + /8 + 7 + 8 = l; and any given equation involving four-plane co-ordinates may be transformed into an equation referred to this system by writing Pi'^'PAFsy>'Pt^ ^0^ «. /3, 7,. 8 respectively,^,, p^, j)^, p^ being the distances of the angular points from the opposite faces. Polar Co-ordinate System. 9, In the system of Polar Co-ordinates, a plane zOx is chosen, and in this plane a straight line Oz is drawn from a fixed point 0. ON CO-ORDINATE SYSTEMS. 7 The position of a point P in space is completely determined, when its distance from the fixed point is given, the angle through which the line OP has reyolved in a plane from Oz, and the angle through which the plane z OP has revolved into its position from the fixed plane of reference z Ox. • These co-ordinates are usually denoted by the symbols r, and <^, and the point Pby (r, 6, ^). Thus, if the longitude of a place be I, the latitude \, and the radius of the earth a, we may take the first meridian for the plane zOx, the axis of the earth for the axis of z, and the position of the place will be expressed by (a, |-\, I). If \' be the latitude of Greenwich, its position is given by (a,J-\', 0). 11. (1) Shew that for every point in a plane through the edge AB bisecting the angle between the planes GAB, DAB, y — S = 0, if the angle be the internal angle, y + 8 = 0, external (2) Shew that for every point in a plane drawn through the vertex A parallel to the opposite face, B^+Gy + Dh = 0; or with tetrahedral co-ordinates, /3 + y+8=^0. (3) If AO be drawn perpendicular to the opposite face BCD, then for any point in AO, Bp _ Cy _ DK-An--. AGOD ADOB ABOC (4) Every point in a plane through CD parallel to AB satisfies the equation in tetrahedral co-ordinates, a + /3==0. 8 ON CO-OKDINATE SYSTEMS. (5) At any point in the straight line joining the middle points of AB and CD, the tetrahedral co-ordinates satisfy the equations a = p, -y = 8. (6) The four straight lines joining the middle points of opposite edges of the tetrahedron of reference meet in a point whose tetrar hedral co-ordinates satisfy the equations V (7) If the equations to a point be I m n p' and AO, BO, GO, DO be joined and produced to A', B, C, Df such that bisects the lines AA!, &c., the tetrahedi'al co-ordinates of the point A' wiU satisfy the equations 2a jg y 8 2F l-m-n-p m n p~ l + m+n+p' and similarly for B', C, D'. CHAPTEE II. GENERAL DESCRIPTION OF LOCI OP EQUATIONS. SURFACES. CURVES. Locus of an equation. 10. If an equation F{x, y, z)=0 be given, in which the variahles are the co-ordinates of any point, the number of solutions of this equation is generally infinite, i. e. the number of points whose co-ordinates satisfy the" equation is infinite: we shall pro- ceed to shew what is the general nature of the distribution of the points, whose co-ordinates satisfy the equation. We shall prove in the first place that no algebraical equation can be satis- fied by every point of any solid figure, but, in the most general case, only by every point in some surface or surfaces. 11. If an equation involve only one of the co-ordinates as x, we know that such an equation F{x) = has a finite or an in- finite number of roots, a, b, e,... separated by definite intervals, and is reducible to the equations x = a, x = h,.,.., each of which, as x = a, is satisfied by every point in a plane parallel to the plane yOz, at an algebraical distance a. Hence, all the points whose co-ordinates satisfy the equation F(x) = lie in a series of planes parallel to yOz at algebraical distances a, b, c,.,. If the given equation involve two only of the variables, as F{y, a) = 0, on the plane yz let the curve be constructed which is the locus of F{y, z) = 0, and let a straight line be drawn parallel to Ox through any point in this cui"ve, every point in this line is such that its co-ordinates satisfy the given equation, and the same is true of all points in the curve, and of no other points. Hence, all the points which satisfy the proposed equation lie in a surface generated by a straight line parallel to Ox, which passes successively through every point of the curve traced on the plane yz : such a surface is called a cylindrical surface, and the cui-ve is called the trace on the plane of yz, and is one of an 10 SURFACES. infinite number of curves, which are called guiding curves to the cylindrical surface. The number of guiding curves is infinite, since, if any curve be traced upon the cylindrical surface, so as to cross every gene- rating line, a line moving parallel to Ox, so as to traverse every portion of such a curve traced in space, would generate the entire cylindrical surface, that curve serving to guide the direction of motion of the generating line. 12. We may notice here, that, if the equation F{y, s) = be reducible to a series of equations of such forms as {y-iy + (z-cy=Q, {my — nzf +{z— c)' = 0, the trace on yz is reducible to a series of points, and the locus of the equation F{y, a) =0 becomes a series of straight lines paral- lel to Ox, passing through those points. In such cases, the locus appears to be different in character from that of the general case, since it is a series of lines instead of being a surface. But, it may be seen that this is only in appearance, since each of the equations whose locus is called a point represents a closed curve of infinitely small dimensions, and the lines are cylinders whose breadths are infinitely small, and the locus of the equation F{y, s) = is as in the general case a series of surfaces : and a similar interpretation may be given in every case. 13. We shall now proceed to the general case, F{x,y,z)=0, in order to examine the position of all the points which satisfy the equation; and we shall find, first, the position of those points which are at an algebraical distance g from the plane of yz, which is the same thing as examining the position of those points which lie on a plane whose equation is x=g. These points are contained in the cylindrical surface whose equation is F{g,y,z)=0. The trace of this surface on the plane a; = ^ is the line which contains all the points of the sur- SUEFACKS. 11 face which lie on that plane ; and if the series of lines be traced corresponding to different positions of the plane a; = ^ for values of ^r from —00 to +00 , we shall evidently obtain a surface which contains all the points which satisfy the equation F{x, y, z) = 0. 14. As an illustration of tracing surfaces, we will take the case of the surface whose equation is {x+yy=az. If x = 0, "jf = az; therefore the trace on the plane of yz is a parabola whose axis is Oz and vertex 0. Similarly, that on zx is another equal parabola having the same axis and vertex. If z = Tc, {x + yY = ah, which is the equation of two planes parallel to the axis Oz, equally inclined to the planes yz, zx ; therefore the trace on the plane s = i is two straight lines equally inclined to the planes yz, zx. Hence, the surface is generated by straight lines, as PQ, which move parallel to the plane of xy, constantly passing through the traces on yz, zx so as to be inclined to those planes at equal angles of 135°. . The shape is therefore a cylindrical surface as in the figure^^ 12 SUEFACES. III. Trace the surfaces represented by the equations (1) os'+y'=ax, (2) si' = ax + by, (3) x''+y''+z^=ax + hy + ce, (4) x'+^=cz, (5) xy = am, (6) {x + z) {y + z) = ax, (7) {ax + by + czy= m' (ax+by + dz). 15. By the introduction of constants, which admit of all values within certain limits, equations may be formed, which will represent all points within certain bounding surfaces. For example, a sphere whose center is (a, h, c) and radius r may be represented by the equation {x-ay+{y-hY+{z-cY = r^ in a rectangular co-ordinate system, if then r be capable of re- ceiving all values from r^ to r^, the equation represents all points contained between two concentric spherical surfaces, whose radii are t-j and r^. Again, if a, b, c be capable of receiving all values consistent with the equation a' + b' + c' = d', the above equation will represent all points contained between two spherical surfaces whose center is the origin and radii r + d and r — d. If c and r be constant, and a, h have all yalues consistent with the equation a^ + V = d\ the equation represents all points contained within a ring, gene- rated by a circle of radius r, revolving about the axis Oe, the center being at a distance d from that axis, and c from the plane of a;^. In the same manner, it may be shewn that {x-oy+{y-^f+{z-r,y=Q, SURFACES. 13 in which a may have any values hetween — a and + a, /3 -h ... +b, and 7 — c ... +c is satisfied by all points which lie within a parallelepiped whose faces are parallel to the co-ordinate planes. Locus of the Polar equation. 16. We shall examine in order the loci of equations which involve one or more of the co-ordinates. (1) If the equation be F{r) = 0. This is equivalent to a series of equations r = a, r='b, any one of which being satisfied the original equation is satisfied : r = a is satisfied by all points at a distance a from the origin, measured in any direction ; therefore the locus of F{r) = is a series of concentric spheres, whose center is the origin. (2) If the equation be F{6) =0 it is equivalent to 6 —a, 0=/3, any one, 6= a., is satisfied by every point of lines through inclined to Os at angles equal to a; therefore the locus of i'' (^) = is a series of conical surfaces, whose common axis is Oz, vertex 0, and vertical angles, 2o:, 2/3 (3) If the equation be F{<^) =0, it is equivalent to ^ = a, ^ = /3, any one, = a, is satisfied by every point in a plane through Oz inclined at angle a to the plane z Ox ; therefore the locus of F{^) = is a series of planes through Os inclined to Oaj at angles a, j8, (4) If the equation involve only r and ^ as -F (r, 0) = 0, since, for all values of ^ the same relation exists between r and 6, the locus of the equation is the surface generated by the revo- lution of a cui-ve traced on a plane through Oz, as the plane revolves about Oz as an axis. (5) If the equation involve only (j) and 0, as F{4>, ^)=0, for every value of ^, there are a series of values of 0, correspond- ing to which if lines through be drawn, every point in these 14 SURFACES. lines will be such that its co-ordinates satisfy the equation, and as (f> changes or the plane through Oz revolves, these lines assume new positions relative to Oz, and generate during the revolution of the plane, conical surfaces, a conical surface being defined to be a surface generated by a straight line moving in any manner with the restriction that it passes through a fixed point. (6) If the co-ordinates involved be r, ^ as in F{r, ^) =0, for each position of the plane through Oz inclined at any angle ^ to z Ox, there are values of r, which are constant for all values of 6, i.e. there are a series of concentric circles, in the plane, the co-ordinates of each point in which satisfy the equation. The locus of the equation is therefore a surface generated by circles having their centers in 0, and varying in magnitude as their planes revolve about the line Oz through which they pass. (7) If the equation involve all the co-ordinates, as F(r, 6, ^)=0, let any value be given to <^, as /8, then, corresponding to this value there is a plane through Oz, and if the locus of F{j; 0,0) =0 be traced on this plane, and such curves be drawn upon all planes corresponding to values of ^ from — oo to + <» , the surface which contains all these curves is the locus of the equation. Curves. 17. Curves in space are called generally curves- of double curvature, because generally they do not lie entirely in one plane. If we take three points very near to one another, these three points lie in one plane but not generally in one straight line, but a fourth point will lie generally on one side or the other of this plane, the bend first in one plane and then in another giving rise to the term double curvature. Equations of curves. 18. Through every curve there can be drawn an infinite number of surfaces, the intersections of any two of which will include every point of the curve. At the same time we must observe that two surfaces, each of which contains a given curve, SURFACES. 15 may not be sufficient to determine the position of the curve, because they may intersect in other points which are not connected with the given curve. Thus, if we take the case of a circle, it is true that it lies entirely in the intersection of a certain sphere and cylinder, but the sphere and cylinder are not sufficient to determine the circle because they may also intersect in another circle, and the circle to be considered is not defined by those surfaces : in this case it is possible to find two surfaces which do define the "circle com- pletely, as for example a plane, and either the sphere or cylinder. 19. If F{x,y, s)=0, and F^{x, y, a)=0 be equations of two surfaces, these surfaces by their intersection determine a certain curve, and if another equation ^ («, y, z) = Q be obtained from those two equations, by any processes of addition or mul- tiplication, the third equation will be satisfied by every point in the curve determined by the intersection of the first two sur- faces, and we may employ this equation and either of the first two to obtain properties of the curve; although the two new equations may represent surfaces which intersect in other points than those of the curve originally proposed. It is often convenient in practice to consider a curve as the intersection of two cylindrical surfaces, whose generating lines are parallel to two of the axes. In this way of considering curves, the equations of the surfaces are of the form ^ (aj, z) = 0, ■f{y,s)=0. As a simple example of the determination of a line by two " surfaces, we will take a straight line parallel to the axis of z. x = a and y=b are the equations of two planes parallel to the planes of yz, zx, which intersect in a straight line parallel to Oz. CHAPTER III. PROJECTIONS OP LINES AND AREAS. DIRECTION-COSINES, AND DIRECTION-RATIOS. 20. Def. The Geometrical prelection of a straight line of limited length upon any other straight line given in position, is the distanCfe intercepted between the feet of the perpendiculars let fall from the extremities of the limited line upon the straight line on which It is to be projected. 21. TJie Geometrical projection of a straight line cf limited length on a given straight line is equal to the given length multi- plied hy the cosine of the acute angles contained between the lines. Let PQ be the line of limited length, AB the indefinite line upon which it is to be projected. Let QBN be a plane through Q perpendicular to AB meet- ing it in N, PR parallel to AB meeting ^^iVin R. Therefore PR being parallel to AB is perpendicular to the plane QRN, and therefore to RN and QR, and QN is perpen- dicular to AB; hence, if PM be drawn perpendicular to AB, MN is the projection of PQ, and QPR is the acute angle con- tained between PQ and AB, and since PRNM is a rectangle, MN= PR = PQ cos QPR. If PQ produced intersects AB, the proposition is obviously- true. 22. Def. The algebraical projection of a line PQ upon an indefinite line AB given in position is the projection estimated in a given direction, as AB. DIRECTION-COSINES. 17 If a be the angle through which PQ may be supposed to have revolved from PB, drawn in the positive direction AB, the algebraical projection of PQ = PQ cos a. If N lies in the opposite direction with reference to M, a is obtuse, and PQ cos a is negative. The algebraical projection of a limited straight line, upon a line given in position, measures the distance traversed in the direction of the latter line in passing from one extremity of the former to the other. This consideration shews, that if all the sides of a closed polygon, taken in order, be projected on any straight line given in position, the sum of the algebraical projections of these sides is zero ; since, in passing round the perimeter of the polygon from any point, the whole distance advanced in any direction is zero. Hence, the algebraical projection of any side AB of a closed polygon, is the sum of the algebraical projections of the remain- ing sides commencing from A and terminating in B. Note. In future, when the term projection is used, the alge- braical projection is to be understood. 23. Let PQ be any line, PM, MN, NQ three straight lines drawn in any given directions so as to terminate in Q, and I, m, n the cosines of the angles which PQ makes with these directions. N Then PQ will be the sum of the projections of PM, MN, and NQ on PQ; .-. PQ=LPM+m.MN+n.NQ. Directton-cosines. 24. The direction of a straight line in space is determined when the angles which it makes with the co-ordinate axes are known. Def, If the co-ordinate axes be perpendicular, the cosines of the inclinations to the three axes are called direction^cosines. c 18 DIRECTION-COSINES. 25. To find the relation between the direction-cosines of a straight line. If I, m, n be the direction-cosines oiPQ, and PM, MN, NQ be parallel to the co-ordinate axes, PM=PQ.l, MN^PQ.m, NQ = PQ.n. Join PN, then, since QN is perpendicular to NM, MP, and therefore to the plane PMN, PNQ is a right angle ; .-. PQ' = PN' + N^ = P]\P + M2P + NQ' ; which is the relation required. Hence the three angles of in- clination cannot all be assumed arbitrarily. 26. To find the angle between two straight lines in terms of their direction-cosines. Let PQ, P Q' be two straight lines whose direction-cosines are (Z, m, w) and (Z', m, n') respectively. Let PM, MN, NQ be drawn parallel to the axes, connecting any two points P, Q, and PP, QQ' perpendicular to PQ', and let 6 be the angle between PQ and P Q'. r Then P' Q, the projection of PQ on P Q', will be equal to the sum of the projections of PM, MN, NQ on P'Q', namely, PM', M'N',N'Q'; DIRECTION-RATIOS. 19 .-. PQ cos 6 = FM. I ' + MN. m + NQ.ri, and since Pif = PQ.l, MN= PQ.m, NQ = PQ.n ; .". cos 6 = 11' -\- mm + nn. Hence sin'^ = {I' + m' + n") (r + m" + «") - (??' + mm' + m')' = (mw' - m'nf + (mZ' - nlf + (?m' - ^'m)^ Directionyratios. 27. Dep. If the co-ordinate axes be not perpendicular to each other, the direction of a line PQ is fully determined, if the ratios of PM, MN, NQ to PQ are given, PM, MN, NQ being parallel to the axes. These ratios are called direction-ratios. 28. To find the relation between t^e direction-ratios of a straight line. In the last figure, let the angles yOz, si Ox, xOy be X, (i, v, and let a, /3, 7 be the angles between PQ and the axes, I, m, n the direction-ratios of PQ. Projecting the line PQ and the bent line PMNQ terminated in the same points on Ox, PQ cos a = PM+ MN cos v + NQ cos fi ; .'. cos a = l + m cos v + n cos ft,, similarly, cos = I cos v +.m + n cos X, and cos y — l cos fi + m cos \ + n. Also projecting PMNQ on P^, Plf cos a + MN cos /9 + NQ cos y = PQ; .•. I cos a + m cos /3 + n cos 7 = 1; .-. l = l^ + m'' + n^+ 2mn cos X + 2w? cos /t + 2lm. cos i/, which is the relation required. 29. The following is the relation which always exists be- tween the cosines of the angles which a straight line makes with three oblique axes. cos"* a sin'' X + cos^ /3 sin" /J- + cos^ 7 sin" v + 2 cos /3 cos 7 (cos /JL cos v — cos X) + 2 cos 7 cos a (cos v cos X— cos fi) + 2 cos a cos /S (cos X cos /* — cos v) = 1 — COS'X — C0SV-C0S%+ 2C0SX COS fl cosv. This may be deduced from the equations of the last article. C2 20 DIRECTION-RATIOS. 30. To find the angle between two straight lines whose direction- ratios are given. Let {I, m, n) and {V, m', n') be the direction-ratios of two straight lines PQ, P Q', 6 the angle between them, and (a, 0, y) the angles of inclination of P' Q' to the co-ordinate axes. Take r a limited length measured on PQ ; then rl, rm, rn will be the intercepts on the axes by planes drawn through P, Q parallel to the co-ordinate planes, and r cos 6 the sum of the projections of rl, rm, m on F Q' — rl cos a' + rm cos /8' + rn cos 7'. Also cos a' = Z' + m' cos v + n cos /*, (Art. 28), cos j8' = I' cos !» + m' -f- w' cos \, cos 7' = V cos /i + m' cos X + n; .'. cos = W + mm' + nn' + 2 {mn + m'n) cos \ -f- 2 {nT + n'l) cos /4 -I- 2 (Zw' + I'm) cos i*, which is the cosine of the angle required. Projection of a Line on a Plane. 31. Def. The orthogonal projection of a line of limited length on a plane is the line intercepted between the perpendiculars drawn from the extremities of the limited line upon the plane. 32. The orthogonal pryectitin of a line upon a plmie, is the length of the line multiplied hy the cosine of the angle of in- clination of the line to the plane. Let PQ be the given line, AB the plane, PM, QiV perpen- diculars upon the plane. Since PM, QN are perpendicular to the plane AB, PM is PEOJECTIONS. 21 parallel to QN, and the plane MPQN is perpendiciilaf to tlie plane AB; join MN, and draw PL parallel to MN; .: zPLQ=^ MNQ = a right angle ; .-. MN=PL = PQ cos QPL, and MN is the projection of PQ on AB, ^ QPL = the inclination of PQ to the plane, whence the proposition. Projection of a Plane Area upon a Plane. 33. Def. The orthogonal projection of a closed plane area upon a fixed plane, is the area included within the line which is the locus of the feet of perpendiculars drawn from every point in the boundary of the plane area. If a series of planes he taken forming a closed polyhe- dron, the algebraical projections of the faces upon any plane are their areas multiplied by the cosines of the angles which their normals, drawn inwards, make with the normal to the plane. 34. The ortJiogonal projection of any plane area on a given plane is the area multiplied hy the cosine of the inclination of the plane of the area to the given plane. Let APB be any closed curve described upon a given plane, and APB' the orthogonal projection upon any other fixed plane, 22 PKOJECTIONS. which is the locus of the feet of the perpendiculars di-awn to the second plane from every point of the cui've APB, The areas APB, AFB' may have inscribed in them any number of parallelograms, such as PQ, P Q', whose sides are in planes PMP, QNQ drawn perpendicular to the line of intersec- tion of the given planes, and parallel to that line, and these parallelograms are in the ratio of 1 : cosine of the inclination of the planes ; therefore the sums of the parallelograms are in the same ratio. Hence, proceeding to the limit when the breadths of these paral- lelograms are indefinitely diminished, the area of the projection oiAPB = area of APB x cosine of the inclination of the planes. 35. ^ the faces of any closed polyhedron he projected on any plane, the sum of the algebraical pryections of the faces upon any fixed plane is zero. Let the polyhedron be cut by a prism having its faces perpendicular to the given plane, and let the base upon the given plane be a, and the areas of two consecutive sections of the faces of the polyhedron made by this prism be /3, 7 ; if a be the algebraical projection of /8, — a will be the projection of 7, the sum of which is zero ; and this will be the case what- ever be the number of pairs of such sections, in case the poly- hedron should be re-entering, and also whatever be the form of the prism. Hence, if prisms be taken in this way so as to include the whole volume of the polyhedron, the truth of the proposition is evident. 36. To find the area of any plane surface in terms of the areas of the projections upon any rectangular co-ordinate planes. Let I, m, n be the direction-cosines of a normal to the plane on which the given area A lies, A^, Ay, A^ the areas of the projections upon the co-ordinate planes of yz, zx, xy. Then, since I is the cosine of the angle between Ox and the normal to the plane, which is the same as the angle between the plane of A and the plane oiyz, A^ = Al, and similarly, A„ = Am, and A, = An ; .: A' = A'{r + m'+ w») = A^ + A," + A.". PROJECTIONS. 23 37. To find the plane upon which the sum of the projections of any numler of given plane areas is a minimum. Let A, A', A!' ... be any number of plane areas, [l, m, n), {V, m', n) ... the direction-cosines of the normals to their planes, (\, /i, i;) those of the normal to a plane upon which they are projected; and let {A,, A„, A,), {A',, A\, A\), ... be the areas of the projections of the given areas upon the co-ordinate planes. Then since IX + m/x, + nv is the cosine of the angle between the plane of A, and the plane upon which it is projected, the projection of A is AQX+ mp, + nv) = A^ + ^„ytt -1- A^v ; therefore the sum of the projections of all the areas upon the plane (\, /*, v) is \S {A^ +fj,t (A) + v'S, (A,) which is to be a minimum by the variation of X,, fi, v, subject to the condition X,= + ^»+ 1/^=1; .-. 2 {A,) dX + t (J,) dfL+t ( JJ dv = 0, and XdX + fidfji, + vdv — 0, must be true for an infinite number of values oi dX : d/i : dv ; X _ fi _ V 1 •"■ ?G4y- 2pg -rpj= v[{S(j.)r+ {S(A)r+{S(^,)n ■ which determine the direction of the plane of projection in order that the sum of the projections of the areas may be a minimum. IV. (1) The sum of the three acute angles which a straight line forms with the co-ordiaate axes is less than 1 80°. (2) The sum of the acute angles which any straight line mates with rectangular co-ordinate axes can never be less than ^ sec"' (-3). (3) Two straight lines are drawn in the planes of osy and ym, making angles a, y with the aaces of x, z respectively j the direction- cosines of the straight line perpendicular to the two are proportional to tan a, — 1, tan y. 24 PROBLEMS. (4) If two straight lines be inclined at an angle of 60", and their direction-cosines be (}, m, n), {I', m', nT), there is a straight line whose direction-cosines are l — l', m — ni, n — n, and this straight line is incliaed at angles of 60" and 120° to the former straight lines. (5) The direction-cosines of a straight line perpendicular to the two whose direction-oosiaes are proportional to I, m, n and m+n, n + l, l + m, are proportional to m, —n, n — I, I —m. (6) If the angles which a straight line forms with the co-ordinate planes be in an arithmetical progression whose difference is 45°, the line must lie in one of the co-ordi)iate planes. If it form angles o, 2'a, 3a, it must lie in one of the co-ordinate planes. (7) Shew a priori that the rational equation connecting the direction-cosines of a straight line can only involve even powers of those quantities. (8) The straight lines whose direction-cosines are given by the equations al + 5m + en = 0, aP + pm'' + yn'=0, will be perpendicular, if a'(J3 + y) + b'{y+a.) + c'{a + ^ = 0, and parallel, if a' b" c' _ - ^- ^ -t- - = 0. "■ P y (9) The straight lines whose direction-cosines are given by the equations al + bm + cn = 0, ^+^ + 2 = 0, will be perpendicidar, if c c and parallel, if ^(aa) ± ^(6^) =fc J{cy) = 0, action-cosines of a line making hose direction-cosines are {I, m, n), {I', m, n"), {I", m", n"), a c (10) The direction-cosines of a line making equal angles with three straight lines, whose direction-cosines are PROBLEMS. 25 are proportional to m {n - n") + m' {n" -n) + ni' (n - n), n {V - 1") + n' {I" -I) + w" {I - V), I (m' - m") + V (m" — m) + T (to — to'). If the given lines are mutually at right angles, the direction- cosines are I + 1' + 1" m + m' +m" n + n + n" (11) Find the direction-cosines of the two straight lines which are equally inclined to the axis of «, and are perpendicular to each other and to the line which makes equal angles with the co-ordinate axes. (12) J£ A, B, C, D be four points in a plane, A', E, C, If their projections on any other plane, the volumes of the tetrahedrons ABGD', A'EG'D are equal. (13) Shew, by projecting upon the base, that the area of the surface of a right cone is iral, a being the radius of the base, and I the length of a slant side. (14) The angle between two faces of a regular tetrahedron is sec"' 3. (15) The straight line which makes equal angles with three straight lines, also makes equal angles with three planes each containing two of the straight lines. (16) If the edges of a tetrahedron ABGD which terminate in D be a, 6, c; and the respectively opposite edges a, V, c ; shew by pro- jecting AB, BG, GD on AD, that the angle between a and a, is (5»+6'=)~(c''+0 cos -i n^— 7 . Hence shew that, if two pairs of opposite edges be respectively at right angles, the third pair will also be at right angles to each other. CHAPTER IV. DISTANCES OF POINTS. EQUATIONS OF A STRAIGHT LINE. Distance between two points, 38. To find the distance between two points whose co-ordinates are given, referred to rectangular axes. / / M p ^ u / ^^ / 1 V — rr Let (x, y, s), («', y\ z) be two points P, Q whose co-ordi- nates are given referred to a rectangular system; and let a parallelopiped be constructed whose diagonal is PQ, and whose edges PM, MN, NQ are parallel to the co-ordinate axes Ox, Oy, Oz; and join PN. Then, since QN is perpendicular to the plane PMN, and therefore to PN, Pg' = P2P+QI^, but PN^=PM' + MN''; .-. P^ = PM'+MN'' + Ng'. PM is the diflference of the algebraical distances of Q and P from the plane yOz, and similarly for MN, NQ: .■.P(r={x-xy+{y'-yr+{z'-z)\ DISTANCE BETWEEN TWO POINTS. 27 If a, ^, 7 be the inclinations of PQ to the axes of co-ordi- nates, x —x = PQ cos a, y'-y = PQ cos^, »' — a = PQ cos 7 ; .•. 1 = cos' a + cos'' yS + cos' 7. The double sign, which appears in the value of PQ, may be interpreted in a manner similar to that adopted in the case of the radius vector in polar co-ordinates in Plane Geometry. If the angles a, jS, 7 define the direction of measurement of the distance PQ of Q from P, the opposite direction is defined by v+a, 17+^, ir+y, and therefore these -angles with an algebraical distance — PQ, equally determine the position of the point Q with reference to P. The distance of the point {x', y\ z) from the origin is 39. To find the, distance, letween two points referred to oblique axes. Let \, fi, V be the angles between the axes; and {x, y, z), (x, y, z) two points P and Q. Let a paraUelopiped be constructed whose diagonal is PQ, and edges PM, MN, NQ are parallel to the axes Ox, Oy, Oz. Now, the projections on PM of the line PQ, and of the bent line PMNQ tenninated in the same points, are equal. Therefore if a, yS, 7 be the angles which PQ makes with the axes, PQ cos a = PM+ MN cos v + JV^ cos fi, 1 similarly, PQ co%p = MN+ NQ cos\ + PM cos v,\ (1), and PQ cos y = NQ+ PM cos fi + MN cos X. J Also PQ is the projection of PMNQ on PQ, .-. PQ = PM cos a -{■ MN cos ^ + NQ cosy (2). Therefore multiplying the equations (1) by PM, MN, NQ we have by (2), PQ' = PM" + MN + NQ' + 2 MN. NQ cos \ + 2 NQ.PM cos /i + 2 PM. MN cos v. 28 DISTANCE BETWEEN TWO POINTS. and PM is the difference of the algebraical distances of Q and P from y Oz, and therefore =x' — x, and similarly MN= y' — y, and NQ = z' —z; .'. P(^= {x - xY +{y'- yY + {z' -z)i + 2{y'- y) {z - z) cos \ + 2 (s' — z) (x' — x) cos /i + 2 {x' — x) {y —y) cos v, whence PQ is determined as required. 40. If I, m, n be the direction-ratios of PQ, PM=l.PQ, MN=m.PQ, NQ=n.PQ; .•. \ = 'P + rr^+n^ + 2'mn cos\ + 2mZ cos /i + 2 Zm cos v, which is the equation connecting the direction-ratios of any line referred to oblique axes, obtained above, Art. 28. 41 . To find the distance of two points whose polar co-ordinates are given. Let (r, 6, ) and (?•', &, <}>') be the given points P and Q. Join OP, Q, QP, and let a spherical surface, whose centre is and radius unity, intersect OP, OQ and Oz in p, q and r. Then, rp=d, rq=6', a.ni^qrj)=^'—^. PQ' = 0P'+ q- 20P.0Q cos pq — r'+r'^—irr cos pq. But cos^g' = cos pr cos'pr + sinpr sinqr cos prq = cos 6 cos 0' + sin d sin 0' cos {(j>' — )}, which gives the required distance. The Straight Line. 43. The general equations of the straight line which will be employed are of two forms : one form is symmetrical and the equations are deduced from the consideration that the position of a straight line is completely determined^ when one point in the line is given, and the direction in which the straight line is drawn. The symmetry of this form gives great advantages, and in all questions of a general nature, the general symmetrical equations will be almost exclusively employed. The other form is un- symmetrical, and tiie equations are deduced from the consideration that a straight line is the intersection of two planes, and is completely determined when the equations of the two planes are given. These equations in their simplest forms are the equations of planes parallel to two of the co-ordinate axes, and are the same as the equations of the projections of the straight line parallel to these axes upon two of the co-ordinate planes. It will be seen that in particular cases it is advantageous to use the unsymmetrical form of the equations. 44, To find the symmetrical equations of a straight line. Let ^ be a fixed point (a, p, y) of a straight line, P any other point {x, y, s), I, m, n, the direction-cosines of AP; and let AP=r. Then the projection of PQ on the axis of a; is a; — a, and it is also Ir ; 1 x — a J • •! 1 y-^ ^-y hence, — 5 — = r, and similarly, = r, = r. 'I ■' m n The equations of the straight line are then x—a._y — ^_z — y I m OT ' ^v a;-«_ y-/3 _g-7 *''' ~L M ~ -N ' if i, M, iVbe any quantities proportional respectively to ?, m, n. 30 TIIE-STEAIGHT LIXE. It should be careftdly remembered that, when the former equations are used, each member of the equations is equal to the distance r between the two points {x, y, z) and (a, P, 7). The equations of a straight line wiU be of the same form if the axes be oblique, the same interpretation being given to r, and I, m, n being the direction-ratios. The projections employed in the above proof will then be the intercepts on the axes made by planes parallel to the co-ordinate planes. 45. To find the non-symmetrical equations of a straight line. If a straight line PQ be projected by straight lines parallel to the axes Oy, Ox, whether rectangular or oblique, on the two co-ordinate planes xz, yz, each projection will be a straight line as pq, p'q, in those planes respectively. Hence the co-ordinates x, z of any point {x, y, z), in PQ being the same as those of the projection of the point in pq, satisfy an equation of the fonn a; = ms + a, and the co-ordinates {y, z), similarly an equation of the form y = nz + b, and consequently the equations of the line may be written X = mz + a, y — nz + b. 46. On the number of constants employed in the equations of a straight line. It may be noticed that the latter system of equationg involves only four constants, whilst the symmetrical system involves six. THE STEAIGHT LINE. 31 • Of the three I, m, n, however, we know that they are connected by the relation P + 711^ + 71^=1, (Art. 25), which renders them equivalent to only two independent constants ; and if we take L, M, N, since these are only required to be proportional to l,7n, 71, one of them may be assumed arbitrarily, and they are still equivalent to two constants only. Also, of the three a, /3, 7, one may be assumed at pleasure ; for, since the straight line cannot be parallel to all the co-ordi- nate planes, let it be not parallel to that oiyz ; then, at whatever distance a from yz we take a parallel plane, the straight line will meet this plane, and we may take the point where they meet for the point a, ;8, 7 ; that is, we may give to a any value we please, and the three a, /S, 7 are consequently equivalent only to two independent constants. 47. To f/nd the equations of a straight ZtVie parallel to a co- ordinate pla7ie. If a straight line be parallel to a co-ordinate plane, as that oiyz, every point in it is at a constant distance from this plane, and we have the equation x= a, therefore its equations will be of the form x = a, y = Tis + b. Taking the symmetrical form, since the line will be perpen- dicular to the axis of a;, I = 0, and therefore L = 0, and the equations of the line assume the form m n X— a ~ M ~ z-y N or, which form implies that x = a. for every point in the line at a finite distance, since the members are not infinite for • such values. 48. To find the equations of a straight line parallel to one of the co-ordinate axes. If the straight line be paxallel to one of the co-ordinate axes, it will be parallel to the two co-ordinate planes passing through 32 THE STEAIGHT LINE. that axis, and consequently any point in it will be at an in- variable distance from each of these planes. Thus, if a straight line be parallel to the axis of x, the distances of any point in it from the planes xs, yz, will be constant, a fact expressed by the equations, y = l, z=c, which will therefore be the equations of the line. As before, the symmetrical form is a; — a_^ — /8_a — 7 49. To find the angle between two straight lines whose equations are given. If the equations of a straight line be given in ihe form a;— a_y — /8_« — 7 then, if I, m, n be its direction-cosines, I m n +V(f+w' + w') +1 L~ M~ N" ^{r + M^+IiT) ~ >/{L'+]\P + N'') ' or, the direction-cosines are +L +M +N H the equations be given in the form X = mz + a, y = nz + b, since these may be written, x — a ti — h' = ^- =0, m n the direction-cosines of the line are +m +n +1 (!)■ In (1) and (2), the ambiguities have the same sign. (2). THE STEAIGHT LINE. 33 Hence, if the equations of two straight lines te X — a. _y — ^ _z —7 x — aL_y — yS' _z — 'f' the angle between them is LL' + MM' + NN' ^°® ^/{L' + JiP + N') ^{L" + M" + N") ''^ '' and if the equations be X = mz + a, y = nz + b, X = m'z + a, y = n'z + h', the angle between them is _i 7nm + nn' + 1 ^°^ V(m» + wH 1) V(w" + w" + 1) ■ 50. Jb _^«cZ '-ar + {y'-^Y+ {z'-if- [l {x'~ a) + m {y'-fi) + n {z'-r,)]-]. If the equations of the line be x—mz-\-a, y ==nz + b, which are equivalent to X —a_y — b ■ = z, m n the distance will be replacing a, jS, 7 by a, h, 0, and I, m, n by tn, n, 1, in the expres- sion already found. 38 THE STRAIGHT LINE. 59. To shew that the shortest distance between two straight lines which do not intersect is perpendicular to each. Let AP, BQ\)& the two straight lines, and let a plane he drawn through BQ parallel to AP, and BB be the orthogonal projection of ^P upon this plane, B being the projection oi A ; therefore AB will be perpendicular to both straight lines, for it meets two parallel lines AP, BB, to one of which, BR, it is perpendicular, and it is also perpendicular to BQ. Let P, Q be any points in AP, BQ, join PQ, di-aw PB per- pendicular to BR, and join QR ; then PQ is greater than PR, being opposite to the greater angle, and PR = AB; therefore AB is less tha^ PQ, or the distance which is perpendicular to both straight lines is less than any other distance. 60. To find the shortest distance between two straight lines whose equations are given. Let the equations of the two straight lines be I m n ' I m n and let X, fi, v be the direction-cosines of the straight line per- pendicular to each, then IK + mji +nv =0, I'X + m'li + n'v = 0. \ fl _ V Hence, mn—m'n nV —n'l hd — I'm ~ V{(»»w' - m'nf + {nV -n'lf + {hn' - I'mY] sin ' 6 being the angle between the lines, (Art, 26). Now, if we suppose P, ^ to be the points (a, ,8, 7), (a', j8', 7'), THE STRAIGHT LINE. 39 the projection of P$ on AB will be \ (a — a') + ;it (jS - /3') + 1» (7—7) , but this projection will be AB itself; hence AB - ^"""'^ ("»w'-m'w) + {^-0) {nV-n'T) +(7-7') {Im'-l'm) ' sin ^ The equations of the straight line on which the shortest dis- tance lies, may be obtained in the following form, u + u' cos 6 Z (as — a) + w (y — /3) + w (a — 7) = and Z'(a3-a')+m' (y-/3') + »'(a-7') = sin"^ ' u' + u cos 6 sin''0 ' where m = Z (a' - a) + m (/3' - /3) + « (7' - 7), and u = Z' (a - a') + m' (/3 - /3') + w' (7 - 7'). 61. The simplest form in which the equations of two straight lines can be presented will be obtained by taking the middle point of the shortest distance between them for the origin, the line on which the shortest distance lies for one of the axes, the axis of z suppose, and the two planes equally inclined to the two straight lines as those of zx, zy. If 2a be the angle between the two straight lines, 2c the shortest distance between them, their equations will thus become y=x tan a, s = c ; and y = — x tan a, z— — c. 62. To find the equations of a straight line referred to four- plane co-ordinates. If (a', y8', 7', S') be four-plane co-ordinates of a fixed point in a straight line, (a, )S, 7, S) those of any other point in it, p the distance of these points, then^,^:^,^:,!::^ p p p p will be equal respectively to the cosines of the angles between the straight line and the normals to the fundamental planes, and if these be I, m, n, r, we shall have a-a' _ /8-/3' _ 7-7' _ g-8' I m n r ■ = Pl a-a'_^-^_7:-7'_8-8: ''^' L ^ M ~ N ~ B ' ^'■^ if L, M, N, B be respectively proportional to Z, m, n, 40 THE STRAIGHT LINE. It is obvious that, since the direction of the straight line, expressed in this system, involves the three ratios L : M : N : B, and in the former system depended only on two, some invariable relation must subsist between L, M, N, E. In fact, we have Aa + B^+ (}Y + m = SV=Aa' + B^'+Cy' + BS', A{ci-a')+B{l3-^')+C{y-'y')+D(B-S')=0; whence AL + BM+ GN+ BE = 0. If tetrahedral co-ordinates be employed, the relation will be L + M+N'+B = 0. 63. It should be noticed that two of the three equations (1) are sufficient to determine the straight line, the third being directly deducible. If, for example, we have a-^a' _ y3-/3 ' y-7' L M ~ N ' each member is equal to A{a-a.')+B{^-^')+G{y-y') AL + BM+CN ' -I){B-S') , . , . ^, S-S' or to -pr HIT? — TTvf ) which IS the same as — =— . AL + BM+ ON' E 64. If the straight line pass through one of the angular points of the tetrahedron of reference, as the one opposite D, (0, 0, 0,p^ the equations become L M N~ R ■ (1) The straight line given by the equations x + 2y + 3z = 0, 3x + 2y + z = 0, makes equal angles with the axes of x and «, and an angle sin"' -j^ with the axis of y. (2) The equations =- = ^ — ^ = — denote thirteen straight x+1 y+1 is + l ^ lines. Shew that four are equally inclined to each other, and con- struct for the rest. PROBLEMS. 41 (3) Find the direction-cosines of the straight line determined by the equations Ix + my + nz = mx + ny + lz = nx + ly + mz. (4) Find the equations of the straight line which passes through the origin and intersects at right angles the straight line whose equations are {m + n)x + {n + l)y + (l + rn)z = a, ('m, — n)x + {n — l)y + {l-m)z = a; and obtaia the co-ordinates of the point of intersection. (5) Find the equations of the straight line passing through the points (b, c, a) (c, a, b), and shew that it is perpendicular to thei line passing through the origin and through the middle point of the line joining the two points, and also to each of the straight lines whose equations are X y z x = y = z, -=^=-. a b c (6) Interpret the equation {x" + y^ + !^ {l" + m' + n') = {Ix + my + nzf, and give a geometrical illustration. (7) The straight lines determined by the equations Ix + my + nz = 0, l(J>—c)yz + m(e — a)zx + n(a — b)xy= 0, are at right angles to each other. (8) The straight lines given by the equations 7 n <* ^ c . Ix + my + nz=v, — -i 1 — =0, " X y z will be at right angles, if -5- -f- — -i- - = 0. ° ° I m n (9) The equations of two straight lines are X _ y _^— c sin a cos a ' X y z + c sin a shew that the distance between two points on these straight lines whose distances from the axis of z are a, b respectively is ^(4c''-t-»' + 6''=F2a6cos2a). 42 PROBLEMS. (10) Shew that the equations a + mz — ny _P + nx — lz _ y + ly— mx I m n . ... ^ x + nB-my y + ly-na z + ma-lfi are reducible to -; = '- = ' = > I m n I, m, and n being direction-cosines. (il) The equations of a straight line are given in the form a — ny + mz P — lz + nx_y — mx + ly X. fi V obtain them in the form fiy — V)8. va — ky X)8 — fia ZA.+TO/X. + WV ZA. + w/t + nv " l\ + m/ji, + nv (12) The equations of the straight line on which lies the shortest distance {2d) between the two straight Hnes, T+-=l, x = 0- and = 1, « = 0, be ' a e " II z X , ax hy - are!j+ = l-Tr = :K~^- h c a dr or ax. ^x. ^ 1111 Shew that w = -5 + ti + 1 • 4. tetrahedral co-ordinates, a + /8 + 7 + S = 0, is the equation of a plane at an infinite distance. 82. To find the condition of parallelism of two planes. Let the equations of two planes be Za +»W)S + M7 + rS = 0, I'a + »w'/3 + w'7 + r'S = 0. These intersect in a line which lies on the plane at infinity, whose equation is a + i8 + 7 + S = 0, using tetrahedral co-ordinates. Hence, these planes intersecting in a straight line are satisfied simultaneously by an infinite number of values of a, /8, 7, S: therefore, employing indeterminate multipliers, ^l + XT +1=0, \m + X'»i' + 1 = 0, \n + \'w' +1=0, \r + XV +1=0, from which we obtain the required conditions of parallelism, l — m m — n _ n—r V — m ~ m' — n' ~ n' — r ' Planes under Particular Conditions. 83. Equation of a plane passing through a given point. Let (a, b, c) be the co-ordinates of the given point, Ix + my + nz =p, the equation of the plane, then since {a, h, c) is a point in this plane Ia + mb + nc =p, 54 EQUATION OF A PLANE. or, eliminating^, I {x — a) + m {y — b) + n {z — c) = 0, is the general equation of a plane passing through the point {a, b, c). 84. Equation of a plane passing through a point determined by the intersection of three given planes. If the point be given by the equations of three planes, M = 0, V = 0, w = 0, passing through it and not intersecting in one straight line, then lu + mv + nw = will be the general equation of, a plane passing through that point, for it is satisfied by the values of x, y, z which are given by the equations M = 0, v = 0, w = 0, taken simultaneously, and therefore passes through the inter- section of these planes, which is the given point; and since this equation is of the first degree, and involves two arbitrary constants, namely, the ratios I : m : n, it ia the general equation of a plane passing through the given point. If the three planes, m = 0, v=0, w = 0, do intersect in a straight line, then these equations, and therefore the equation lu + mv + nw= 0, will be simultaneously satisfied for all points lying in that straight line. Hence, lu + mv + nw = 0, cannot be the general equation of a plane passing through a given point. The position of a point is not, in this case, completely determined by the given equations, but only the fact that it lies on a certain straight line. 85. Equation of a plane passing through two given points. Let (a, b, c), (a', V, c') be the given points : the equation of a plane passing through {a, b, c) is l{x — a)+m,{y — b)+n{s — c)=0. If this plane pass also through {a, b', c), we shall have l{a'-a) + m {b' -b) + n {c' - c) =0, and eliminating n, we obtain the equation l[{z-a) {c'-c)-{z-c) (a-+a)} + m{(i/-b) {c'-c)-{z-c){b'-b)}=0, EQUATION OF -A PLANE. 35 or, altering the arbitrary constant, ^ (X — a z— c'\ (y — h z — c\ '„ for the general equation of a plane passing through two given points. This equation may be written symmetrically, . X — a , y — h a-c„ X -, + iJi.frr-r + v- =0; a — a "^ b — c —c \, fi, V being connected by the equation \ + fj, + v = 0. 86. Equation of a plane passing through two given points, lying on given pkmes. If the straight line passing through the ' two points be given by the equations m = 0, « = 0, the equation lu + mv = will represent a plane passing through the points of intersection of M = and v = 0, and therefore through the given points, and since this equation involves an arbitrary constant (? : to), it will be the general equation of a plane passing through the two given points. Or, the equation lu + mv.+ nw = 0, if I, m, n be connected by a relation of the form al+bm + cn = 0, will represent a plane passing through the two points given by the equations M = 0, v = 0, M = 0, u = a, v = i, w = c. 87. Equation of a plane passing through three given points. Let {a, b, c), (o', V, c), {a", b", o") be the three given points, l{x — a)+m{t/ — b)+n{3 — c)=0, (1) the equation of a plane passing through {a, b, c). If this plane also pass through (a', b', c') and (a", b", c"), we shall have l{a'-a)+m{b'~b)+n{G'-c)=Q,^ (2) I {a" -a)+m {b" -l)+n (c" - c) = 0, (3) 56 EQUATION OP A PLANE. and eliminating I, m, n between (I), (2), and (3), we obtain {x - a) [h (c' - c") + V (c" - c) + h" (c - c')} + {y-h)[c {a' - a") + c' (a" - a) + c" (a - a')} -\-(^s-c) {a (J'- &") + a' (J" - J) + a" (i - S')} = 0, (4) as the equation of the plane passing through three given points. The coefficients of x, y, z in this equation are the projections on the co-ordinate planes of the triangle formed by the three given points, call these A„, Ay, A,; then 05.4^, will be equal to three times the volume of the pyramid whose base is A^, and vertex the point {x, y, z). Hence, equation (4) asserts that the algebraical sum of the pyramids whose bases are the projections of any triangle on the co-ordinate planes, and common vertex any point in the plane of the triangle, is constant for all positions of this point. The equation here obtained becomes nugatory if h (c' - c") + h' (c" - c) + h" (c - c') = 0, c («' - a") + c {a" -a)+ c" {a - a') = 0, and a (6' - b") + a {b" -b)+ a" (5 - J') = ; which are equivalent to {b - V) (c" - c') - (c - c') (&" - V) = 0, (c - c') {a!' -a') - (a - a') (c"- c') = 0, (a- a!) {V- b') - (6 - b') {a"- a') = ; a — a'_b — b'_c — c' ' Zj ^ — J? IT' — Z' I" 3 a —a 0—0 c —c which are the conditions that the three given points should lie in a straight line. 88. To find the equation of a plane passing through a given point, and parallel to a given plane. If (a, b, c) be the given point, and I, m, n the direction-cosines of a normal to the given plane, the equation of the proposed plane will be l{x — d) 4wi(^-6)+M(a-c)=0. 89. To find the equation of a plane passing through two given points, and parallel to a given straight line. EQUATION OF A PLANE. 57 Let (a, b, c), {a, b', o') be the given points, I, m, n the direc- tion-cosines of the given straight line ; then the equation of any plane passing through {a, h, c), («', b', c'), is X, fjb, V being connected by the equation \ + /i + i' = 0, (2) and if this plane is parallel to the line whose direction-cosines are I, m, n, its normal is perpendicular to this line, hence IK .^ «V* ^ - ^0; (3) o' — a b' — b c' — c and eliminating \,/t, v between (1), (2), and (3), we obtain the required equation, ' x—a f m w \ y—'^f ^ ^ ^ 1 ^~^ ( ^ ^ ^ -n ^ifZ^yb'-b c'-c) '^ b'-b[c'-c a'-a) '^ c'-c\a'-a b'-bj~ This equation becomes identical if -; = t; — 7 = - — , which are the conditions that the given straight line may be parallel to the line joining the two given points. The equations (2) and (3) are in this case coincident, or every plane passing through the two points will necessarily be parallel to the given straight line, as is otherwise evident. The required equation will then be the equation of any plane passing through the two given points. 90. To Jind the equation of a plane passing through (f g'&jen point, and parallel to two given straight lines. If the direction-cosines of the two straight lines be ^»N^)/^ , and V, m, n', and the co-ordinates of the given point a, V^ the equation of the plane will be (mn-m'n) {x-a) + {nl'-n'l) {y-b) + (Zm'-Z'm) (s-c) = 0. (Art. 60). If - = ^ = ^ , this equation is satisfied for all values of I' m n x,y,z; or, if the given straight lines are parallel, there are an infinite number of planes satisfying the given conditions, the direction of the normal to the required plane being in- determinate. 58 EQUATION OF A PLANE. 91. To find the equation of a plane equidistant from two given straight lines, not in the same plane. Let the equations of the two given straight lines be X — a _7/ — ^ _z —y I m n (1) ^=2^'=f^=/. (2) I m n the middle point of the line joining {x^, y^, a,) and {x^,2/i, ^a). Then, 2X = a;, + a;, = a + a' + Zr + ?V', 2Y=y^+y^=^+^'+mr+m'r, 2Z— Si + gj = 7 + 7' + «>• + nV, and eliminating r, and r', \ce obtain for the locus of (X, Y, Z), the equation (2X- a - a') (wiw' - m'w) + (2 F- /3 - )S') (mf - n'T) + (2Z- 7 - 7') (ZW - ZW) = 0. (3) The plane represented by this equation bisects all lines joining any point of (1) to any point of (2), and therefore bisects the shortest distance between them ; and since the direction-cosines of the normal to (3) are proportional to mn' — m'n, nV — n'l, Tmi — Tim, the normal is parallel to the shortest distance between the lines, (Art. 60). Hence this plane bisects at right angles the shortest distance between the lines. YI. (1) Find the equation of the plane passing through the poinds (a, 6, c), (6, c, a), (c, a, V), and the equations of the planes each of which passes through two of the points, and is perpendicular to the former plane. (2) The equation of a plane passing through the origin, and con- taining a straight line x — a. y — P _ z — y I m n ' I \m nj m \n I J n \l m/ PROBLEMS. 69 , Henoe, find the equations of tie straight line passing through the origin, and intersecting two given straight lines ; and examine the case in which the straight lines are paxallel. (3) The equation of a plane passing through the origin, and con- taining the straight line whose equations are a; + 2?/ + 3s; + 4 = 2a; + 32/ + 4« + 1 = 3a! + 4y + « + 2 is SB + y - 2s = 0. (4) The equation of a plane passing through the origin, and con- taitiing the straight line a + m« — wy P + nx — lz y + ly — mx I m n ' is (P + m" + n") {ax + Py+ ys) = (Ja + »ij8+ my) {Ix + my+nz). (5) Shew that the co-ordinates of a point, which divide the distance between the points (a, j8, y), (a, /3', •y'), in the ratio A' : \, Xa + W \B + Kff , Xv-i-XY are -r — 77- , -h: — ^4- , and -^ y- . (1) Hence, shew that the locus of a point dividing the distance between any two points on the two straight lines y — P _» — y x — ay — Pl_ z — y in the ratio \' ; X, is the plane whose equation is {mm, —mn)\x t — y- 1 -H &c. = 0. (2) Also, that the equation Ax + £y + Cz = I) represents a plane, according to Euclid's definition. (6) Shew that if the straight lines x _ y _ e X _ y _ s x _y _e a P y'aa hfi cy' I m n' lie in one plane, then - (6 — c) + -^ (c - a) + — (a - 6) = 0. (7) Find the equation of the plane which passes through the two parallel lines x — a y — P ^~y . a! — a'_y — /8'_« — y' I m, n ' I in n ' and explain the result when — ^ — = = - — - . 60 PROBLEMS. (8) The equation of a plane passing through the two straight lines x-a y-b _ g-e x-a' __ y-V _ z-d ~ir ^~V~~~e~' a ~ b ~ o ' is (6c' - b'c) X + (ca' -(/a)y + {ah' - a'b) « = 0. Give a geometrical interpretation of the equations. (9) Shew that the three planes lx + my + nz=0, (m + n)x + (n + l)y + {l + rn)z = 0, x + y + is = intersect in one straight Hne X _ y _ z m — n n—l l — m' (10) Determine the conditions necessary in order that the planes ax + cfy-¥ Vz = 0, 7i,Si> may be written Pi P^ Pi P4. If a 5 /3i! 7i) ^1 1^e tetrahedral co-ordinates, this equation becomes 98. To find the equation of a point which divides the straight line joining two points, whose equations are given, in a given ratio. 64 FOUR-POINT CO-OEDINATE SYSTEM. Let the equations of the two points P, P be Za + »«)8 + W7 + j-S = 0, and I'a + m'/3 +m'7 + rh = 0, and let ^ be a point in PF, such that PQ: QP ::p.: X; therefore for every plane through Q, whose distances from P and P are tt and tt', Xtt + /att' = 0, (Art. 95) ; .-. X. ^°' + "^ + 7.+ ^^ M-. V"y^'-f = 0, (Art. 94), l + m+n + r "^ I +m+n +r ^ ' which is therefore the equation of Q. 99. To shew that the stra^ht lines joining the middle points of opposite edges of a tetrahedron intersect and bisect each other. The equation of the middle point of AB is a + /3 = 0, and of the middle point of CD is 7 + S = : therefore the equation of the middle point of the line joining these is a +^ +7+8=0, which for the same reason bisects the lines joining the middle points of the other opposite edges. 100. The student is recommended to examine carefully the processes employed in the following applications of Point-Co- ordinates. Let Za + »»/3 + 717 + rS = be the equation of any point U, and let planes be drawn through this point and each of the edges • and let {ab) denote the point in which the plane HOB meets AB, and similarly for the other edges. The point E lies in the straight line joining (db), for which la + m0 = 0, and {cd), for which 7nr/ + rS = 0, since its equation is of the form L {h + m^) + il/(?w7 + rS) = 0. FOUR-POINT CO-ORDINATE SYSTEM. 65 Since, for {db), la. +»n;8==0, {ad), la. + rh =0,, (Jc), JW/8 4- «7 =0)1 {cd), W7 + j-S = oj • the straight lines joining these pairs of points meet BD in a point h'd' whose equation is m^ = rS, and the equation of bd is m^ + rS = ; therefore h'd', hd divide BD haimonically-. Similarly, the line joining {ah), {ac) ; and {hd), (cd) intersect BO in b'c', for which m0 =a ny ; and the straight line {ah) {cd) meets the plane passing through A and the points h'c', h'd', in the point whose equation is 2la. + 2m/3 - M7 - rS = 0, since this equation may te written 2la. + (m/3 - ny) + {m^ - rS) = 0. Again, the equation 2h. + «i/3 + ny = being of the form La + MB +2^ {la + m^ + ny + rS) = 0, represents a point in plane ABD, and, being of the form L {la + 7W/3) +M{la + ny) = 0, lies on the line joining {ab) , {ac) , and obviously lies in the plane ABC. Let the straight line AJE, BE meet the opposite faces in A', B', the equations of these points are m^ + W7 + j-S = 0, and la + ny + rS = 0, and therefore A'B' intersects AB in the point la — m/3 = 0, the same point in which {ac) {be), {ad) {bd), meet AB. ■ 101, To find the equation of a point at an infinite distance. Let the equation of the point be h + m^ + W7 + rS = ; since the point is at an infinite distance, the distance la. + m/Q + nr^+rh l + m + n + r from any plane, whose co-ordinates a, ^, y, B are finite, is infinite ; .•. l + m + n + r = 0. 66 FOUR-POINT CO-OEDINATE SYSTEM. Then, since I, m, n, r are proportional to the tetrahedral co- ordinates of the point, (Art. 99), if a,, /Sj, %, S, be these co- ordinates, we have «i + /3i + 7. + Si = 0, which is the same result as was given in (Art. 81). 102. To find the relations between the constants in the equation of a point, in order that the point may lie in the different portions of space cut off by the indefinite planes which form the faces of the fundamental tetrahedron. Let the equation of a point P be h. + »i/3 + W7 + rS = 0, and let Z + ot + n + r = s. If AP or AP produced meet the opposite face in Q, the equation of Q is m^ + ^7 + rS = ; and if e be the pei-pendicular from Q on any plane through P, whose co-ordinates are a, /8, 7, 8, (?ra + w +»•) e + Za = 0, (Art. 93), a and e being estimated in the same direction. If P be between A and Q, a. and e have opposite signs ; m + n+r . ... « • -u . ,1 . . J IS positive, or j is between 1 and co . If P be 'm. AQ produced, a and e are of the same sign, a is greater than e ; 7?2 -4- Vh -f- V 8 .'. -J : is less than — 1, or ^ is between and — 00 . If P be in QA produced, a and e are of the same sign, a is less than e ; .'. J is between and — 1, or ^ is between 1 and 0. I. For points within the fundamental tetrahedron, ^ , — , - , - are all between 1 and co ; I m n' r ' .'. I, m, n, r are all of the same sign. FOUR-POINT CO-ORDINATE SYSTEM. 67- II. For points between BGB, and ABG, AGO, ADG pro- duced, s s J is negative, - , - , - between 1 and oo . III. For points within the solid angle formed by BGA, CDA, DBA produced, Y is between and 1, — , — , - negative. IV. For points between AGB, BGB, BA G, ABB produced, — and J are negative, - and - are positive. And similarly for the nine other compartments. 103. The results of the preceding article may be obtained, if we assume a knowledge of Tetrahedral Co-ordinates, by observing that the tetrahedral co-ordinates of the point are I m n r (Art. 97). 104. To find the distance between two points whose equations are given. Let the equations of the two points 0, 0' be la + m/9 + M7 + rS = 0, and let l+in + n-\-r = s, ?' + m' + w' + r' - s'. • p2 68 POUR-POINT 00-OEDINATE SYSTEM. Then, i{ AO, Aff meet BCD in a, a', and Ba, Ba' meet CD in b, b', we have from the triangles OA 0', aAa', AOf + AO'^-OO" „/™ Aa' + Aa"-aa" 2A0.A0' =^°'0A0= ^^^^^, . -I, ^ Oa Aa AO , O'a Aa' AO' ^^* —=T = —l' ^""^ — = — =737- -('-?)(;-?)-»'-('- 3 ('-r)"-. From which form it is manifest that the final result will be an expression in terms of the squares of the edges, and we shall obtain the result by investigating the coefficient of AB', and de- ducing the rest by symmetry. Now the only terms which can involve AB' are contained in Aa" and Aa'^, and, as before, AB'+Ba^-Aa" .„ AB" + Bb" - Ab" also 2AB.Ba 2AB. Bb ab Bb m 8 — 1 Ba 8-1- m' .: Ad' = ■■Aff-U- m \ s — l) AR^. i '^,AB'+.. s ■ writing down only the terms in which AB" appears. Similarly, Aa'" = -^, AB'+ FOUR-POINT CO-OEDINATE SYSTEM. 69 • = (?-f)(?-7)^^+ which gives the distance required. 105. If a, 0, 7, S and a', /3', y, S' be tetrahedral co-ordinates of the points 0, 0' we deduce the expression for the square of the distance, 00" = {a' -a) {^-^')AB' +...... which gives the distance between two points, when their tetrahe- dral co-ordinates are given. 106. To find ths relation between the CQ-ordmates of a plane. Let a, ^, 7, B be the co-ordinates of a plane, X, /i, v, p the cosines of the angles between the direction in which the co-ordi- nates are measured, and the perpendiculars from A, B, C, D on the opposite faces, and let Aa be the perpendicular on BCD, ■BTj, ■nr„, ■ST^ those from B, 0, B on the plane drawn through a parallel to the ^ven plane, JEF the trace of this plane on BCB. Project Aa on the normal to the given plane ; .-. ^,X = a - ^ + ■sj-j , = a - 7 + 57 I . (1) = a — S + t^rf ■' 70 FOUK-POINT CO-ORDINATE SYSTEM. Now since Wj, isr., w^ are proportional to the perpendiculars on EF, from B, C, D, 'sr,.aCD + i!T,.aDB+vTa.aBC=0;* and aCB + aBB + aBC =BCB; .: by {l),p^\.BCB = oi.BOB-^.aCB-y.aBB-S.aBC; also, p,.BGB=2>,.4CB = p^.ABB=p,.AB0; .-. \ = ^ - ^ cos {CB)-^ cos (BB) - - cos {BG), Pi Pi Pa Pi denoting by ( GB) the angle between the faces which meet in GB, and similarly for the rest. Similar values may be obtained in the same way for fi, v, and p. If P be the foot of the perpendicular from A on the given plane, the perpendiculars from P on the faces of the tetrahedron will be a\— Pj, a/i, av, ap, ,.^^2^ZPl + ^^±+y^Z + S.^=0; Art. (81) Pi Fi Pi Pa i'i Pi Pz A or, — + — + — 2 + — 5 Pi Pi Pa P, - ^ cos ( GB) - ^ cos {BB) - M cos {BG) PiPi PiPa PiP, -2yl,os{AB) -?^ COS {AG) -?^ COS {AB) = 1, PaPi PiPi PiPa which is the relation required. ♦ The equation of a straight line, referred to trilinear co-ordinates, is Pi Pi ' P3 where m,, wj, Wj are the perpendiculars, estimated in the same direction,. from the angular points of the fundamental triangle upon the straight line, and PuPup, those upon the sides. Now, taking EFfoT the straight line, and BCD for the fun- damental triangle, for a, a point in that line, — : £- ; X y, aCD : aDB • aBC Pi Pi Pa .: ■i!r,.aCD + TS,.aDB + x!!3.aBC=0, whence the equation in the text. FOUE-POINT CO-OKDINATE SYSTEM. 71 107. To find the relations between the co-Ordinates of a plane at an infinite distance. If the plane be at an infinite distance, the difference between any two of its co-ordinates will vanish compared with either of them ; whence, if a, ;S, 7, 8 be the co-ordinates of the plane, we must have a = ^ = y = S, the relations required. 108. Since each of the co-ordinates is of infinite magnitude, the expression for \ in the last article will give us = --- cos {CD)-- cos {DB) - - cos [BG), Pt Pi Pa Pi which is the equation arising from the projection of the faces meeting in A upon BCD. From this and the three similar equations, we may eliminate P\iPiiPz^ andjj^, and obtain the relation which subsists between the cosines of the inclinations of the faces. 109. To find the perpendicular from a given point upon a given plane, referred to Tetrahedral Co-ordinates. Let (a, ;8', 7', S') be the co-ordinates of the point, lo. + JMyS + M7 4- rS = 0, the equation of the plane. If a,, /8j, 7,, Sj be the point-co-ordinates of the plane, the equation of the plane may be written ««, + /3/3j + 77^ + hZ, = 0, Art. (81), and the perpendicular required is equal to a'«, +*/9'A + 7'% + S'S,. Art. (94). ^ , ±,/K + -?^'cos((72)). I m n r /J_ I ^ ^ ^ 7TF+_r!^cos(a2>)-. therefore the perpendicular required is la + W/8' + nlane, whose co-ordinates are given, makes with the faces of the tetrahedron. If \, fi, V, p be these cosines, a, jS, 7, S the co-ordinates of the given plane, we have deduced in the previous article \ = - - ^ cos (Oi)) - ^ cos {DB) = - cos {BC), Pt Pa Fs Fi and similarly for fi, v, p. The relation between \, fi, v and p is formed immediately from the consideration that the equation Fx P2 P^ Fi ' when the plane moves parallel to itself to an infinite distance, becomes Pl P, Fb Pi If the plane be the face BCD, \=1, fi = — cos CD, v = — cos DB, And p = — COB BG; .-. --- cos CD- - cos DB-- cos B0= 0, Pi P2 P, Pi as before. Art. (108). 111. To fmd the condition that two 'planes whose co-ordinates are given may he parallel. If (a, /3, 7, 8) (oj', /8/, 7j', S') be the two planes, a-a' = i8-/3' = 7-7' = S-8', since the perpendicular distance between the planes is constant. 112. To find the eqvMion of a line. Let the equations of two points in the line be M = Za + m^ + m7 + 1/8 = 0, v = l!a. + W/8 + n'y + v'h = 0, PEOBLEMS. 73 then^ for every point in the line joining them, \m + /*« = 0, for some value of \ : /i. Hence, if be an equation in which L, M, N, E involve only one variable in the first .degree, this is the equation of any point lying in a certain straight line, and may therefore be considered as the equation of a straight line. 113. To find the equation of a plane. Let M = p, u = 0, w = be the equations of three points in the plane, not in the same straight line. Then "Ku + ixv + vw = for any point in that plane with certain values of \ : /i : v. Hence, if be an equation in which L, M, N, B involve any two variables in the first degree, the equation is that of any point lying in a certain plane, and may therefore be considered as the equation of the plane. VIII. (1) The equation of the center of gravity of the face ABC is a + )8 + y = 0. Hence, shew that the lines joining the vertices with the centers of gravity of the opposite faces meet in a point. (2) The equation of the center of the circle circumscribing the triangle ABG is a sin 2^ + yS sm 25 + y sin 20= 0. (3) The co-ordinates of the plane passing through the centers of gravity of the faces AG£>, ABB, and ABC are given by the equations (4) If P be any point on BD, Q, R points in AC, such that AQ : QC :: DP : PB :: CB : BA, then PQ and PB will intersect the lines joining the middle points of BG, AD, and AB, CD respectively, and divide them in the same ratio as .4C. 74 PROBLKMS. (5) If through the middle points of the edges BO, CD, DB straight lines be drawn parallel respectively to the opposite edges, these straight lines will meet in a point ; and the line joining this point with A will pass through the center of gravity of the pyramid. (6) The equation of the center of gravity of the surfeoe of the tetrahedron is [A + B + G+D){a + P+y + h')=Aa + B^-^Cy + Dh. (7) Shew that the equation of the centers of the eight spheres which touch the fiwes or the faces produced of the fundamental tetrahedron are (8) The center of the inscribed sphere lies on the line joining the centers of gravity of the volume and of the surface of the tetrahedron, and divides it in the ratio 3:1. (9) The points B, G, D are joined to the centers of gravity of the opposite faces, and the joining lines produced to points 6, c, d, so that B, b, &c., are equidistant from the corresponding faces, prove that the co-ordinates of the plane bed are given by the equations -2a = ;8 = y = 8, and that this plane divides the edges AB, AC, AD in the ratio 1 : 2. (10) If points be taken in the lines joining B, G, D ix> the centers of gravity of the opposite faces, dividing them in the ratio m : n, the plane containing these points divides the edges AB, AC, AD in the ratio m : 2m + 3n. (11) If through any point P straight lines AF, BP, GP, DP be di'awn meeting the opposite faces in a, b, c, d, the straight lines AB, ab intersect, and their point of intersection and the point in which Gd meets AB divide AB harmonically. (12) The straight lines joining D to the intersection oi AB, ab, and A to the intersection of DB, db, will intersect in a point lying on Be. CHAPTER VII. TEANSFOKMATION OF CO-OEDINATES. • 114, The investigation of the properties of a surface repre- sented by a given equation, is often rendered more convenient by referring it to a different system of co-ordinate axes, in the choice of which we must be guided by the nature of the investigation proposed. We proceed to obtain formulae by means of which such transformation may be effected. 115. To change the origin of co-ordinates from one point to another, without altering the direction of the axes. Let a, p, y be the co-ordinates of the new drigin referred to the primary system, x, y, z, x', y', z' co-ordinates of any the same point P referred to the first and second systems respec- tively. Then the algebraic distance of P firom the plane of ys, measured parallel to the axis of x, is equal to its algebraic dis- tance from the plane of y's', together with the algebraic distance of that plane from the pliane oiyz. But these distances are x, x', a respectively. Hence as = cc' + a, and similarly y=y' + ^, z = z' + y, are the formula required. 116. Since the formulae thus obtained involve three arbitrary constants, we can generally by this transformation make the co- efficients of three terms in the resulting equation vanish, but as ■the coefficients of the terms of highest dimensions are unaltered, none of the three terms, so eliminated, can be of the same dimen- sions as the degi-ee of the equation. Thus, in an equation of the second degree, we can generally destroy the terms of one dimen- sion in X, y, and z; in an equation of the third degree, three of the tenns of two dimensions, and so on with equations of higher orders. If, however, the terras, whose coefficients we destroy, 76 TEANSFORMATION OP CO-OKDINATES. differ by more than one dimension from the degree of the equa- tion, the equations for determining a, 0, 7 in order to effect this result, will rise to the second, third, or higher orders, but in a surface of the w* degree, the coefficients of the terms of w — 1 dimensions in the transformed equation will involve a, 0, 7 only in the first power. For, if f{x, i/,z) = be the equation of the n*" degree, 'the transformed equation will be f{x' + a, y' + /3, a' + 7) = 0, or/(„,A,) + («f+y| + «'D+ and the coefficients of x"'\ x"'~'y', &c in the transformed equation will be each of which, since /(a, /3, 7) is a function of w dimensions, will be of the first degree only in a, j8, y. Hence, if we equate three of these quantities to zero, we obtain three equations of the first degree in a, jS, 7, which, if they be independent and consistent win give the point to which the origin must be transferred in order to destroy the three corresponding terms of the equation. 117. To transform from one system of co-ordinates to another system having the same origin, hoth systems being rectangular. Let Ox, Oy, Oz be the first system; Ox', Oy', Oz' the second; a^, S,, c, ; a^,l^,c^; a^,\,c^; the direction-cosines of Ox', Oy', Oz', referred to Ox, Oy, Oz; x, y, z; x, y', z' ; co- ordinates of the same point in the two systems. Then the algebraic distance of the point from the plane of yz vsx; but measured successively parallel to Ox', Oy', Oz', this same distance is a,x' + a^' + a^'. Hence X = a^x + a^y' + a^' , ^ and similarly, y = h^x' + b^y' + b^z, I (1) z = c^x' + c^y + c/, I the formula} required. TKANSFOEMATION OP CO-OEDINATES. . 77 • The nine constants introduced in these results are connected by six equations of condition, expressing that the two systems of co-ordinates are rectangular, for since Ox, Oy, Oz are mutually at right angles, we have the system of equations < + 6/ + c/=l, (A) and by reason of Ox', Oy , Oz' being also at right angles, the system «A + *A + ''i,C3=0, 1 The number of disposable constants in this transformation is therefore only three. The relations (A), (B) subsisting among the nine constants involved in these formulae may also be written K^K^K=\A (A') by considering Ox, Oy', Oz' the primary system of axes, in which case the direction-cosines of Ox, Oy, Oz, will be {a^, a^^a^, (&j, 5j, &j), (Cj, Cj, Cg). The equations (A') and (B') expressing the same facts as the equations (A) and (B), are of course deducibl& from those. Either systera may be obtained from the identical equation a' + / -1- s" = a;" -1- y'^ + s'^ substituting for x, y, z their equivalents given in equations (1), or similarly for x,y', z'. 118. The relations between these constants may also be ex- pressed in the following convenient form. From the equations a^a^ + J A + cfi^ = 0, «„«, + W + C3C, = 0, we obtain immediately «! _ ^1 ^ ^1 . h<^z-h^i ''^^^-('."2 o.,b,-a^\' 78 TRANSFOEMATION OF CO-ORDINATES. each member of these equations is therefore equal to «+V+0' = {« + J/ + c,') K' + v + V) - («,«3 + \h + oaY}^ by equations (A), (B). In a similar manner, we obtain a» h„ c„ = ±1, a. b, c. ±1, = + 1. By using the equations (B') in a similar manner,, we obtain *!C, - *,c, J3C, - 5jC, JjCj - b.,c^ ' which shews that the ambiguities in the three systems of equa- tions here obtained, must be taken all of the same sign. Any two of these three systems of equations may be taken as completely expressing the relations between the nine constants : the third system being immediately deducible from the other two. 119. Euler's formulce for transforming from one system of rectangular co-ordinates to another having the same wigin. There being in the formulae already obtained for this purpose, nine constants connected by six invariable relations ; it must be possible to obtain formulae to effect this transformation which shall involve only three constants. The three chosen by Euler for this purpose are (1) the angle which the intersection of the planes of xy and aiy' makes with the axis of x, (2) the angle made by the same straight line with the axis of x, {3) the angle between the planes of xy and a;'^'. Let Ox, Oy, Oz be the original, Ox', Oy, Oz' the transformed axes of co-ordinates ; Ox^ the intersection of the planes of ay, TRAKSPOKMATION OF CO-OEDINATES. 79 x'y: X 0x^ = 0, x'Ox^ = ^, s Oa' = -v^, which is the same as the angle between the planes of xy, xy'. The transformations may be effected by successive trans- formations, each in one plane, (1) through an angle 0, in the plane of xy, from Ox, Oy to Oa;„ Oy^; i (2) through an angle i/r, in the plane of y^s, from Oy,, Oz to Oy„Oz'; (3) through an angle ^, in the plane of y^x, from Ox^, Oy^ to Ox', Oy. The formulae for these transformations are, using the same suffix for any one of the co-ordinates as for the corresponding axis, x=x^ cos 6 — yi sin 6, ) y = x^siad + y^ cos 0, J y^=^y^cos^jr-z'smf,l a = y, sin T^r + s' cos •^j ) x^= X cos ^ — y' sin ^. y^— X sin ^ -ysin^, ] + y cos 1^. J 80 TRANSFORMATION OP CO-ORDINATES. from which we ohtain, by successive substitutions, x = x' (cos 6 cos ^ — sin 5 sin ^ cos i/r) — y' (cos ^ sin /r with the plane of xy, and whose trace on that plane makes an angle 6 with the axis of x, is (aj'cosS— yslni9cosi/r)° {as'sm6—y'cos,6ca&->^y_ (7+ysini^)' ^ + P - ? ' the equation of a curve of the second order, which will be a circle, if . cos" 5 sin"^ cos'i/r sin" 5 cos"i/rcos"^ sin'ilr -, cos •^ cos 6 sin 9 cos i/r cos sin and 5 72 — V. a Hence, cos i/r, cos 0, or sin must be equal to zero, and we obtain the systems of solutions, ,-r\ . „ cos'^ sin"^ 1 (I) cosV^ = 0, -^jr-+-jj-=--^: jj ^1 COs'ilr sin'i/r (II) cos6l = 0, ^=-^--7^' or cos i|^ = + (III) sin = 0, i, = S^-!i^ or cos ■yfr = ± Of these solutions, the first gives impossible values of 0, and the second or third impossible values of '^, as a > or < h. TEANSFOEMATION OP CO-OEDINATES. 83 Suppose a>h, the only possible circular sections are given sin 6 = 0, cos i/r = + . // 1 or there are two systems of circular sections made by planes parallel to the axis of x, and equally inclined to it at an angle cos" 123. Transformation from a system of rectangular co-ordi- nates to a system of oblique co-ordinates, having the same origin. If (ai&jCj), {a^i^c^), (aSa^aCj) be the direction-cosines of the second system, referred to the first, we shall have the equations, as in rectangular co-ordinates, X = a^x' + a^y + a^z', y = b^x + b^y + h^z, e = c^x + cj + c^z ; but the six equations of condition which in that case subsisted will now be reduced to the three V + K + < = 1. .< + K + < = 1. and we have six disposable constants remaining. 124. Transformation from one system of co-ordinaies to an- other having the same origin, both systems being oblique. Let Ox, Oy, Oz; Ox, Oy, Oz be the two systems; On, On, On" the normals respectively to yz, zx, and xy, and let nx denote the angle nOx, and so for the others. Then the distance of a point whose co-ordinates in the two systems are respectively {x, y, z), {x, y', z) from the plane of yz, is x cos n/x, and is also a;' cos nx + y cos ny + z cos nz. G2 84 TRANSFORMATION OF CO-ORDINATES. Whence x cos nx = x' cos nx' + y cos ny + z' cos nz', and similarly, y cos w'.y = as' cos n'x + y cos ny + «' cos w'»', z cos w"s = aj' cos n'x + y cos n'y + z cos w"»', the required formulae, involving in this form twelve constants, but as they may be written in the form X = a^x + aj + a^', y = \x + \y + &8«', z = c^x + c^y' + Cjs', COS W3C where a, = and similarly for the others, we see that ' cos nx ' really only nine constants are involved, and these are connected by three equations on account of the angles between the original axes being fixed, so that there are still six disposable constants only. 125. Transformatwn from any one system of axes to any other. If we wish in any of the above transformations of the direc- tions of the axes also to remove the origin, we may first remove the origin to the point (a, /S, 7), retaining the directions of the axes. This will give x = x^ + a, y = y^ + 0, z = z^ + y, a;,, y,, gj being the co-ordinates of a point {x, y, z) referred to the system of axes through the new origin parallel to the primary system. Now changing the direction by transformations of the form x^ = a^x' + a^y' + a^z', &c., we see that the most general transformation possible is obtained by formulae of the form x = a + ffljOj' + a^y' + a/, y=^+b,x' + \y' + h/, z = y + c^x' + c^y + c^' . TEANSFOEMATION^OF CO-ORDINATES. 85 126. To shew that the degree of an equation cannot he changed hy transformation of co-ordinates. We can now prove the important proposition, that the degree of an equation cannot he altered by any transformation of co-ordinates : the degree of an equation meaning the greatest number which can be obtained by adding the indices of the co- ordinates involved in any term. For let Ad'y^z" be a term in an equation of the w"" degree, such that p + q + r = n: this will be a type of all the terms of the w* degree involved in the equa- tion, any one of which may be obtained by assigning to A,p, q, r suitable values. Now on any transformation this term becomes 4 (a + a,x'+ aj+ a^zj (^ + \x'+b,y'+b/y (7 + c,x'+ c,y'+ c,z'Y, and no term in this product rises beyond the degree p + q + r or n. Hence the degree of an equation cannot be raised by transformation of co-ordinates. Nor can it be depressed, for if by any transformation the degree be depressed, then on retrans- formation, the degree of the equation so depressed would be raised to its original value, which we have seen to be im- possible. 127. To transform from rectangular to polar co-ordinates. In the cases in which polar co-ordinates are required to be used, we may first transform the axes so that the axis of z is parallel to the line from which is measured, and the plane of sx parallel to the plane from which be taken for the three- plane system, and I, m, n be the sines of the angles which the 86 TEANSFOEMATION OP CO-ORDINATES, edges DA, DB, DG make with the planes DBO, DCA, DAB respectively, we shall hare a = lx, /8 = my, i P^ PJ for the formnlse of transformation. 129. To transform from one four-point co-ordinate system to another. If the equations of the fundamental points of the second system, referred to the first, be h. + m^ + wy + rS = 0, &c., and a, )8, 7, 8 ; a', /S', 7', S', be the co-ordinates of any plane in the two systems, «- l + rn + n + r ' ^•' (^^- »^^ from which equation the formulae required for transformation can be deduced. VIII. 1. If the expression ax'+ by'+ c«'+ 2ai/z + 2h'zx + 2c'xy become by transformation of co-ordinates, 0(8'+ y8/+ ya* + 2ay« + 2^«a5 + 2/«y, shew that a+h + c = a + P + y, and bc + ca + ah- a"- V- c"= ^y + ya + a^- a"- ^'- y", both systems of axes being rectangular. 2. If (^iWjW,), (Ijn^^, (}/n^n^ be the direction-cosines of a sys- tem of rectangular axes, and if =--(- — -1- — = 0, and rr+ — -t- — = 0, then will •=-^- — -1 — = 0, and a -.h : c :: l.ll, : m.mjm. : n.n.n,. l^ m, n, ' " 18 3 18 3 3. If al'+ hm'+ cn'= = al^+ hm'+ cn^ = C1I3 + bm^+ m^, shew that l^-m^ : ?/- V : li-m^ :: m^-n^ : m^-n^ : m^-ni, and that h ('»>«'3+ %»»2) + ^8 ("^aW,-!- niiw,) + Z, {m,Wj+ nijW,) = 0. PROBLEMS. 87 4. Transfoi-m the equation y» + «a; + ajy = a', referred to rectan- gular axes, to an equation referred to another system, one of which , makes equal angles with the original axes. 5. Shew that, by the same transformation as in the last problem, the equation x' + y' + z" + yz + zx + xy = a' is reduced to the form lx^ + y' + ^=2a\ 6. The equations of the straight lines bisecting the angles be- tween the straight lines given by the equations lx + my + nz = Q, ax'+2hxy + c^=(i, are Ix + my + Ji« = 0, a^ {ard - 6 («"+ V)} + xy {a {m'+ n'J-c («.'+ 1')} - y' {cmn -b('nv'+n')} = 0. 7. The equations of the straight lines bisecting the angles be- Jtween the straight lines given by the equations Ix + my +nz=0, ax" + by" + cz" = 0, are lx + my + ms = 0, x' {b — c)+y'{c — a) + !^ (a-b) = ^{l'(b-c) + m'{c-a)-n'{a-b)} + ^{-l'(b-c\ + m'(c-a) + n'(a-b)} mn I \ ' ^ ' + ^{l''{b-c)-m"{c-a) + n'{a-b)}. 8. Shew, by transformation of four-point co-ordinates, that the center of gravity of a tetrahedron is also the center of gravity of the tetrahedron formed by joining the center of gravity of the faces. 9. Shew, by the same metbod, that the center of gravity of the surface of a tetrahedron is the center of the sphere inscribed in the tetrahedron formed by joining the centers of gravity of the faces. CHAPTER VIII. ON CERTAIN SURFACES OF THE SECOND ORDER. 130. Before we proceed to discuss the general equation of the second order, we think it advisable for the student to render himself familiar with some of the properties of the surfaces which are represented by the general equation. We shall therefore introduce him to the equations of these surfaces in their simplest forms, in which the axes of co-ordinates being in the direction of lines symmetrically situated with regard to these surfaces, the nature and properties of the surfaces will be more easily deduced. The student will thus be enabled more clearly to understand the methods adopted in the general equations. For this purpose we shall give geometrical definitions of the surfaces, and deduce equations from those definitions : and we shall shew vice versd how from these equations the geometrical construction of those surfaces can be deduced. The Sphere. 131. To find the equMtion of a sphere. Def. a sphere is the locus of a point, whose distance from a fixed point is constant. The fixed point is the center and the constant distance the radius of the sphere. Let (a, b, c) be the center of the sphere, r the radius, {x, y, z) any point on the sphere ; ... (^x-aY+{y-hY + {z-cY = r'. This equation may be written in the general form a? + f + s' + Ax+By-\-Cz + D = Q), the equation required. 132. Since the general equation of the sphere contains four arbitrary constants, the sphere may be made to satisfy four spe- cific conditions. It may be seen from geometrical considerations that, when four conditions are given, there may be only one sphere, or a •ON CERTAIN SURFACES OP THE SECOND ORDER. 89 • limited number, or an infinite number of spheres, which satisfy the equations; at the same time the four conditions must be consistent with the nature of a sphere, and if this be the case and the conditions be independent, there must be a limited number of spheres satisfying those conditions. For example, if four points be given through which a sphere is to pass, no three points can lie in one straight line, and if four points lie in one plane, they must also lie in a circle, otherwise no sphere could pass through them, and if such a condition is satisfied, an infinite nuniber of spheres can be constructed each of which contains the circle in which the four points lie ; if the four points do not lie in a plane so that the four conditions to be satisfied are inde- pendent, the sphere is completely determined. Again, if four planes be given, each of which is to be touched by the sphere, no three of these must have one line of intersec- tion, and the four cannot pass through one point, except under a condition, and in that case an infinite number of spheres can be drawn, touching the four planes. In other cases, eight spheres can be drawn satisfying the conditions. Equation of a sphere under specific conditions. 133. To find the equation of a sphere passing through a given point. Let (a, b, c) be the given point, and the equation of the sphere x^ + y^+z' + Ax-\-By-\-Cz + D = 0; .■..a^ + V+c^+Aa + Bl+Cc + D = (i; ... ai' + f + z^ + A{x-a)+B[y-l) + G{z-c) = a^+W+e is the equation required. If the point be the origin, the equation becomes a?-\-y^ + s' + Ax + By+Cz = Q, and the sphere may satisfy three more conditions. 134. To find the equation of a sphere which passes through two given points in the axis of z. 90 ON CERTAIN SURFACES OF THE SECOND ORDER. Let Cj, Cj be the distances of the given points from 0; when x = 0, y = Q, the equation must become {z — cj (a — Cj) = ; therefore the equation of the sphere is £8= + /+ (a-cj (a-c,)+^ar + % = 0. If the sphere touches the axis of z, c^ = c^ = y,* a? + y^ + z'' + Ax + By - ^ryz +>/ = 0. 135. To find the equations of sfpheres which touch the three axes. Let the equation of the sphere be se' + f + z'' + Ax + By+Cz + JD = 0. Since it touches the axis of x, let a be the distance from the origin ; therefore when t/ = 0, a = 0, cc' + Ax+I) = 0, the roots of which are each equal to + a ; .-. A = ± 2a and JD = a\ m Similarly, y" + By + o" is a complete square ; .-. B=± 2a and C=±2a, and the equations of the spheres which satisfy the given con- ditions are a;" + / + a" + 2ax ± 2ay ± 2az + a" = 0, which are eight in number for any given value of a, corres- ponding to the different compartments of the co-ordinate planes. 136. To find the equation of a sphere touching the plane of xy in a given point. Since the sphere meets the plane of xy in the given point (a, h, 0) only, when a = 0, the equation must reduce to {x-ay+{j,-lY = Q. Therefore, the -equation of the sphere is {x - a)=+ iy-iy + z^+Gz = 0. * The symbol E> denoting identity, is employed by continental writers, because it is frequently coavenient to distinguish between identity and equality. ON CERTAIN SURFACES OF THE SECOND ORDER. 91 137. Interpretation of the expression {x-a)'+{y-bY+{s-cy-r'' in the equation of a sphere. Let the equation of the sphere be {x-ay+{y-Vf+{z-cY-r'=(i, and {x', y\ z) be any point Q, C the center of the sphere, and let a straight line through Q intersect the sphere in the points P, P', and its equations be x — x' y — y' z — z' I TO n therefore at the points P and P' {lp + x' - af + {mp + y' -ly + {np + z' -cf -r^ = Q, if Pj, Pj, be the roots of this equation, ftp,= (a,'-ar+(y-J)«+(«'-cr-r»; therefore the left side of the equation for any point {x, y, z') is QP. QF, ox -QP. QF, according as Q is without or within the sphere. If Q be without the sphere it is the square of a tangent drawn from Q to the sphere. If Q be within, it is the square of the radius of the smaji-. circle on the sphere whose center is Q. f Cor. All tangents drawn from an external point tq the'' sphere are equal. On the Relations of two or more Spheres. 138. To find the'equation of the radical plane of two spheres. Def. The radical plane of two spheres is the locus of points, through which if lines be drawn intersecting the spheres, the product of the distances of the points of intersection from these points, measured in one direction, is the same for the two spheres. Let the equations of the two spheres (A) and (B) be (x-ay + {y-by+ (z-cy -r' = u = 0, and {x - a'y+ {y -b'y+{2- c'y - r" = u'=0. 92 ON CEETAIN SURFACES OF THE SECOND ORDEU. The equation of the radical plane is therefore u — u' = 0. 139. To shew that the six radical plunes of four spheres inter- sect in one point. Let M = 0, m' = 0, m" = 0, m"'=0 be the equations, in this form, of the four spheres. The equations of the six radical planes are given by » t II III u = u =u =u , which intersect in one point determined by these equations. Def. The point of intersection of the six radical planes is called the radical center of the four spheres. Poles of Similitude. 140. Def. If a point be found such that the tangents drawn to two spheres are proportional to the radii of the spheres, such points are called Poles of Similitude. If the Pole of Similitude be in the line joining the centers of the spheres produced, such a pole is called the External Pole of Similitude. If the pole of similitude be in the line joining the centers it is called the Internal Pole of Similitude. If the spheres intersect the internal pole in a point through which if chords be drawn to each other, the rectangles under the segments are proportional to the squares of the radii. Both poles of similitude are the vertices of the cones which envelope both spheres, if they do not intersect ; if they intersect, this is true of the external pole alone. Properties of the Poles of Similitude of four Spheres. 141. A sphere may be defined as the envelope of planes which are equidistant from" the center; and if four-point co- ordinates be employed, the relation between the co-ordinate of such planes, which expresses this fact, is called the equation of the sphere. The equations of the four spheres whose centers are the fundamental points A, B, G, D, and radii r^, r^, r^, r^, are /3 = r , 7 = r,, S = r. ON CEETAIN SURFACES OP THE SECOND OEDEE. 93 The external and internal poles of similitude of the spheres (A) and (B) have equations a. R - + — = 0, and similarly for the rest. (1) The external poles of (AB), (AC), and (AD) lie in a plane whose co-ordinates are connected by the equations a _ j3_ 7_ S which evidently contains also the external poles of (SC), {CD) and {BA). (2) The co-ordinates of the plane containing the external poles of {AB) and {A G) and the internal pole of AD satisfy the equations a_ j8 _ 7 _ S T T T T ' 1 '2 ' S i and the same plane evidently contains the external pole of (BC) and the internal poles oi{BD) and {CD). (3) The co-ordinates of the planes containing the external poles of {AB) and the internal of {A G) and {AD) satisfy the equations a _/3 _ 7 _ S '•i~»'i!~ '•s" '•4' and this plane evidently contains the external pole of {CD) and the internal poles of {BG) and {BD). Hence one plane contains the six external poles, four planes contain each three external and three internal poles, and three contain each two external and four internal poles. The poles of similitude lie in eight planes, each of which pass through six poles of similitude situated three and three in four straight lines. The planes are called planes of similitude. Thus for the six external poles a _j8_ 7 _ S , /3 7 a 7 /a /S and = — "" »•„ r„ r, r, [t^y-"-' 94 ON CERTAIN SURFACES OP THE SECOND ORDER. therefore the pole of (BC) lies in the line joining those of (AB) and {AC). Similarly it lies in the lines joining those of (BB) and {DO). Hence, the six external poles lie in the sides of a plane quad- rilateral, as in the figure. Cylindrical Surfaces. 142. It has been seen that the locus of an equation F{x, y) = 0, which involves only two of the co-ordinates, is a cylindrical surface, of which the generating lines are parallel to the axis of the co-ordinates omitted. We shall now shew how to obtain the equation of certain cylindrical surfaces in which the generating lines are in a general direction. 143. To find the equation of the cylindrical surface, whose generating lines are in a given direction and guiding curve an ellipse traxxd on the plane ofzx. Let the equations of the guiding ellipse he ^ + ^ = 1, ands = 0, (1), and il, m, n) the directions of the generating lines. Let the equations of any generating line be nx=lz + a\ ny: At the point of intersection of the generating line with the guiding curve, the values of x, y, a in (1) and (2) being the same, we obtain as a general equation, after eliminating x, y and z, C + S = «', (3), ON CERTAIN SURFACES OP THE SECOND ORDER. 95 and since this is true for all positions of the generating line, eliminating a, j8 between (2) and (3), {nx-hy {ny-mzf _ ^, a' + i>' ~ ' is true for every point in the cylindrical surface, and is therefore its equation. Conical Surfaces. 144. Def. a conical surface is a surface generated by a straight line which constantly passes through a given point, called the vertex, and is subject to any other condition. 145. To find the, equations of a conical surface, whose vertex is the origin, and of which a guiding curve is an ellipse, whose center is in the axis of z, and plane parallel to the plane ofxy. Let the equations of the guiding ellipse be a? if' -2 + 1^ = 1, and z = c, (1), those of a generating line in any position, x = az, y=^z. (2). Eliminating x, y, z the point in which the generating line meets the guiding curve, 1 Since this equation is true for every position of the generan^^y line, eliminating a, ^ from (2) and (3), d'^V~&' which is the required equation of the surface. 146. To find the equation of an oblique circular cone. Let the equations of the guiding circle be x' + f^c', z = 0, (1), and the co-ordinates of the vertex be a,0,b; and let the equations of the generating line in any position be x — a = a.{z — b)\ $Vf-- in 2, = y3(.-J)j ^^^' 96 ON CERTAIN SURFACES OP THE SECOND ORDER. eliminating x, y, z the co-ordinates of the points in which the lines (1) and (2) intersect, {a-ahf + ^V = c\ (3), and this being true for every position of the generating line, we obtain from (2) and (3) {a{z-h)-h{x- a)Y + hy = c\z- hf; or, [az -hxf-\- by =^c'{z- hf, the equation of the oblique circular cone. 147. To find the circular sections of the oblique circular cone. The equation of the cone {az-lxY + by=c^{z-by may be written in the form 6" (ar* + / + a' - c') = a [2abx + {V +^-a^)z- 2&c''}. If a section be made by a plane whose equation \a z = k, the points in the curve of intersection satisfy the equation F {a? + f + z'- c') = k [2abx + (Z." + c'-a')z- '2bc'}, which shews that the curve lies on a sphere (Art. 131), and is therefore a circular section. If a section be made by a plane whose equation is 2abx + {V + c" - a") » - ibc^ =/, the points in the curve of intersection satisfy the equations V{a? + y^+B'- the variable ellipse remains always similar to a given ellipse, which is the trace on the plane of xi/. The surface may therefore be generated by the motion of a variable ellipse, whose plane, &c. (See Def.) The Hyperholoid of one Sheet, 152. To find the equation of the hyperholoid of one sheet. Def. The hyperholoid of one sheet may be generated by the motion of a variable ellipse, which moves so that its plane is always parallel to a fixed plane, and which changes its form so that its vertices always lie in two hyperbolas traced on perpendicular planes, to which its plane is perpendicular, these hyperbolas having a common conjugate axis. H2 100 THE HYPEEBOLOID OP ONE SHEET. Let A Q, BB be tlie hyperbolas traced on the two perpendi- cular planes taken for the planes of zx, yz, 00 their common semi-conjugate axis, being the direction of the axis of z. Let QPB be the variable ellipse in any position, P any point {x, y, z) in it, QN, BN its semi-axes. Draw Pif perpendicular to QN. Then, MN=x, PM=y, ON=z, and QN^'^BN'' 1; also, since Q, B are points in the hyperbolas, OA = a, OB=b, and 00=0, if or. Q2P_^ BN^ 2 ~t" 7 2 2 — > a c which is the equation of the hyperboloid of one sheet. 153. To construct the surface which is the locus of the equation, ^2+|j2 g2-l. THE HYPERBOLOID OP ONE SHEET. 101 Let the surface be cut by a plane whose equation is » = 7, then the projection of the curve of intersection upon the plane of asy has for its equation, which is the equation of an ellipse, whose semi-axes a, /3 are given by the equations, therefore, the vertices of the ellipse lie respectively on the hyper- bolas, which are the traces of the surface on the planes of zx, yz. a B . Also since - = t > tliis ellipse is always similar to the ellipse which is the trace of the surface on the plane of xy. Hence, the locus may be generated by the motion of a variable ellipse which moves, &c. (See Def.) 154. If the surface be cut by a plane parallel to the plane of zx, whose equation is a; = a, the curve of intersection will be an hyperbola, the equation of whose projection on the plane of yz, will be If a < a, the semi-axes /3, 7 will satisfy the equation T2 — 2 — 2 • c a - Hence, the extremities of the transverse axis 2jS will lie on the ellipse which is the trace on the plane of xy. as ^ 2 If a > a, we shall have to = -4 = -« — 1. c a Hence, the extremities of the transverse axis 27 will lie on the hyperbola, which is the trace on the plane of zx. 155. To find ike form of the surface at an infinite distance. If z be increased indefinitely, 102 THE HTPERBOLOID OF ONE SHEET. Let this surface and the hyperboloid be cut by a straight line drawn parallel to Oz through a point {x',i/', 0), and «,, a, be the corresponding values of z, a« ■*" i» - c" ^. 1- J2 p2 + 1 , 1 "g = 1, and «! — «2 = ■ C ' * ' z^ + z/ if oj' or y' or both, and therefore a, and «2> be indefinitely increased, 2!j — ^j diminishes indefinitely, and ultimately vanishes ; is the equation of an asymptotic surface, which lies farther from the plane of an/ than the hyperboloid. This asymptotic surface is a cone, for, if it be cut by any plane whose equation is - = - cos 0, all the points of intersection lie in the planes ^ = + - sin ^. The surface is therefore capable of being generated by a straight line which passes through the origin, and is guided by the ellipse whose equations are a^ If' -o + ij = l, and z = c. a b The figure shews the position of the conical asymptote rela- tive to the hyperboloid. ABab is the principal elliptic section, A'B'a'h', A"B"a"b" the sections of the hyperboloid and cone made by a plane parallel to that principal section, at a distance OG=c. t y<:^ -tT^.^ ^""""^'^^Zi'''^ r ^ "Cir lL - j-^^ THE HYPEEBOLOID OF TWO SHEETS. 103 The Hyperholoid of two Sheets. 156. To find the equation of the hyperholoid of two sheets. Dep. The hyperbolold of two sheets may be generated by the motion of a variable ellipse, which moves so that its plane is always parallel to a fixed plane, and which changes its form, so that its vertices lie always on two hyperbolas traced upon two perpendicular planes, having a common transverse axis, perpen- dicular to the fixed plane, to which the plane of the ellipse is parallel. J5/ Let A Q, AR be the hyperbolas traced on two pe planes, taken for the planes oizx, xy, and having the common semi- transverse axis OA, and let QPR be the variable ellipse in any position, whose axes are QM, BM, parallel to the plane oiyz. Take P any point {x, y, z) in the ellipse, and draw PN per- , pendicularto RM, then OM=x, MN=y, and NP^s; there- fore, since P is a point in the ellipse, n + RM'^ QM- = 1J and if a, c and a, I be the semi-axes of the two hyperbolas, A Q, AR, RM'_a? _ QM', " -wo -"a 1 — o 3 104 THE HYPEEBOLOID OP TWO SHEETS. which is the equation of the hyperboloid of two sheets. 157. To construct the locus of the surface whose egiuation is a» b' c" ~ Let the surface be cut by a plane whose equation is a; = a ; the equation of the projection on the plane of ys of the curve of intersection is which, if a > a, is the equation of an ellipse whose semi-axes ^, y are given by the equations therefore the vertices of the ellipse lie in two hyperbolas, whose equations are ^-^ = 1 and^-i'-l which are the traces of the surface on the planes of scy, zx, having .By. a common transverse axis in the line Ox : and since if = - . this c ' ellipse is always similar to a given ellipse, axes 2h, 2c. Hence, the locus may be constructed by the motion of a . variable ellipse which, &c. (See Def.) 158. If the surface be cut by a plane parallel to the plane of xy, whose equation is a = 7, the curve of intersection will be an hyperbola, the equations of whose projection on the plane oixy, will be THE HYPEEBOLOID OF TWO SHEETS. 105 which, may be written -2—^ = 1, whose transverse and conju- gate semi-axes will satisfy the equations Hence, the tranverse axis will have its extremities in the hyper- bola, which is the trace on the plane of zx, and the hyperbolic section will be similar to the trace on the plane of x^, 159. To find the form of the hyperhohid of two sheets at an infinite distance. x^ v^ z^ If x be increased indefinitely, the equation — = tj + -i + 1 shews that y, or z, or both, are also increased indefinitely, and the equation becomes 1 \ =-^ + "s ultimately. Let the hyperboloid, and the surface represented by this equation, be cut by a straight line parallel to the axis of x, drawn through the point (0, y', s'), a;,, a;,, the corresponding values of a;, are given by the equa,tions, a» ~ 6" "^ c» ' .-. ' ;■ ^ = 1, and a;. - a;. = — — - ; a' ' " ' ajj + a?! therefore, a;, - a;, diminishes indefinitely, and ultimately vanishes as y', or a', or both, increase indefinitely; hence the hyperboloid of two sheets continually approximates to the form of the surface whose equation is ^=|5 + J, which is therefore called an asymptotic surface. 106 THE ELLIPTIC PAEABOLOID. Also, if this surface be cut hy a plane whose equation is ^ = - cos 0, all the points of intersection lie in the two planes - = + - sin ; and the surface can therefore be generated by straight lines drawn through the origin, which intersect the ellipse, whose equations are p + -5 = 1, x = a. c This asymptotic surface is therefore a cone on an elliptic base, and lies nearer to the plane of yz than the hyperboloid, since a; ' > a;A Its position relative to the hyperboloid is shewn in the figure in which BC is the section made by a plane parallel to yz through the extremity of the transverse axis, ^nd DE, de are sections of the hyperboloid and conical asymptote, made by a plane parallel to yz. The Elliptic Paraboloid. 160. To find the equation of the elliptic paraboloid. Def. The elliptic parp.boloid may be generated by the motion of a parabola, whose vertex lies in a parabola traced upon a fixed plane, to which its plane is always perpendicular, the axes of the two parabolas being parallel, and the concavities turned in the same direction. THE ELLIPTIC PAEABOLOID. 107 Let 03 0^ be the plane on which the fixed parabola 0^ is ti-aced, Ox the axis of OQ; QR the axis of the moveable parabola QP, P any point {x, y, s) in the parabola. Draw P^ perpendicular to QR, and QU, NM to Ox, then since P is a point in QP, if I, V be the latera recta oi OQ and QP, PN^=r.QN, and QTP = l.OU; .:^ + ^=OU+QN= 0M=- x, which is the equation of the elliptic paraboloid. 161. To construct the locus of the equation, Y + j, -X. Let the locus be cut by a plane, whose equation is y = ^8, the projection of the curve of intersection upon the plane of zx has for its equation, which represents a parabola whose axis is parallel to the axis of X, the co-ordinates of whose vertex are ^ , ;8, ; therefore the vertex of the parabolic section lies in the parabola whose equation is y* = Ix, which is the trace on the plane of a>y ; there- fore the locus may be constructed by the motion of a parabola, whose vertex, &c. (See Def.) lOS THE HYPERBOLIC PARABOLOID. The Hyperbolic Paraboloid. 162. To find the equation of the Hyperbolic Paraboloid. Dep. The Hyperbolic Paraboloid may be generated by the motion of a parabola, whose vertex lies in a parabola traced upon a fixed plane, to which its plane is perpendicular, the axes of the two parabolas being parallel, and the concavities turned in opposite directions. Let xOy be the fixed plane upon which the parabola is drawn. Ox the direction of the axis of the parabola : let QR be the axis of the moveable parabola QP, parallel to Ox, measured in the direction contrary to Ox. Draw PN perpendicular to QR, and Q U, NM to Ox ; then, if P be any point {x, y, z) in QP, OM—x, MN=y, and NP=z. Let I, V be the latera recta of OQ, QP; therefore, PIP =1' . QN, and QTP = l.OU, OJj^ PAT" and ^ - —■ =0U- QN= OM; I •• I l'~ ' which is the equation of the hyperbolic paraboloid. THE HYPERBOLIC PARABOLOID. 109 163. To construct the locus. of the equation 1 l'~^- Let the locus of the equation be cut by the plane, whose equation is y = /3: the projection of the curve of intersection upon the plane of aa; has for its equation, which represents a parabola, whose axis is measured in the direction contraiy to Ox, and the algebraical distance of whose vertex from the plane yOs is y ; therefore the section by the plane y = ^ is a parabola, whose latus rectum is Z' and the co- ordinates of whose vertex are, ~ , ^, 0; or, the vertex lies in a parabola traced upon the plane of xy, whose equation is y^ = Ix. Hence the locus may be generated by the motion of a para- bola, whose vertex, &c. (See Def.) 164. The locus may also be generated by the motion of a hyperbola ; for if it be cut by a plane parallel to that of ys on the positive side, whose equation is x =a, the equation of the projection of the curve of intersection on the plane of yz will be y — j; =a, whose transverse and conjugate semi-axes, ;8, 7, will satisfy the equations /S* = Za and 7* = Z'a, the extremi- ties of the transverse axis lie in the trace on the plane of asy, and the conjugate axis is equal to the double ordinate of the trace on the plane of zx corresponding to x — — a. If it be cut by a plane on the negative side oiyz, the section will be an hyperbola whose transverse axis is in the direction of Oz. 165. To find the form of the hyjperbolic paraboloid at an infinite distance. If y and z be indefinitely increased while x remains finite, l'x\ l=jy-+ -^)=Y ultimately ; 110 THE HYPERBOLIC PARABOLOID. and if these planes and the hyperbolic paraholoid be cut by a straight line parallel to Oy, drawn through a point {x', 0, z) y^, y^ the corresponding values oiy are given by the equations, l~ I" ^°^ 1-1'^°^' y.°-y.° = a;, or. ■■ ^ ' ' Ix' y, + y,- Therefore, if x' remain finite or small compared with y, or y^, y^ — y, diminishes as e' increases and ultimately vanishes ; and the two plafies whose equations are -^ = ± -777 give the form of the surface at an infinite distance for finite values of x, or for values of x which are small compared with y or e. These planes will not form an asymptotic surface, except for points at which x vanishes compared with y or «, since y^ — y^ will not ultimately vanish in that case, and similarly ioTz^-z^. THE HYPEEBOLIC PARABOLOID. Ill The figure is intended to sliew the position of the asymptotic planes with reference to the hyperbolic paraboloid. Ox is parallel to the axis of the generating parabola, of which OB is one position in the plane of ex. PAp, FAp' are opposite branches of a hyperbolic section per- pendicular to Ox, the asymptotes of which RGB!, rCr are sec- tions of the asymptotic surface, AA', the transverse axis, being parallel to Oy. LL', 11! are the traces on the plane of ys of both the para- boloid and its asymptotic surface. QBq^ is a branch of a hyperbolic section on the negative side of Ox, SC, sC's, the asymptotes are sections of the asymptotic surface, and the transverse axis BO' is parallel to Os. 166. To shew that the elliptic and hyperbolic paraboloid are particula/r cases of the ellipsoid, and the hyperboloid respectively. Q^ if z^ be the equation of an ellipsoid or hyperboloid, and remove the origin to the point (- a, 0, 0). The transformed equation is . 171. The projections of the generating lines upon the principal planes are tangents to the traces on those planes. The equation of the trace on the plane of zx is a' c^ ~ ' and that of the projection of a generating line on the same plane - = cos 6* + - sm a, GENERATJJjTG LINES. 117 and the points of intersection are given by the equation a' z -a + 1 - (cos 6^ + -sin &f = 0, C ^ ~ C g2 _ 2a or -5 cos' ^ + — cos ^ sin ^ + sin" 6 = 0, c which, giving coincident values of a, shews that the projection is a tangent to the trace upon the plane of zx. Similarly, the projection on the plane of xt/, and the trace on that plane, intersect in points given hy the equations ™2 ,.2 -C03 5+fsin^=l, -5 + 75 = 1, a a whence [- sin ^ — ^ cos j = 0. Hence the points of intersection coincide, or the projection is a tangent to the trace on the plane of xi/. 172, To shew that two generating lines of the same system do not intersect. The equations of two generating lines of the same system are - = COS » + - sm p, f = sm 5 + - cos 5 ; a ~ c c and - = cos 5'+ - sin 6', f = sin 6'+ - cos ff. a c c If the two lines meet, we have at the points of intersection, = cos 5 - cos 6' ±- (sin 5 - sin ff), c and = sin 5 - sin 5' + - (cos Q — cos &) ; c and the condition of intersection is (cos 6 - cos &Y + (sin 5 - sin Oy = ; which cannot he satisfied unless 6 = &. Hence, generating lines of the same system do not intersect. 118 GENERATING LINES. 173. To shew that generating lines of opposite systems inter- sect. The equations of two generating lines of opposite systems are - = cos p + - Sin a, f- = sm p + - cos p ; a ~ c c and - = coa6'+-Bmd', f = sin 5' + - cos 5'. a c ~ c If the two lines meet, we have on the points of intersection, = cos - cos 6' ±- (sin 6 + sin 6'), c ^ and = sin5 — sin^'T -(cos^ + cos^'), and the condition that they may intersect is CDS'* e - cos" 0' + sin= e - sin" 6' = 0, which being identically true, shews that any two generating lines of opposite systems intersect. 174, No straight line lies on an hyperhohid which does not helong to one of the two systems of generating lines. For, if possible, let a straight line (0) lie entirely on the hyperboloid, then since each system generates the whole hyper- boloid, (C) must meet an infinite number of straight lines of each system ; let two of these {A) and {B) of opposite systems intersect ( 0) in two different points, in which case a plane can be drawn intersecting the surface in three straight lines ; but the section of a surface of the second order by a plane must be a curve of the second degree, therefore no such line as {C) can eidst. 175. To shew that a hyperhohid may he generated hy the motion of a straight line, intersecting three fixed straight lines, which do not intersect. Since any generating line intersects aU the generating lines of the opposite system, let us take three fixed generating lines of the same system : these will therefore not intersect. If a GENEEATJJfG LINES. 119 straight line now be supposed to move in such a manner as always to intersect these three straight lines, it will trace out the hyperboloid of which they are genejrating lines. For, the three points in which the moving line meets the three, fixed lines are points of the hyperboloid, so that it meets the hyperboloid in three points, which is impossible, unless the straight line lies altogether upon the surface, since the equation determining the points of intersection of a straight line with a surface of the second order, being a quadratic equation, cannot be satisfied by more than two roots without being satisfied by an infinite number. The straight line, therefore, in its different positions, will trace out the hyperboloid. 176. To find the Iocms of the intersections of two generating lines of opjposite systems, drawn through the points in the principal elliptic section, whose eccentric angles differ hy -a constant angle. Let 6+ a, and — a, be the eccentric angles of the principal elliptic section, differing by a constant angle 2a. The equations of the generating lines of opposite systems are - = cos(0 + a)± -sin(^ + a), |=cos(5+a) + -cos(^ + a); c(i CO c and - = cos(0-a) + -sin(0-a), | = sin(5- a) + - cos (^-a). Q. o At the points of intersection. = cos 9 sma + - sin d cos a, .'. - = ± tan a. c c Also - = cos 5 cos a + - cos sin a = cos 6 sec a, a c ? = sin cos a + - sin ^ cos a = sin 5 sec a ; h ~ c a? . V^ 2 .-. ^ + f. = seca. Therefore the locus of the intersections of the two pajrs of opposite systems is the two elliptic sections, parallel to the 120 GENERATING LINES, plane of xy, which intersect the traces on the planes of zx, yz, at points whose eccentric angles are + a, and 6 is the eccentric angle of the ellipses at these points. 177. The geometrical construction of the locus may he made as follows; let PQ, PQ', andP'^, P'Q', he generating lines at points P and P in the principal elliptic section ; PT, P'T their projections on that principal plane will be tangents to that ellipse; therefore (Art. 171) QTQ' will be the line of inter- section of the planes containing the generating lines, and will be perpendicular to the principal plane. The co-ordinates of T are a cos 6 sec a and I sin 6 sec a, as is easily shewn by means of the eccentric angle of the ellipse, hence, ■ = cos" 6 sec' a + sin' 6 sec' a - 1 = tan' a, .-. QT= c tan cc = Q'T, or, the loci of Q, Q are elliptic sections parallel to the principal elliptic section, at distances c tan a from its plane. 178. The figure is meant to be a representation of the positions of sixteen generating lines of each system^ correspond- QENERATI|G LINES. 121 ing to eccentric angles differing \>j | . ABab is the principal elliptic section, A'B'a'h' and A"B"a"b" are the parallel elliptic sections which intersect the conjugate axis of the hyperholoid at its extremities 0', 0", the axes of which sections are in the ratio ^2 : 1 to the axes of the principal sections. The generating lines through the extremities of the axes Aa, Bb intersect these two ellipses at the extremities of Jheir latus rectums, as L', K\ and i", K\ and they are parallel to the asymptotes of the principal hyperbolic section through £6 : those through the extremities of the latus rectums as L, K, pass through the extremities of the axes of the two ellipses. 122 GENERATING LINES. The two ellipses AB'a! and A'B'd' are the loci of the inter- sections of opposite systems of generating lines drawn through the extremities of conjugate diameters of the principal elliptic section. The figure serves to represent that the intersection of gene- rating lines of opposite systems drawn through points in the principal elliptic section, whose eccentric angles differ by a constant angle, lie in an ellipse parallel to the principal plane. As for example, such pairs of generating lines as LB , PD, and BL', PP. 179. To find the generating lines of a hyperbolic paraboloid. The equation of the hyperbolic paraboloid, l l'~^' is satisfied by the values of x, y, z for every point in the line whose equations are ■^ + — =- m y^ A. 1.-^0: (2) ^jl^^lV- a *' ^^> whatever be the value of a. Therefore by giving a all values, we obtain two series of straight lines, all of which lie entirely in the surface ; these are the two systems of lines which are rectilinear generators of the paraboloid. The equation (1) shews that in the two systems all the gene- rators are parallel respectively to the two asymptotic planes, whose equations are i^+— =0 180. To shew that generating lines of a hyperholie paraboloid of the same system do not intersect; and that those of opposite systems do intersect. Let the equations of two generating Ijnes of the same system be s/l^ >JI'~'J1!' •Jl^'JV a ' GENEEATIlJp LINES. 123 If the two lines intersect these equations are simultaneous, „ Q therefore —jp- = 0, "which is impossible since a is not equal to j8. Hence they do not intersect. Changing the order of the signs in the amhiguities in the second set of equations, we have the equations of a line in the system opposite to that of the first. If then the straight lines intersect, oe. and the consistency of these equations proves that two generating straight lines will always intersect if of opposite systems. 181. It may be shown by the reasoning employed in Art. 174, that no straight line can lie on the paraboloid which does not belong to one of these systems, and, as in Art. 175, that the paraboloid may be generated by the motion of a straight line which inter- sects two fixed straight lines, and is parallel to a fixed plane ; also by a straight line which intersects three fixed straight li/t^^ which are themselves all parallel to the same plane. It appears also from the latter construction, that if ^nv straight line intersects three fixed straight lines, which are 3^/ parallel to the same plane, the intersecting straight line will in all its positions be parallel to another fixed plane. 182. To shew that the prelections of the generating lines on the principal planes, are tangents to the principal sections. Since the equations of the generating lines are y j-£ ^ X+-L = :^'^ ^/i^ ^iv~ <>ji" ^/i-^/r a • The equation of their projections on the plane of sx, is 2z _ V? _a_ 124 GENERATING LINES. V which, being of the form z = mx — — is the equation of a tan- gent to the parabola a" = — Z'a;, and similarly for the projection on the plane of xt/. 183. The figm-e is intended to represent the manner in which the hyperbolic paraboloid is capable of being generated by straight lines. HAK, H'J!K' are portions of the branches of a hyperbolic section made by a plane parallel to that of yz, catting Oz on the positive side; EGE', DCD' are the asymptotes. FBF', QB' Q' are portions of the branches of the hyperbolic section parallel to yz on the negative side of Ox. The two sections are so chosen that the generating lines through B, the extremity of the transverse axes of one section, pass through A, A, the extremities of the transverse axis of the other. 5, the corresponding equations are m' m' 1 - , the section being parallel to the greater real axis. III. The equation of the hyperboloid of two sheets is a' b' c'~ If J > c, the corresponding equations are ^-Y=-^-j=-^^,andm = 0, b^'^a' c' b' ^''^c" the section being parallel to the greater conjugate axis. 185. We may also determine the plane circular sections as follows. The equation of the central surface being ax^+h/' + cz'^l, we may write it in the form b {x' + y' + z') + {a-b) x'-{b-c) s»= 1, 128 GENERATING CIRCLES. or {s/{a — h)x + >/{b — c) z] {\J{a—'b)x — \/{b — c) «} If, therefore, a, b, c be in order of magnitude, the equation is satisfied by the points of intersection of the planes whose equa- tions are V(a — b)x± >J{b —c)z = k, and the sphere whose equation is b{a?-\-y^ + z')-\-k y{a - J) a; + V(6 -c) «} - 1 = 0. Hence, plane circular sections are parallel to the mean axis in the ellipsoid, to the greater transverse axis in the hyperboloid of one sheet, and to the greater conjugate axis in the hyperboloid of two sheets, and there are two systems of such plane sections, all having the same inclination to the principal planes containing these axes, in opposite directions. It is obvious that these are the only circular sections, since a plane not parallel to one of the axes, as that of y, being of the form Ix + my + ns = p, could not reduce the expression (a — S) ar* — (J — c) sf', to a linear form, for the points of intersection with the surface, which is requisite in order that they may lie upon a sphere. 186. Any two circular sections of opposite systems lie in one The equations of the planes of two circular sections of opposite systems are y{a-b)x-^J{b-c)^-^c][^/{a-b)x + ^J{b-c)z-^<■]==(i^, or, {a-b)a?-{h-c) z^-{k+J^{a-b) x-{k-k') ^{b-c) z+kk'=0. Hence, they intersect the surface in a sphere whose equation is I (a^+y' + s") _ 1 + (A + ^') sl{a-b) x+{k-k') s/{b-c) z-kk'=0. 187. To find the circular sections of the paraboloids. The equation of any paraboloid is ^ + - = 2a;. c GENERATING CIRCLES. 129 Let the equation of the cutting plane be Ix + my + nz =p, and the equations of a diameter be a'-'go _ y-yo _ g-go _.. — — — r J (ajj, y„, «J being the center of the circular section ; .*. iK + mfju + ni/ = 0, and lx^+my„ + nz^=p. At the points in which the diameter meets the surface, c therefore the equations ^^+ -^ — 2\ = 0, ' be m/i + nv + 1X = 0, are true for an infinite number of values of \ : /* : v; hm en I . iwi" + cw* 2 Qm? + en") + ^ ' ^Also ^ + — is constant = M, suppose ; .-. MK^ + (m- i) /i» + [M- i) »^ = 0. Hence, eliminating \, we obtain the following equation which is true for all values of the ratio /* : v, M{mfi + nvY + P [(M- i) /** + (M- ^ v j = ; = 0, M cannot = 0, since h and c are not infinite ; we must have, therefore, tw or w = : suppose in = 0, then M^ t > and Mn' + (M-l^l^^O; 1 P \ ■=■ = 0, and b and c must be of the same sign. c K 130 GENERATING CIRCLES. Hence, tte hyperbolic paraboloid has no circular section; and, for the elliptic paraboloid, r 1 n' , « - = r = 5 , and m = 0. c b — c Similarly, we have the system r 1 «i» T = - = 7 ,n = 0. c c — o One only of these is possible, namely, the first, if h> c. Hence, the circular section is parallel to the tangent at the vertex of the principal parabolic section which has the greater latus rectum. The co-ordinates of the center of a circular section are given by the equations (1), which may be written, 0- + V(&-c) ^ 2(J-c)Vc+i'\/i' The equations of the locus of the centers of all circular sections are s = ± 2 V{c (S - c)}, and «^ = 0. The elliptic paraboloid may therefore be generated by the motion of a variable circle parallel to either of two fixed planes, whose center remains on two corresponding lines, parallel to the axes of the principal parabolic sections. 188. We may also find the positions of the circular sections of the elliptic paraboloid as follows. Arrange the equation of the paraboloid in the form te' + y' + g' a;' ./I 1 and let the surface be cut by a plane whose equation is of one of the forms '-^V(^V^. the points of intersection lie on one of the spheres whose equa- tions are that is, the sections by those planes are circular. GENERATING CIRCLES. 131 Also, these are the only circular sections, for no plane, except a plane parallel to one of the axes as that of y, could, hy the combination of its equation with the equation of the paraholoid, reduce the expression -r — «° ( — t) *o ^ linear form for the points of intersection with the surface, which is necessary in order that they should lie on a sphere. 189. Any two circular sections of an elliptic paraboloid, of (ypposite systems, lie on one sphere. For, the equations of the planes of two circular sections of opposite systems are or {x-h) {x-h') -«» (I - l) + (Je-k') z a/(^-^) = 0. Hence, they intersect the paraboloid in two circles lying on the sphere whose equation is {x-h){x-h')-\-if-\-z'-'ihx+{h-h')z/J(^\=0. X. (1) The equations of the generating lines of the surface yz + xc + xy + a^ = 0, J drawn through the point ( 0, am, j , are ;<; (1 =fc »») = am, -y = ^ (mz + a). (2) At any point where the planes a; + y + « = ±a meet the sur&ce xy + yz + zx + a' - 0, the two generating lines of the surface are at right angles to each other. (3) If be the angle between the generating Unes of the hy- perboloid «fl "*" ta "*" *9 ~ > a c E2 132 PROBLEMS. which pass through a point at a distance r from the origin, and it p he the perpendicular from the origin upon the plane passing through them, siiew that 2 a6c cot <^ = p (r* - a" - 6° + c"). (4) The tangent of the angle between the generating lines of the surface a ' which pass through the point (a;„, y„, »J, is a — b (5) Generating lines of the hyperboloid as* y' «° are'drawn through points in the plane of xy, whose eccentric angles are a, /3 : shew that their points are given by the equations a + B , . a+B . a-B a-li' a cos — jp- sm —~- ± c sin —^ cos „ Z 'U J 2 Also, the shortest distance (S) between two of the same system is given by the equation 4sm"— 2^ sm'-^^ cos' -^ cos"— ^^ 8^ = 7? ■*■ 6^ "^ 7 (6) The eccentric angles of the points in which the principal hyperbolic sections are met by any generating line are complemen- tary, and that of the point in which it meets the principal elliptic section is equal to one of these. (7) In the hyperboloid x' a' " ¥ c' no three generating lines can be mutually at right angles, unless i i-i a''^b''~c" and, if this condition be satisfied, an infinite number of such systems exist. (8) The generating lines of the hyperboloid a' b' c. If the directrix be parallel to an imaginary axis the modulus and modular conies are real. If the directrix be parallel to the real axis the modulus is real, and the modular conies imaginary. III. For the hyperboloid of one sheet, write — c" for c' in the equation of the ellipsoid, when i> a. If the directrix is parallel to either axis the modulus and modular conies are real, the less modulus corresponding to the. focal hyperbola. 196. To find the relation between the moduli in central surfaces. Let e, e be the moduli corresponding to the two directions of the directrix. MODULAB GENERATION. 141 Then __ = ^_„and-^ cos' CO sin' o) a' — c' which is the general relation between the moduli. 197. To find the modular conies for non-central surfaces. I. For the elliptic paraboloid, in which e = cos oj. The equation (1) reduces to sin' (o{7/ + l3 cot'©)' + a' = - 2aa; + a' + /S" cot' to, w' z' and comparing it with the equation ^ H — = 2x, we shall have, ^, ri being the co-ordinates of a focus, ^ a' + /3'cot'a) „ ', ?= f^ , i;=(Scot'coso)< 1, Surface at first an hyperboloid of two sheets, j)assing through a cone, to an hyperboloid of one sheet, conjugate axis perpendicular to the directrix. Focal conic at first an hyperbola, transverse axis perpendicular to the directive axis, passing through the asymptotic limit, two straight lines to an hyperbola, transverse axis pa- rallel to the directive axis. e = 1, Surface an hyperbolic paraboloid. Focal conic a parabola. e> 1, Suppose an hyperboloid of one sheet, confi--^ gate axis parallel to the directrix, incluVy ing an hyperboloid of revolution. > Focal conic an ellipse, transverse axis parallel to the directive axis. If o) = the ellipsoid becomes an oblate spheroid. The pro- late is inadjnissible because the directrix cannot be parallel to the directing plane. The hyperboloid of revolution of two sheets is lost between e = 1 and e = cos w. Umbilical generation. 199. The locus of a point, the square of whose distance from a fixed point bears a constant ratio to the rectangle under 1*4 UMBILICAL GENERATION. the perpendicular distances from two directing planes, is a surface of the second degree. This locus contains ten disposable constants ; three dependent on the position of the fixed point, three on the position of each of the directing planes, and one more, namely, the constant ratio. The fixed point will, therefore, not generally be unique, but may be on any point of a curve locus. The fixed point is called an umbilical focus, the intersection pf the planes a directrix^ and the constant ratio the umbilical modulus. 200. To find the locus of a point, the square of whose dis- tance from a focus is in a constant ratio to the rectangle under the distances from two fixed directing planes. Let the foc^s 8 be taken for the origin, the planes bisecting the angles between the directing planes being parallel to the planes of xy, yz. Let also m be the inclination of the directing planes to the plane of xy, a, 7 the co-ordinates of any point in the directrix, and e the constant ratio. From any point P, let PQ, PR be drawn perpendicular on the directing planes ; .-. SP'^ePQ.PB, the equations of the directing planes will be [x — a) sin w ± (a — 7) cos , ^ 7ecos"fi) P = ;— 5— and c = — — ^ 5—, ^ 1—e sm* 0) 1 + e cos" w and the equation of the surface, referred to its center, is . , \ » /, 2 \ 2, 2 1—e sin" ft) ^2 1 + e cos" ft) ^ (1 - e sm" ft)) a^+ (1+e cos" o,) «"+/= -^-^^ f- ,eos"ft) ^- «" m" a" Comparing this with the equation -a+^ + -a=l, we obtain a" (1 - e sin" ft)) = i" = c" (1 + e cos" w) ; 146 UMBILICAL GENERATION. and g £_ _ 1 the equation of the umbilical focal conic which is confocal with the section by its plane, and passes through the foci of the other sections, and also the umbilici. Again. J' = ^ + a = f(iH.^^)=-^,^, the equation of the umbilical dirigent conic, which Is obviously, the polar reciprocal of the focal conic with respect to the principal section in the same plane. 202. In the case of the cone, a and y vanish, and. the equa- tion becomes (1 - e sin'' CO) d' + f+ (1 +e cos'w) z^ = 0, and comparing this with the equation e sm 0) = 1 + -5 , and e cos w = -^ — 1 ; and the dirigent passes through the focus which is at the vertex. 203. To find the umbilical focal and dirigent conies in the case of non-central surfaces. If e sin" (0 = 1, the equation (1) becomes y' + es" 4- 2aa; — 2e cos" « . 7s = a' — ey" cos" w. Comparing this equation with the equation |- + - = 2x, b c UMBILICAL GENEEATION. 147 if ^, f be the co-ordinates of the focus, o- = —b, e = - , and sin" w = -^ . c _^lso e ^ 67° cos" 6) sin" ft) -g" _ 5°-y'cos'' S* = -cos''ft).26(^--) = -2(J-c)(f-|), the abscissa of whose focus is - ■ — ^ =~, therefore the umbili- ^ Jl u Z cal focal conic is a parabola, confocal with the parabolic section in whose plane it lies, and having its vertex in the focus of the other principal section. If f ', f ' be the co-ordinates of the directrix, O — C I which is the reciprocal of the focal curve with respect to the section z" = 2cx. 204. Surfaces capable of generation hy the umbilical method. With a real focus and directrix the only surfaces which can be generated are the ellipsoid, the hyperboloid of two sheets, and the elliptic paraboloid, which' is the limit of both ; also the corre- sponding particular cases of these surfaces, viz. a cone, the limit of the hyperboloid of two sheets, a point or evanescent ellipsoid, and an infinitely slender cylinder or evanescent elliptic para- boloid. The surfaces of revolution capable of being generated by this method are the prolate spheroid and the hyperboloid of revolution of two sheets. Def. a surface of the second degree shall in future be de- nominated a Oonicoid. l2 148 PROPERTIES OP CONICOIDS. Properties ofcontooids deduced iy the modular and umbilical methods of generation. 205. If a section, of a conicoid he made hy a plane contain^ ing two directrices, the sum or difference of the distances of any point of the section from the corresponding foci is constant. Let P be any point of the section whose plane contains the directrices QD, Q'D', and let F, F' be the corresponding foci. Draw QPQ perpendicular to the directrices and DPD' paral- lel to a directive plane. Then, since the modulus is the same for both foci, FP : PD :: F'P : PD'; .-. FP : F'P :: PB : PD' :: PQ -.•PQ', and FP ± F'P : PQ ± PQ :: FP : PQ. Now, PQ + PQ' or PQ ~ PQ' is constant, according as P is or is not between the directrices, and FP : PQ is constant, since PQ : PD is so ; .-. FP ± F'P is constant. 206. If a straight line he drawn through any point in a directrix intersecting a conicoid in any two points, the line joining the corresponding focus with the point in the directrix hisects the angle hetween the focal distances of the points of intersection, or the supplement of that angle. Let the focus F correspond to the directrix DQ, and QPP' intersect the surface m P, P; then, FP: PQ :: FP: PQ; .-. FP : FP :: QP : QP, which proves the proposition. Cor. If QP be a tangent to the surface, QFP is a right angle. 207. ^ a straight line touching a conicoid meet two parallel directrices, it makes equal angles with the lines drawn from the point of contact to the corresponding foci. For, if P be the point of contact, Q, Q the points in which PROPERTIES OP CONICOIDS. 149 the tangent meets the directrices, F, F' the corresponding foci, since the modulus is the same for both foci, FP : PQ :: F'P : PQ'. Also the angles PFQ, PF' Q' are right angles, therefore the triangles are similar, and the angles QPF, Q'PF' are ecLual. 208. ^ a cone, having its vertex in any directrix, envelope a conicoid, the plane of contact passes through the corresponding focus, and is perpendicular to the line Joining it with the vertex. For if V be the vertex, and F the focus, VP any side of the cone touching the surfaces in P, PFV is a right angle. Hence, the locus of P which is the plane of contact is a plane through i^perpendicular-to VF. 209. If the vertex of a cone be any point in a focal curve of a conicoid, and the iase be any plane section of the conicoid, the line joining the focus with the point in which the directrix meets the plane of section is an axis of the cone. Let the plane section cut the directrix in F, and FP be a tangent at P to the section, then FP is perpendicular to FF, which is therefore an axis. CoR. 1. The second plane of section of the cone intersects the corresponding directrix in the same point as the first plane. Cor. 2. If the first plane section passes through the directrix, the second will also pass through the directrix, and in this case, since there wiU be an infinite number of axes of the cone, it will be one of revolution. 210. ^ the vertex of an enveloping cone of a conicoid be a point on a focal curve of the surface, the cone is one of re- volution, and its internal axis is the tangent to the fbcal curve at the vertex. Let Fbe the vertex of the cone, VP, VP' the tangents to the trace of the surface on the plane of the focal curve, then PP' is a tangent to the dirigent conic at the foot of the corresponding directrix, (Art. 193) ; and since the plane of contact is perpendi- cular to the plane of the focal curve, it will contain the corre- sponding directrix, and therefore by Cor. 2 of the last article will be a cone of revolution. 150 PROPERTIES OF CONES OF THE SECOND DEGREE. Also, since the tangent at V is perpendicular to the directrix, and to the line joining Fand the foot of the directrix (Art. 193), it is perpendicular to the plane of circular section, and is the internal axis of the cone. Properties of Cones of the Second Degree. 211. The sines of the angles, which any side of a cone makes with a focal line and the corresponding dirigent plane, are in a constant ratio. Let a plane pass through any directrix DQ and the corre- sponding focus F, and let P be any point in the section of the cone made by this plane : V the vertex of the cone. Draw PR, PQ perpendicular to the dirigent plane and directrix. Then FP : PQ and PR : PQ, and therefore FP : PR are constant ratios ; and BF being perpendicular to VF, therefore VF is perpendicular to the plane of section, and PFV is a right angle. PF PR Hence, the ratio of the sines proposed is —^ : p= , and is therefore constant. 212. The product of the sines of the angles which any side of a cone makes with the directing planes is constant. If Fbe the vertex of a cone, Pany point on the cone, PL, PL' perpendicular on the directing planes through V, then by the umbilicar generation of the cone, (Art. 202), PV^ is propor- PL PL' tional to PL . PL'; or -^. -^ is constant, which is the property enunciated. 213. The tangent plane of a cone makes -equal angles with the planes through the side of contact and each of the focal lines. For, let the tangent QPQ' perpendicular to the side FP meet the dirigent planes in the points Q, Q', and take F, F' the foci corresponding to the directrices through Q, Q' ; then FQ is perpen- dicular to VF, and also to PF, and therefore to the plane VPF; also VP is perpendicular to P^and PQ, and therefore to FP; hence FPQ is the inclination of the planes VPF, VPQ, and being equal to F'PQ , the proposition is proved. RECIPROCAL CONES, 151 Cor. If a sphere be described having its center at the vertex, and meeting the focal lines in 8, H, and the side of the cone in P, the great circle touching the curve of intersection in P makes equal angles with the arcs SP, HP. 214. The sum of ike angles which a side of the cone makes with the focal lines is constant. The statement of this proposition amounts to saying that the sum of the arcs SP, HP, in the corollary of the preceding article, ia constant. This may be shewn immediately by limits, as in the case of the plane ellipse. Reciprocal Cones. 215. If a cone he constrviCted whose sides are perpendicular to the tangent planes of any given cone, the tangent planes will also he perpendicular to the sides of the given cone. For, let two tangent planes be drawn to a cone A, then two corresponding sides of the other cone B, perpendicular to those tangent planes, will be perpendicular to their line of intersection ; the line of intersection of the tangent planes to A is, therefore, perpendicular to the plane containing the corresponding sides oiB, Proceeding to the Hmit, the line of intersection becomes a side of the cone A, and the plane containing the sides of 5 a tangent plane to ^ultimately. Whence the truth of the propositipn.- From this reciprocal property the cones are called recipro- cal cones. If aa? + hy^ + cs" = be the equations of a cone, a c is that of the reciprocal cone. 216. The directing planes of any cone are perpendicular to the focal lines of the reciprocal cone. The directing planes of the cone aa? + hy^ + cs' = are in- clined to the plane of zx at angles whose tangents are ve-^)' 152 EECIPEOCAL CONES. and the equations of the focal lines of the reciprocal cone a? «" «' - ^ 2* /x — +% + -=0, are j--t — =0, a o c a—b b—c which, therefore, are perpendicular to the directing planes. Hence, the focal lines of a cone correspond to the directing planes of the reciprocal cone. * 217. The curves in which a sphere whose center is in the common vertex of reciprocal cones intersects the cones, are called reciprocal spherical conies. The reciprocal property connecting the two may be stated thus : " Every point of a spherical conic is the pole of a great circle which touches the reciprocal spherical conic." 218. If two lines correspond respectively to two planes per- pendicular to them, drawn through the common vertex of reci- procal cones, the plane which contains the two lines corresponds to the line of intersection of the corresponding planes. Hence, theorems relating to any cone have reciprocal theo- rems in the reciprocal cone. 219. The following examples of theorems and their reciprocal theorems will be sufficient to illustrate the method of derivation. Theorem. The intersections of a tangent plane to a cone with the directing planes make equal angles with the side of contact. Reciprocal Theorem. The planes containing a side of a cone, and the two focal lines, make equal angles with the tangent plane along the side of the cone. Theorem. The sum or difference of the angles, which a side of a cone makes with the focal lines, is constant. Reciprocal Theorem. The sum or difference of the angles, which a tangent plane to a cone makes with the directing planes, is constant. Theorem. The product of the sines of the angles, which any side of a cone makes with the directing planes, is constant. Reciprocal Theorem. The product of the sines of the angles, which any tangent plane of a cone makes with the focal lines, is constant. PEOBLEMS. 153 Such reciprocal theorems are easily translated into the cor- responding theorems for reciprocal spherical conies. XI. 1. • If two conicoids have a common focus S, and a common directrix, and if a tangent to one of the surfaces at P meet the other surface in Q, Q' and the directrix in B, SF will bisect the angle 2. If the surfaces in (1) be condirective, the angle QSQ' will be constant. 3. A diameter of constant length revolving in a given central surface describes a cone, a tangent plane to which cuts the surface in a curve of which one axis is the line of contact ; and the right lines in which the tangent plane cuts the directing planes make equal angles with the side of contact. 4. A spherical triangle has a given area, and two sides in a given direction, prove that its base touches a spherical conic, and is bisected by the point of contact. 5. If two planes be drawn through any point of a cone parallel to the directing planes, the side of the cone passing through the point makes equal angles with the tangents to their sections of the cone. State the reciprocal theorem. 6. A paraboloid is cut by any plane, prove that the cyKnder passing through the curve of section, and having its sides parallel to the axis of the paraboloid, will be condirective with the surface. 7. A cone and a conicoid are concentric and condirective, prove' that their curve of intersection lies in a sphere, and a cyhnder con- taining this curve and having its axis parallel to an axis of the conicoid, will be condirective with both. 8. K a conicoid be intersected by a sphere whose center lies in a principal plane, the cylinder containing their curve of inter- section, and whose axis is perpendicular to the given plane, wiU be condirective with the given surface. 9. Every sphere inscribed in a cone of revolution circumscribing an ellipsoid, will cut the ellipsoid in a plane curve. 10. When a series of ellipsoids are inscribed in a cone of revolu- tion, so as to touch it along the same curve, .., a_i3wia_ eu ''"U be constant for all. 154 PROBLEMS. 1 1. The, distance between a focus and the corresponding directrix of the section of an ellipsoid, made by the plane of contact with any enveloping cone of revolution, is constant. 12. If a sphere intersect an ellipsoid in two plane curves, the sphere and ellipsoid will have two common enveloping cones, whose vertices lie on opposite branches of the umbiL'cal focal curve. 13. Any point in the plane of a focal curve of an ellipsoid will be the focus of two plane sections perpendicular to that, which will be real only when the point lies within the trace on that plane and without the focal curve. Also, if the point lie on the focal curve, these planes will coincide and will contain the normal to the focal curve at that point. 14. The foci of a series of parallel sections of an ellipsoid perpen- dicular to the plane of a focal curve will lie on an ellipse which touches the trace on that plane, and the focal cui-ve. 15. The cone described with its vertex on one focal curve, and its base on the other, with respect to any conicoid, will be a cone of revolution. This cone will intersect the conicoid in two planes which both pass through the directrix corresponding to the vertex of the cone, and each through a fixed point in the common axis of the focal curves. 16. If a section of an ellipsoid be taken passing through a focus P, and the corresponding directrix, and if P' be the point on the trace of the surface such that the eccentric angles of P, P' in the focal curve and the trace respectively are equal; D, D' the ex- tremities of the diameters conjugate to these points, the eccentricity of the section is -^ . rr-= , being the center, a, p, the semi-axes of the focal curve, and a, b, of the trace of the surface. CHAPTER XI. DISCUSSION or THE aENEEAL EQUATION OP THE SECOND DEGREE. 220. OuE object in this chapter is to investigate the origin and axes of rectangular co-ordinates, to which, when referred, the equation of a surface represented by a proposed complete equa- tion of the second degree, will assume its simplest form, and to investigate the relations among the coefficients which discrimi- nate the various kinds of surfaces capable of being represented hj this equation. The extent of the simplification which may be effected by transformation of co-ordinates may be anticipated from the fol- lowing considerations. By a change of origin we introduce three arbitrary constants, which will enter only into the coefficients of the terms of one dimension in aa, y, and z, (Art. 116), and by which, unless a cer- tain relation between the coefficients subsists, these coefficients may be made to vanish. By a change of the direction of the axes, we introduce nine constants, connected by six equations, or three independent arbitrary constants, which will enter into all the coefficients of the transformed equation. We may determine these constants by assuming any three relations among the coefficients, provided such relations furnish equations independent of and compatible with the six necessary relations connecting the constants of trans- formation. The three relations which we shall choose are, that the coefficients of yz, zx, and xy in the transformed equation shall severally vanish ; and if these relations, together with the six connecting equations, furnish real values of the constants of transformation, the desired simplification will be effected. The method which we shall adopt requires a preliminary investigation, which is effected in the two following articles. 156 DISCUSSION OP THE GENEHAL EQUATION 221. To determine the condition necessary in order that the equation U=aa? + hf + cs' + Idyz + IlVzx + Idm/ = 0, may represent two real or imagirui/ry planes. If a be finite, the proposed equation is equivalent to [ax + dy + h'zf = (c" -ah)f+2 {h'c' - aa') yz + (6" - ca) s" = v, ox ax + c'y + b'z = + i^v. But, if the equation represent two planes, x must be capable of being expressed as a linear function of y and z in two ways, which can only happen when i; is a complete square with respect to y and z. The condition for this is {b'c - aa'Y = {c" - ab) {b" - ca), or a {abc + 2a'&'c' - aa^ - bV^ - cc'^) = 0, or aba + Idb'd - aa" - bV - cd^ = ; since a was assumed to be finite. The important fanction of the coeflScients of i[7 which forms the left-hand member of this equation, we shall, after Mr Salmon, denote by -ffC^/). 222. When the equation 17= represents two planes, to de- termine their line of intersection. The line of intersection of the two planes will manifestly lie on the plane ax + dy + b'z = 0, and by symmetry will also lie on the planes dx + by + a'z = 0, b'x + a'y + cz = 0. If we eliminate z from the first two of these equations, and X from the last two, we obtain the equations X {b'd -aa')=y {da' - bV) = z {a'V - cd), which are the equations of the line of intersection. It is obvious that H{U) is the eliminant of these three equations. 223. To determine the coefficients of the In-ansforrmd equation, when U= is referred to such a system of axes that the pro- ducts of the variables vanish. OP THE SECOND DEGREE. 157 Assume a, j8, y to be these coefEcients ; then since, by this transformation, U becomes and since the expression a;^ + ^^ + s" is unaltered by any trans- formation, the expression h(x' + y' + z')-U=V will become h{a?+f + s") - ax^ - /3/ - yz" = W, •whg,tever be the value of h. Hence, if we give h such a value that F shall be separable into linear factors, TTwill also, for the same value of h, be separable into linear factors. Therefore the equations H{r) = ih-a) {h-h) {h-c) -a"{h-a) _ h'^ (h-b)- c" (h-c)- 2a'b'c' = 0,' and S{W)={h-a.){h-^){h-y)=0, have the same roots. Hence, a, /8, 7 are the roots of the equation (A -a){h- 1) [h-c)- a'" {h-a)-h" (h-b)- ^Qi-c) -2c^b'c'=(i. .This equation is called the "Discriminating Cubic." We shall, for the present, assume what wUl hereafter be proved, (Art. 228), that its roots are always real ; and proceed to investi- gate the position of the axes to which the equation is now referred, with respect to the original axes. 224. To determine the equations of the axes, to which U must he referred, in order to assume the form aa!= + /3/ + 73=. When h=oi, W becomes (a — /8) ^ + (« — 7) «'> and the two planes represented by W= intersect in the new axis of x. Hence, the new axis of x is the line of intersection of the two planes represented by F= 0, with the same value of h. Its equations are therefore, referred to the original axes, X [Vd +«'(«- a)] = y [c'a' + 5' (a - b)] = z {a'V + c' (a - c)} ; and similarly, the equations of the new axes oiy and z are X [h'd + a' (/3- a)] =y {da! + V (/3- b)] = z [a'b' + d {fi- c)}, X [I'd + a' (7 - a)] = y [o'a + 6' (7 - J)} = ^ [a'V + c' (7 - c)}. 158 DISCUSSION OF THE GENEBAL EQUATION If two of the roots of the discriminating cubic, as a, j8, be equal, two of these systems of equations appear to become coin- cident. They are in reality indeterminate, as might be inferred from the circumstance that W, which in that case becomes will not be altered by any transformation of x and y in their own plane. This case, and the case of three equal roots, will be separately considered afterwards. 225. To find the direction cosines of the new aooes, referred to the old. It will be convenient here to denote (a-J)(a-c)-a", [a-c) {oL-a)-h'\ [pL-a){a.-h)-d\ by \, /A, V respectively, and Vc' + d{ai- a), da + 5' (a - 1), dV + c' (a - c), by X', /*', V respectively. We shall then have, if/(A) = be the discriminating cubic, /(a) =0, which may be written in any of the forms W' = /iV, yx/i' = v'X', vv = \V> X'" = fw, /i'" = vK, v^ =■ \/i. Also, X + fi + v=f'{a). Now, if I, in, n be the direction cosines of the new axis of x, l\' = m/i = nv ; .". Pfiv = mVA. = n\/i, r m' w° r+OT' + w' ^ 1 ^ 1 °'' \- ,jL-v~ X + ii + v~f'{ai) (a-/3)(a-7)" _ j^ (a-5)( a-c)-a" , (g-c) (a-a) -a" Hence, 1^ = -^ — X. , — v— , »» = ^-7 — a^ , x , n (g— g) (a — &) — c" (a-/3)(a-7) ' which may be written P^4-^ = 0,&c.,sinceXH-^ = 0. da da da Similarly for the direction cosines of the axes of y and z. 226. To find the conditions that the discriminating cubic may have two equal roots. If the equation /(A) = has two roots equal to a, we shall have /(a)=0 and/(a)=0. OP THE SECOND DEGREE. 159 These gi-ve us, with the same notation as in the last article, X + fi + v = 0, W'^fi'v', (1/1= v'X', w' = X'/i. Hence, we have Ai Lb V This equation can only be satisfied by the vanishing of two of the quantities \', /*', v. Assuming then X' = 0, /i' = 0, and substituting in/' (a) = for a — a and a - 5, we obtain, if a and b' be finite, c' (a— c) +a'b'=0. Hence if /(A) = have two -roots equal to a, we have the three equations a' {..-a)+b'c' = b' (a-b) + cV == c' (a - c) + a'6' = 0. If c' also be finite, this supplies the two conditions be' _. c'a a'b' a' V c' ' The equations of the new axes of x and y become in this case indeterminate, and since lit t I rii be , ca ab by the conditions already fulfilled, the equations of the new axis of z will be a'x=b'y = c'z. (Art. 224). If a, b', or c vanish, these conditions in this form become indeterminate; we will therefore obtain equivalent equations which shall remain finite in this case. Since each of the §qual quantities a , b — rr , o r > is equal to a, we have 11 r II ITI 00 ca , ab —r — a — a, -Yr='b — a, — r-=c — a, a b c and hence, a" = (b-a) (c-a), 5'==(c-a) (a-a), c"={a-a) {b-a) ... (1). From these equations we obtain 7.'2_V2 c"-a" , a"-b" + a= — : \-b = J— + c = a. >—c c—a a—b 160 DISCUSSION OP THE GENERAL EQUATION From equation (1) we see that if a' = 0, a = 5 or a = c, and consequently c' or J' = : if then d and V = 0, we have a = c and c'^ = (a — c) (5 — c). If a, h', and c' all vanish, we must have two of the three o— a, h — a, c — a zero, or two of the three a, b, c equal, which is otherwise obvious. 227. To find, the conditions that the discriminating cubic may have three equal roots. In this case, the transformed expression for TJ will be of the form but this is incapable of being affected by transformation, and must therefore have been originally in the same form. We shall there- fore have a = J = c, a' = b' = c' = 0. These conditions may of course be deduced from the cubic itself We shall, in fact, have the equations « + S + c = 3a, Jc - a'" + m - J'' + a5 - c'' = 3a', abc + 2db'd - ad^ - bb'^ - cc"^ = a». Hence, (a + i + c)" = Sa" = 3 (Jc - a" + ca- &''+ a5 - d"), or {a-bY+ (6-cr+ (c-a)'+ 6 (a" + 5" + c'^) =0, which, for real coefficients, necessitates the before-mentioned conditions. 228. To shew that the discriminating cubic has always real roots. We will first obtain a different form of the cubic, and one which is more convenient for its general discussion. . ^ , <' 1 1 ca , ab Lieth-a = p -, h-b = q rr, h - c = r - -j- . On substitution, the equation becomes be ca ab . ,,. u=n'^--^qr - -jrrp-— i?2' = (1), OF THE SECqjjTD DEGEEE. 161 or. 1=. a T, + - V A — a + V6 h-l + -T-7 + - ca 6 V A-C + dV or, a'6'c' .'(.-„+^) j-(*-»+^') .'(*-«+^) ac dV Now, assume that a , h— ^n- , and c r- are in de- a o c scending order of magnitude, and that a'b'c is positive. We may now trace the changes of sign of u for different values of h. They are exhibited in the following tables h cc a b- c — - b' a'b' p i r u + + + + + + - - + + - - - We thus see that, when db'c is positive, there is one value of h greater than the greatest of the three 00 , ca , ab a r ) — 77- , and c — » a c and one lying between each consecutive pair. Similarly, when a'b'c' is nega|;ive, we may shew that there is one root less than the least of the three, and one between each pair. There are then always three real roots. 229. To determine the surfaces wMcJi may be represented by the general equation of the second degree. ' We may now assume that any surface which can be repre- sented by an equation of the second degree may also be repre- sented by the particular form of this equation, 162 BISCUSSION OF THE GENERAL EQUATION and proceed to discuss the nature of these surfaces. I. If a, /8, 7 he all finite, we may by a change of origin, destroy the tenns involving the first powers of x, y, and z, and reduce the equation to the form It has already been seen that this equation represents an ellipsoid if ^ , ^ , ^ be all negative, an hyperboloid of one sheet if one of them be positive, and an hyperboloid of two sheets if two be positive. If all three be positive, the locus is im- possible. Also, if S vanish, the locus is either a cone, or a point, according as a, ^, 7 have not, or have all the same sign, II. If a = 0, and a" be finite, one of the co-ordinates of the center becomes infinite, and we cannot refer the surface to axes through its center. We may however determine the origin in this case, by making the constant term vanish, and the equation 80 transformed becomes iSy + 7a' + 2a"a; = 0, which represents an elliptic, or hyperbolic, paraboloid, according as /3, 7 have or have not the same sign, III. If a = and a" = 0, we can reduce the equation to the form ^/ + 7a' + S = 0, which represents an elliptic cylinder, if ^ , ^ be both negative, an hyperbolic cylinder if one be negative, an impossible locus if both be positive. If S vanish, the locus is two planes, or a straight line, according as /8, 7 have not, or have the same sign. One of the co-ordinates of the center is in this case inde- terminate, or, there is a straight line every point of which is a center. OF THE SECOgD DEGREE. 163 IVi If a = 0, 13 = 0, and a", /3" Be finite, we may, by a change of origin, reduce the equation to the form 7a''+2a"a3 + 2y3"y = 0, which, again, by a change of the direction of the axes of x and y in their own plane, may be further reduced to the form 7a=+SSa! = 0, and therefore represents a parabolic cylinder. In this case, there is a line of centers at an infinite distance. v. If a == 0, /3 = 0, a" = 0, /3" = 0, the equation, being a quadratic in s only, represents two parallel planes. 230. We have now shewn that the only real surfaces repre- sented by any equations of the second degree are the ellipsoid, hyperboloid of one or two sheets, including cones as a limiting case; elliptic and hyperbolic paraboloids; elliptic, hyperbolic, and parabolic cylinders ; «ind a pair of intersecting or parallel planes. We shall proceed to discuss the conditions to be satis- fied by the coefficients of the general equation of the second degree, in these several cases. These may be most conveni- ently investigated by considering the properties of the center, or centers, of each surface. We will first consider the case of cen- tral surfaces. 231. To mvesttffate under what conditions the general equa- tion of the second degree will represent any one of the central surfaces. The general equation of the second degree is f{x, y, z)=ax'+ by' + cs" + 2a'yz + 2b' sx + 2c'xy + 2a"x + 2b"y + 2c" s + d=0. Let a, ;8, 7 be its center ; which is determined by the con- dition that it bisects all chords passing through it. Hence the equation /(a + Zr, /3 + mr, 7 + nr) = must give equal and opposite values of r, for all values of the M2 164 DISCUSSION OF THE GENERAL EQUATION ratios I : m : n; ox l-f- + m-^ + n-J^ = 0, for all values of da dp ay I '. m I n. This leads to the equations ^=0, ^ = 0, ^= 0, da dp dy or aa + c'^ + b''y + a" = 0^ c'a+b^+a'y+b" = ol (A), &'a+a'/8+C7 + c" = oJ for determining the center. Since for central surfaces this must be at a finite distance, we have the condition abc + 2a'b'c' - aa'^ - bb'" - cc'" > or < for all central -surfaces. This might have been anticipated from the fact that the roots of the discriminating cubic are in this case all finite. Now remove the origin to the center, and the equation be- comes ax' + bf+ eg' + Idyz + Ih'zx + 2(^y +/(«, ^, y) = 0, and /(a, ^,y)=\ (a^+/3 ^'+7^ )+ a"a + 5"^+ c'^+e?, or / (a, )8, 7) = a"a + J"/3 + d'y + 0, A'B'C>0. L>0, M> 0, N^O, three positive roots, and thl sii^S face is an ellipsoid. "v.:^ > , i^O, M<0, ^<0, one positive, and two negative- roots, and the surface is an hyperboloid of two sheets. (2) S<0, A'B'C>0. L>0, M> 0, ^> 0, Inconsistent with the other as- sumptions. i > 0, ifcf< 0, JV< 0, one negative and two positive roots, and the surface is an hyperboloid of one sheet. i < 0, -¥ < 0, ^< 0, three negative roots, and the locus is impossible. 166 DISCUSSION OP THE aENEKAL EQUATION (3) H>0, A'ffC'<0. L>0, M>0, N>0, three positive roots ; an ellipsoid. L>0, M^O, N<0, one positive and two negative roots ; an hyperboloid of two sheets. L<0, M<0, N<0, inconsistent with the other as- sumptions. (4) H<0, A'ffO-KO. i > 0, M> 0, N'^ 0, two positive and one negative root ; an hyperboloid of one sheet. i<0, M<0, N<0, three negative roots; locus im- possihle. Hence, the locus will be an ellipsoid, if ^>0, A'BO'>0, L>0, M>0; S>0, A'B'C'<0, L>0, M>0, ]Sr>0: an hyperboloid of one sheet, if E<0, A'EC'>0, L>0, N<0; H<0, A'B'C'kO, L>0,M>0: an hyperboloid of two sheets, if S>0, A'B'O'>0, M<0, N<0; H>0, A'FC'kO, L>0, N<0: and impossible, if |^^ ^^^^ a'B'G'<0, M<0, N<0. 232. To find the conditions that the general equation of the second degree shall be an elliptic or hyperbolic paraboloid. These surfaces have each one center at an infinite distance. The equations determining the center are ax + c'g + Vz + a" = 0, c'x + by +(iz+ b" = 0, b'x + a'i/+ cz + c" = 0. If these determine one point at infinity, we must have then cihc + 2a'h'c -aa^-lb'^-cd^ = (i (1), and a" (Jc - O + b" {aV - cc') + c" {cW - bV) > or < 0, OP THE SECOND DEGREE. 167 of which the latter may, by means of the former, he expressed in the form «" 2," -" IT' 'f^ — ~ 177+-7T; ;> or <0 (2). DC —aa ca —bo ao —cc ^ '■ The paraboloid will be elliptic, or hyperbolic, according as the two finite roots of the discriminating cubic are of the same or opposite signs : that is, according as Ic .+ ea + ah- a"- 6'"- c" > or < 0. But the three quantities 6c— a'°, ca—b'", aJ—c'^ must be of the same sign, in order that the condition (1) may hold (Art. 221)^ Hence, for an elliptic paraboloid be — a" > 0, and for an hyper- bolic paraboloid be — a'^ <0, the conditions (1) and (2) being also necessary. 233. To find ike conditions that the general equation of the second degree shall be an elliptic or hyperbolic cylinder. In this case, there is a line of centers at a finite distance, or the three planes whose equations determine the center must intersect in one straight line at a finite distance. The conditions that this may be the case are that dbc + 2a'b'e' - ao^^-bb'^-cc"' = (1), „" y c" 5'c'-aa'^c'a'-66'^a'&'-cc' and the three quantities J'c' — ad, c'a! — bb', a!b' — cc', must be finite; for (Art. 221) if one yanish, at the same time that (1) holds, the others will also vanish, and in that case the three planes determining the center will be parallel. The equations of the line of centers, when conditions (1) and (2) hold, may be found to be X [b'c' - aa') - a'a" = y {c'a' - bt') - b'b" = s (a'b' - cc') - c'c". The cylinder will be elliptic, or hyperbolic, according as he — a'^ < 0, for the same reasons as in the last article. 234. To find the condition that the general eguation of the second degree shall he a parabolic cylinder. 168 DISCUSSION OF THE GENERAL EQUATION. The parabolic cylinder has a line of centers at an infinite distance, and therefore the three planes determining the center must be parallel. The coiaditions for this are four conditions which however are equivalent to the three h'c' = ad, c'a' = lV, b'c' = c<: (1). These are, as we have seen, the conditions for the vanishing of two roots of the discriminating cubic. It is also necessary that the planes shall not become coinci- dent, as in that case a plane of centers would exist, and the surface could only represent two parallel planfes. Now the planes will become coincident, if in addition to the above conditions, we have h' ' n" r' h" — =Tn, and t; = -77 , or if a'a" = b'b" = c'c". Oi O c Hence, for a parabolic cylinder, a'a", h'b", c'c" must not be all equal, and the conditions (1) must hold. 235. To find the conditions that the general equation of the second degree may represent two planes. ;The conditions for two parallel planes are, by the last article, * t' I fit 77/ iir t I II TiTit I 11 be =aa, ca=ob,ab=ca; a a =bb =cc . The conditions for two planes not parallel may be obtained from the consideration that such a surface is the limit of a hyper- bolic cylinder, when its line of centers lies on the surface. Now the centers being given by the equations ax + c'y 4- b'z + a" = 0, dx +by-\- a'z + b" = 0, J'a; + a'2^ + ca + c" = ; if X, y, e, be a point on the surface, satisfying these equations, we shall have, multiplying by x, y, z, adding, and simplifying by using the equation of the surface, a"x + b"y + c"z-^d = 0. PROBLEMS. 169 Hence, the conditions will be ahc + 2a'&'c' - ««"* - hh'^ - cc'^ = 0, h" b'c' - aa "*■ c'a - W "^ dV - cc' ~ ^' oc— aa ca —00 ao —cc We must also have Jc - a'" < 0. If Jc - a"' > 0, the surface will he an evanescent elliptic cylinder. XII. 1. Find the nature of the surfaces represented by the foHowing equations : (2) a;"-22/'+.2«' + 3;!sr-a;y-2a; + 7y-5«-3 = 0, (3) x^ + y" +,2 {yz + zx + xy) = a', (4) =««, (5) x' + 2(ifz + zx + xy) + 2,{z-y-l) = Q, (6) of + 6yz- 2z {x + y) = a', (7) x''+{l-m)yz + {l + m)x(y + z) = ax. In (5) s hew that the eccentricity of the priuqipal elliptic section is V 2 — 1/2, and in (7) examine the cases m = —i, 5m = 1, and m=l respectively. 2. The equation 7x'+ 8y' + iz'- 7yz - 1 \zx - Ixy = a" represeiis an hyperboloid of one sheet, whose greater real axis makes with me axis of z an angle tan~^;y2. \ 3. The equation af + ^ + Zs^ + ^yz + zx + xy-'Ix~\iy-25z + d=(i will represent an ellipsoid, a point, or an impossible locus, according as fi? < = > 55. 4. The equation ax'+ly'+2^+ 12yz + 6zx + ixy + 2a"x + 2b"y + 2c"z + d=0 ■will in general represent an elliptic paraboloid, a parabolic cylinder, or a hyperbolic paraboloid, according as «> = »„ + WJ-) = must have roots equal and of opposite signs, or the coefficient of r must vanish. This gives the condition ,df df df ^ dx„ ay„ «2o or I {ax^ + c>„ + J'Z(, + a") + m [dx^ + Sy„ + de^ + h") + n {b'x„ + a'^o + cza + c") = 0, the equation of the diametral plane. This plane passes through the point whatever be the values of I, m, n, or every diametral plane passes through the center, as is otherwise obvious. 245. To determine the principal planes of any conicoid. A principal plane is perpendicular to the system of chords which it bisects (Art. 167) ; hence we shall have, for the direc- tion of a principal plane, the equations al + cm + i'n == si, c'l+ hm + an = sm, h'l + a'm + en =sn; s being some constant. Eliminating the ratios Z : m : m, we obtain the equation («-a)(s-6)(s-c)-a"'(s-a)-J''(«-6)-c'^(5-c)-2aW = 0, the discriminating cubic already discussed. This supplies three values of s, and to each corresponds one system of values of l;m:n; or there are in general three, and only three, principal planes. The reduction of the general equation of the second degree may be effected in this manner, remembering that if a surface be referred to a principal plane as that of xy, the equation can only contain even powers of z. DIAMETEAL SURFACES. 181 It will be necessary however to supply the proposition that these three principal planes are mutually at right angles, as we shall now prove. 246. To shew that the three princtp«,l pla'nes of any comcoid are mutually at right angles. Let Sj, Sj, Sj be the three roots of the discriminating cubic, and let the corresponding values of I, m, n be denoted by the same suflSxes. We shall then have a?! + c'm^ + h'n^ = sj^, c'l^ + hm^ + a'wj = s^m^, h'\ + a'iWj + cWj = SjWj. Multiplying by Zj, wij, n^ and adding, we obtain . \ {al^ + c'jWj 4 J'Wjj) + Mij {c% + Im^ + a\) + n^ {b'l^ + a'm^ + cm,) or \ . sj,^ + mj . s^n^ + w, . s^n^ = s^ (Z,4 + m^m^ + n^n^, whence (s^ — s^) {IJ,^ + m^m, + n^n^ = 0. Hence, if the roots of the discriminating cubic be unequal, the three principal planes are mutually at right angles ; and the equation of the surface referred to them as co-ordinate planes, must assume the form *' Y Ax^ + Bf + Cs' + d = 0. i , It may readily be shewn also that A, B, are the roots , of the cubic; for the coefficient of a" in the transformed equation will be al^ + Im^ + cw/ + 'ia'm^n.^ + ^h'nj.^ + lic'l^m^, which is equal to Zj {a\ + c'm^ + 5'mJ + m^ {c'\ + bm^^ + a\) + n^ {b'l^ + a'm^ + cwj ; that is, to s, (Z/ + m^ + n^) or s, : and similarly B = s^, G = s^. The discussion of the cases of vanishing, or of equal roots^ may now be proceeded with as in Chap. xii. 247. Another method of determining the principal planes, and the lengths of the axes, may be noticed, depending on the fact that these lengths are the maxima and minima values of 182 DIAMETEAL SUEPACES. radii drawn from the center. Let the equation of a central conicoid be ax' + hy*+ cz' + 2a'yz + 2h'sx + 2c xy = d, (1) then, if xyz be the co-cjrdinates of the extremity of a radius, of length r, we shall have a? + f+z'' = r\ (2) and for the principal axes, r must be a maximum or minimum by variation of a;, y, z, subject to (1). This gives the equations, employing an undetermined multiplier X, ax + c'y H- h'z = Xa3, c'x ■\-hy + a'z = Xy, h'x + ay + cz='Kz, (3) Multiplying by x, y, z, and adding, we have by (1) and (2), d = Xr', and eliminating x, y, z from (3) we obtain the cubic equation -a'(^-»)-J-(|-i)-c-(^-c)-2»W.O, the roots of which are the lengths of the principal axes. We shall be able to obtain from (3) the equations of any principal axes, in the form , (d . Vd\ ,, Id , , c'a'N , Id dV\ which, taking the three values of r", are the equations of the three axes. These coincide, as they ought to do, with the equa- tions found ia Art. 224, Plane Sections of Contcoids. 248. To determine the natwe of the section of a conicoid made hy any given plane. This may of course be done by the substitutions of Art. 121, but for surfaces of the second degree whose plane sections will be curves of the second degree, simpler methods may advan- tageously be employed. If it be required only to discover to what species of conies the section belongs, we may eflfect this immediately by taking any orthogonal projection of the curve PLANE SECTIONS OP - CONICOIDS. 183 of section, since an ellipse, parabola, or hyperbola, will be pro- jected into a curve of the same species, though in general of different eccentricity. The only exception is when the plane of section is perpendicular to the plane of projection, but as no plane can be perpendicular to all the co-ordinate planes, there is at least one of the co-ordinate planes which may, in any pro- posed case, be taken as the plane of projection, and which will not be perpendicular to the plane of section. As an example of this method, we may take the section of the paraboloid ay' -i- Jz' = a; made by the plane Ix + my -|- wz = 0. The equation of the projection of the curve of section on the plane of yz is I {ay^ + hz^) + my + nz = 0, which is always an ellipse, or always an hyperbola, according as a and I have like or unlike signs. If Z = 0, the exceptional case above mentioned arises, and taking the projection on zx we have the equation {r^a + ni'b) »" = ni^x, or the section is parabolic, unless r^a + m^S = 0, when it reduces to a straight line. ' Hence, in the paraboloids, all sections pa- rallel to the axis of the principal sections are parabolas, and all other sections ellipses for the elliptic paraboloid, and hyperbolas for the hyperbolic paraboloid. If, however, a more exact determination is required, we may conveniently use the angle between the asymptotes, real or im- possible, as fixing the species of the curve. Let the equation of the conicoid be ' fix, y, z) = aa?+ If + cs" + '2a yz + 2l'zx + 2c xy + 2a" x + 2h"y + 2c"a +d=Q, and that of the plane of section be lx + my + nz = Q. Then if (a, /S, 7) be any point in the plane, and (X, jj., v) the directibns of a line drawn in the plane through this point, we shall have the equation /(a + \r, /3 + /ir, 7-1- !/»•) = 0, (l) to determine the points where the line meets the curve of section, \, n, V being subject to the condition l\ + mfjk + JW' = 0. (2) 184 PLANE SECTIONS OF CONICOIDS. If now (X/iv) be the direction of an asymptote, one value of r in (I) must be infinite, or the coefficient of r" must vanish. This gives for the directions of the asymptotes oX" + J/i' + 01^ + 2a' iw + 2 JVX + 2c'\/i = 0- (3) The equations (2) and (3) completely determine the direc- tions of the asymptotes, and the section will be elliptic, para- bolic, or hyperbolic, according as these directions are impossible, equal, or possible and imequal. If \ : /tj : V,, and Xj : Ms = "a ^^ ^^^ ratios obtained from (2) and (3), then, a> being the angle between the asymptotes, W + Z^ii^+fi^a this, being a symmetric function of the roots, will always be real, and the sections will be elliptic, parabolic, or hyperbolic, according as it is > = or < ] . The ratio of the squares of the axes of the section will be cos « + 1 : cos cb — 1, and the species of the section will be completely determined. Thus in the paraboloid ay^ + fe" = x, we shall have the equations a/*" + 5i^ = 0, Z\ + mfi + nv = 0, and we may readily form the equations _ . w+z^^Ms+^A _^ 'j{i}^i^2- i^^vy } Hence cot w = ,, , ,,.. ■ .,., ; .^ , — sfr • (1) For hyperbolic sections, ah must be negative, and I finite. (2) For parabolic sections, « = 0, and therefore Z= 0. (3) For elliptic sections, ah must be positive, and I finite. (4) For rectangular hyperbolic sections, a{n^ + r)+h{V + m?)=0. (5) For circular sections, cot m = V(— 1)) or a" («' + ly + V (Z" + m^ + 2a& {wV - P (P +»«'+»*'')} = 0, PLANE SECTIONS OF CONICOIDS. 185 and this may be obtained in the form {(r + w") a-{r+ m") by + iahmV = 0, and since by (3) dh is positive, we hare the systems 2 m=0, — = j- ; m = 0, j- = a b — a' ' b a — h' Of these only one is possible, and the results coincide with those of Art. 187. It appears from (1), (2), (3) that the elliptic paraboloid has no hyperbolic sections, and the hyperbolic paraboloid no elliptic sections, and that all sections parallel to the axis in either sur- face are parabolas. To the latter there is one case of exception, namely, when Z = 0, and aw" + bm' = 0, in which case cot m assumes an indeterminate form. We have however for all sec- tions satisfying the condition aw' + bm" = 0, {a + b)l which gives for Z=0, w = -, or a rectangular hyperbola. The section is however in this case really two straight lines, one at a finite distance, and one at infinity. For we may write the equation of the plane of section y^Ja±z V(- b) = k, and when this meets the surface ay^+bz^=x, we shall ha-s^ h{y^a + s ij{-i)\ = x, or the only points at a finite distance in which the plane mS^^ the surface lie on the straight line j = 2y'Ja-k-k+2z >J[- b). These sections are parallel to the asymptotic planes of Art. 165, and are themselves asymptotic in the same sense. In the central surface aaf + by^ + c;^ = k, the equations for the directions of the asymptotes are a>^ + biJ?-\- 01^=0, IX + mfju + nv^O. The first of these shews that every asymptote is parallel to a generating line of the asymptotic cone; hence for parabolic sections, in which the directions of the asymptotes coincide, the 186 PLANE SECTIONS OF CONICOIDS. cutting plane must be parallel to a tangent plane of the asymp- totic cone. The section of an hyperboloid made by such a tangent plane will be two parallel straight lines, since the center of the hyperboloid must be a center of the section, and the sec- tion parabolic. We shall, in general, have the equations W 1"8 _ /*#i _ M + cm^ cP + an^^ am? + lV — amn — hnl — elm a (m'+ m") + & K + + c C^' + m^) ^\iM;K-IJ;V,y+ •• •••••••••1 "^j-4*P + „. + «-)(i-H.f+f)}' Henc cot » = ■.y+"? + M.-+i-) + c(i--fy-) _ (1) The section will be hyperbolic, parabolic, or elliptic, according as I'bc + in'ca + n'ab is negative, zero, or positive. This of course coincides with the conditions that the parallel plane through the center shall meet the asymptotic cone in real and different, coincident, or impossible straight lines. (2) For a rectangular, hyperbolic section, a {m^ + n')+b{ri'+r) +c {r + m") = 0, unless at the same time Pbc + m'ca + n'ab = 0, but it is easily shewn that these conditions are inconsistent for any real finite values of a, b, c. (3) For a circular section, • [a K + w=) + J (w' + V) + c (Z'+ m')Y = 4 (Z" + to" + n=) {nc + m'ca + n'ab) , or Z*(J-c)'+... +2«iV(a-&)(a-c) + = 0. If a, b, c be in order of magnitude, this may be written [I' {b-c) + m' {c-a) - n' {a -b)]' ■¥ iVm' (a -c) (6-c) = 0, PLANE SECTIONS OP CONICOIDS. 187 whence we have two systems of circular sections, I = 0, T = ; m = 0, = 7 ; a — c — a a — c o — c of which only one is possible. Compare the results of Art. 184. It appears from (1) that all sections of an ellipsoid are ellipses, but that for the hyperboloids we may have all three species of conies. 249. The sections, made hy parallel planes, of similar and similarli/ situated conicoids, are similar. This appears from the fact that the equations determining the direction of asymptotes involve only the coeflScients of the terms of two dimensions in x, y, and a, so that the asymptotes of a section of the surface a{x- aT+ 1 {y - /3)» +c{z-if = d, made by any plane, are determined in direction by equations depending only on a, h, c and on the direction of the plane. Hence the sections by parallel planes of all the surfaces repre- sented by this equation for different values of a, ^, y, and d will be similar : and a like proof holds for non-central surfaces. In the similarity here determined, we consider an hyper- bola and its conjugate as similar curves. Indeed, in the sections of the same surface by a series of parallel planes which cut it in hyperbolas, though the asymptotes of all the sections are parallel, the curve of section lies, for one portion of the surface, in one pair of angles made by the asymptotes, and in the other portion in the second pair, the two series being separated by the plane section for which the curve degenerates into two straight lines. 250. To determine the area of a section of a central coni- coid made hy a given plane. Let the equation of the conicoid be ax' + hy^ + ca" = 1, and of the given plane Ix + my + nz =p. Let (a;„, y^, aj be the center of -the section, and r the length of any central radius whose direction-cosines are \, ^, v. 188 PLANK SECTIONS OF CONICOIDS. We shall then have the equations g5^^ = gio= F^, „ (Art. 184), l m n l_ m n ^ ^ " a c and J_ ^~<^'^o'~%o''~<'go'' _ B/, a 7 3 ax Then, for the lengths of the principal axes, we shall have -5 = aX," + bu," + cv' P a maximum or minimum by variation of \, fi, v, subject to the conditions * 1=X' + /*»+«/', = lK + mil + nv. This gives us the equations aXdk + Ifidii + cvdv = 0, "KdX + fid/j, + cdv = 0, IdX, + md/i + ndv = ; or, using two undetermined multipliers 7e, k', {a+k)\+ne=0, {b + k)/i+mk' = 0, (c + Je) v+n^' = 0. Multiplying by X, fi, v and adding we obtain, -j + ji; = 0, and therefore \ _ I fj, m V n F~l ' k'~l ,' k'~l ' —J — a -» — o -! — c P P P whence the equation -j-a — — S — -c /> /» P whose roots are the lengths of 'the semi-axes. The rectangle under these lengths is then // P+m^ + ri? \ y \Vbc + m^ca + n^ctbJ' and the area of the section is PLANE SECTrONS OP CONICOIDS. 189 or, subatituting for a!„, y,, s„, '^ V VZ^Jc + m'ca + r^ah) \ Vic + m?ca + n^ahj ' (1) In the case of the ellipsoid, this may be put into a con- venient form as follows. Let the plane move parallel to itself till the elliptic section vanishes, and let w be the perpendicular upon it from the center when in that position. The center of the section being then a point on the surface we shall have a c and if A, A' be the. areas of a central section, and of a parallel section at a distance p, we have ^■=^ ('-!)• (2) Similarly, in the case of the hyperboloid of two sheets, (3) In the hyperboloid of one sheet, the section can never vanish, but if we take ct the perpendicular on the parallel plane for which the section in the conjugate hyperboloid of two sheet/ vanishes, we shall have A' = A (-5)- (4) If we take two conjugate hyperboloids aa?+jy+cs''=+ 1, and the asymptotic cone to both, aa?-\-li^-\-cz' = Q, the area of the section of the latter may be found, from those of the former, by making a,.h, c infinitely large, preserving their ratios. Hence if A^, A^, A^ be the sections of the three surfaces made by any plane cutting them all in ellipses, and A the area of the parallel central section of the hyperboloid of one sheet, we shall have a,=a[i,.^), a = ^(^-i), A = a^,, whence A^ + A^ = 2A^, or the section of the cone is an arithmetic mean between the sections of the two hyperboloids. 190 PLANE SECTIOifS OP CX)NICOIDS. Also, if Fbe tHe volume of the cone cut off by a plane touch- ing the hyperboloid of two sheets, we shall have Now A = ,n„ — ; — 5 5-^fx- . and w' = h -r- H — • tj{l*hc + m^ca + v?ah) ' a h c' 1 TT .'. V= - ■■ , ■ , and is constant. 3 >^{aoc) 251. To find the area of a section of an elliptic paraboloid made hy a given plane. Let the equation of the paraboloid be ^ +-7- = 2a;, and of the given plane Ix + my + nz =p. Let (a!„, y„, »„) be the center of the section, and r the length of any central radius of the section whose direction-cosines are \, fi, v. We shall then have the equations . \ , =J^=-g!L-=}. (Art. 187). ma + w + «p — ma — no I Also .»(^V^ = 2.„-2(.^!^ = !^!^±^^±^. \a 0] " a I Hence, for the semi-axes, we shall have — + -r = « a maxi- ' ah mum or minimum by variation of /*, v subject to the con- ditions V + /i'' + j/'' = l, Z\ + »w/A+mi; = 0. This leads, as in the last article, to the equation 2 2 I, mr n . ^ +1 + ^i — r-=0- \—au \ — ou If M, , M, be the roots of this equation, m,Mj = — ™-t — = 75-7 , and the area of the section is wi'ffl + w'5 + Up ' 'ir>J{ci)) TT ■ tvjS^''<""t^ <"'''•+"■'+«• PROBLEMS^ 1 9 If the plane move parallel to itself till the section vanishes, and 7s be the perpendicular upon it in that position, we shall have = rt^a + m^S + 2?^, and the area of the section may be written in the form lir KJicih) (^ — ot). Hence the areas of all sections are equal, which are made by planes at equal distances from the tangent planes parallel to them respectively. XIII. (1) If a, j8, y be the angles between three equal conjugate radii of an ellipsoid, shew that cos' a + cos' ^ + cos' y, and cos a cos j8 cos y, are constant ; and that a + ;8 + y wiU always be equal to it, if a certain relation holds between the lengths- of the axes. 3? w" «" (2) In the hyperboloid -2 + xa — 5 = ^j ?hew that if a? and 6' be each > c', the equation of the surface may be put into the form y' + s? — x' = d', and if a, )8, y be the angles between (w«), (mx), (xy) . ^,. , „ > (a' + 6')(a'-c') (6'-e') in this case, cos a (cos 18 cos y - cos o) = -^^ — -^ — ,„ ' I.. ', ' \ r- I I 2 (a' + 6' - c )" (3) If (a;,2/i»i), {x^y^z^, {x^y^z^ be the extremities of three a;' y' z' conjugate diameters of the ellipsoid — + ts + t = 1, the equation of the plane passing through these points is (4) Also the locus of the center of gravity of the triangle formed by joining their extremities is €. yl ?! - 1 „= + j. + g. -3. and the locus of the intersection of planes drawn through their ex- tremities parallel to the conjugate planes respectively, is or V c' (5) The section of a hyperbolic paraboloid by a plane which makes angles o, j8 with the planes of the principal parabolic sections; whose latera-recta are a, b, will be a rectangular hyperbola if asin°a = 6sin'/8. 192 PEOBLEMS. (6) The do-ordinates of the vertex of the section of the paraboloid ^ +-r=x, a b made by the plane y cos a + » sin a =^ are given by the equations a; _ y _ a _ p p a cos a 6 sin a a cos'' a+h sin' a ' cib and the ]atus-rectum of the section is a cos" a + h sin' a ' (7) In a paraboloid of revolution, the eccentricity of any section is the cosine of the inclination of the plane to the axis of the surface, and the foci of the section are the points of contact with spheres in- scribed in the surface. (8) The section of the surface y» + s!c + xy + a' = (i, by the plane Ix + my + nz =p, mil be an ellipse, parabola, or hyperbola, according as f + m' + w' < = or > 2 (mn + nl + Im). "What will be the condition for a rectangular hyperbola? (9) If e be the eccentricity of the section of the cone made by the plane Ix + my + nz=p, shew that e* {mn — I' + nl — m." + Im — n')' •l-e'~ '{I' + m' + n') (mn + nl + Im) (10) Shew that, if any section of the surface given in (9) make angles a, P, y with the co-ordinate planes, the eccentricity of the aection will be constant if cos a -t- cos/3 + cos y be constant. Shew also that the section will be hyperbolic if oosa+cosj8+cosy lie between + 1 and - 1, paraboUo at these limits, and elliptic for all other possible values. (11) ■ The section of the conicoid aa;' -(-,%» + cz' + 2m'yz + 2b'zx+ 2 , degree is evidently the number of homogeneous products of four things of n dimensions, and is therefore _ 4.5... (4 + w-l) _ (w+l).(w + 2) (w + 3) _ 1.2 ...n ^ 1.2.3 ' but in estimating the number of constants with reference to the number of conditions which the locus can be made to satisfy, we must diminish this number by one, since the equation is un- altered if we divide by any one of the constants. The number of disposable constants so obtained is (w + 1) (w+2)(w+3) ._ n{n'+Gn + n) _ 1.2.3 ~ 6 -9^ J- 02 196 DEGREES OF SURFACES AND CURVES. Thus (w) arbitrarily chosen points will completely determine the position and dimensions of a surface of the w"" degree. A surface of the n* degree is also determined by <^ (w) inde- pendent linear equations of any kind between its coefficients. 258. All surfaces of the w*** degree which pass through (n) — 1 given points; and since, by giving proper values to the ratio \ : /*, this surface may be made to pass through any additional point which is not common to the two surfaces m = 0, v — O, this equa- tion will be the general equation of all surfaces which contain the ^ (w) — 1 given points, if \ : /t receive all values from — co to + oo . But this equation is also satisfied by the co-ordinates of all points which lie on the curve of intersection of m = and v = 0, which is therefore a common curve of intersection of all surfaces containing the ^ (n) — 1 points, and is of the degree n". 259. By reasoning similar to the above it can be seen that, if a surface be of such a nature that m points or m linear equa- tions of condition completely determine it, we may assert, that if m—1 such conditions be given, all surfaces of this kind which satisfy these conditions will have a common curve of intersection. 260. Conversely, if <^ (w) — 1 points be given, we may elimi- nate from the general equation of the surface of the w*'' degree all the constants but one, which will enter into the resulting equation in the first power only. This equation will then be of the form u + \v = 0, where u, v are of the n"* degree, and \ an undeter- mined constant. All surfaces represented by this equation will INTERSECTION^ OF SURFACES. 197 pass through the curve given by the equations M = 0, v = 0; which curve is therefore completely determined. For example, eight points determine a curve which is the complete intersection of two conicoids. In the case of complete intersection of surfaces the nature of the curve is not given when the degree is given, except in the case of prime numbers, in which it is a plane curve. For example, a curve of the twelfth degree might be the com- plete intersection of pairs of surfaces of the degrees (1, 12), (2, 6), (3, 4), and these different species, belonging to the same degree, would require a different number of given points completely to determine the surfaces. The following proposition serves to obtain the number of given points sufficient to determine a surface of the w* degree which, by its complete intersection with a surface of a lower degree, gives a curve of the nq^^ degree : this is given by Plucker, but may also be proved directly by a theorem given by Cayley, Nbuvelles Annales, xii. p. 396. 261. All surfactb of the w*'' degree which pass through {n)-{n-q)-l given points of a surface of the q*^ degree cut this last surface in one and the same curve. Of ^ («) — 1 given points, ^ {p) lie on a surface of the p^ degree whose equation is Mj, = 0, and if the rest, "viz. .^(«)-<^'(p)-l,* lie on a surface of the ^ degree, where n =p + q, whose equa- tion is Mj = 0, then u^u^ = is one of the surfaces which contain the ^ (re) — 1 points, and may be obtained by giving a certain value to the ratio X : /* in the equation \u+iJbv = 0, so that The curve of intersection of all the surfaces of the w**" degree containing these points lies on the surfaces u^ = and m, = 0. Hence if ^ (w) — ^ (w — g') — 1 points be taken on any fixed sur- face Mj = 0, all surfaces of the w* degree, which pass through these points, intersect the surface of the g^ degi-ee in the same curve. Thus, if 2=1, the proposition is reduced to the following; 198 INTERSECTIONS OF SURFACES. All surfaces of the w* degree which pass through {n + 1) (w + 2) 1.2 given points in a plane determine a fixed curve of the n^ degree. If 2 = 2, the proposition becomes, All surfaces of the m*'' degree which pass through « (ra + 2) points on a conicoid, intersect the conicoid in the same curve. A curve of the sixth degree, which is the complete in- tersection of surfaces of the second and third degrees, is deter- mined by 15 given points; if the surfaces be of the first and sixth, 27 points are required. One of the tenth degree requires 65 or 35, according as it is a plane curve or lies on surfaces of the second and fifth degrees, 262. When it is said that a curve is determined by a cer- tain number of points, these points must be supposed arbitrarily taken, for it is possible to select the same number of points, which would not be sufficient. Thus, a plane cubic is generally deter- mined by 9 points, but, if those be the nine points of intersection of two of such curves, an infinite number m^ be drawn through them. A curve of the fourth degree of one species can be deter- mined completely by 8 arbitrary points, but if these given points are the intersections of three conicoids which have not a common curve of intersection, taking these surfaces two and two, we may obtain three curves of that species passing through the same eight points., 263. If a curve of* the p*** degree be the complete inter- section of two surfaces of the m^ and w*** degrees, so that p = mn, and if m>n, then the number of points which determine the curve is (w— 1) («— 2), p> p. Hence, as n increases, while mn remains constant, p increases. INTEESECTIONS.DF SUEFACES. 199 t I n. If therefore mn can be divided in a different manner as ni the nearer w is to w the smaller is the number of points requisite to determine a curve of the mm"' degree. Thus mn = 60, 60.1 requires 1890 points, 30.2 960 30.3 630 15.4 451 12.5 334 10.6 250 Nouvelles Annates, xi. p. 361. 264. An extension of the theorem of Art. 261, is also given by Plucker, as follows. If of the {n) — 1 given points, ^{p)+m points lie on a surface of the p^^ degree, and the remainder ^{n) —^{p)—m — l exceed the number which is suflScient for determiaing a surface of the (w— p)* degree, so that ^ (m) - ^ (j?) -m- 1 = ^ {n-p), and therefore, m«l>in)-^ip)-i>in-p)< P^^-Pl^^+'\ it follows that, if of (w) — 1 points through which surfaces of the w"* degree pass ^ {p) + m lie on a surface of the p*^ degree, and V (n-p) (m + 4) m<^—^ ^-^ '-, the curves of intersection of all the surfaces lie on two surfaces respectively of the degree p and n —p. 265. The theory of partial intersections of surfaces is dis- cussed by Salmon in Vol. v. of the Quarterly Journal. Without an examination of such partial intersections it is not possible to analyze different species of curves of the same degree. If we considered only complete intersections of surfaces, curves of the third degree could only be considered as plane curves, whereas it wiU be seen that they may also be partial intersections of conicoids. 266. In order to find the surfaces which may contain a curve of the m"" degree, it is observed that through ^ (k) points a 2(K) INTERSECTIONS OF SURFACES. surface ot the i* degree can te made to pass. Now, the total number of points which are common to a proper curve of the m"' degree with such a surface, supposed not to contain the curve entirely, are mh, since this is the number of points in which h planes intersect the curve ; and the law of continuity makes the statement general. If <\> (Je) = mh + 1, one such surface can be drawn containing the curve, if ^ (A) >?kZ; + 1, two surfaces of the A,"* degree can be drawn, and therefore an infinite number. Thus, for a curve of the third degree, if A = 2, ^ (A;) = 9 > 3.2 + 1, hence an infinite number of conicoids may be drawn containing any curve of the third degree. Again, if ^ (Jc) = mh + \, one surface of the 'M^ degree con- tains the curve and the simplest surface of a higher degree can be found by trial ; which will be of the {h + 1)"* degree, whenever {h + 2) (A + 3) > 2m. Hence, all curves of the fourth degree can be obtained by the intersections of proper surfaces of the second and third degrees : and all curves of the sixth degree from surfaces of the third and fourth degrees. Modifications are required if the surfaces are not proper sur- faces, Salmon gives as examples of this modification, a plane curve of the third degree through which it is possible to de- scribe an infinite number of conicoids, but since each conicoid must necessarily consist of the plane of the curve and an arbitrary plane, the intersection of the plane and conicoid will not deter-- mine the curve: again, if a curve of the fifth degree, which, according to the above laws, ought necessarily to be determined by surfaces of the third degree, lie entirely on a conicoid, all the surfaces of the third degree which contain the curve may be a compound of the conicoid and a plane, and we must recede to surfaces of the fourth degree to determine the curve. If a curve be given of the m* degree, and h, I be the lowest degrees of surfaces upon which it can lie, any surface of the M^ degree constructed to pass through mh + 1 points will contain the curve, and similarly for the other surface. If ml+l points known to lie on the curve be given, and l>h, all the rest can be found. INTERSECTIONa»OF SURFACES. 201 267. The number of arbitrary points through which a curve of the m* degree can be drawn cannot exceed a certain superior limit which is easily determined, for suppose h arbitrary points be given and a cone be constructed containing the curve, and having its vertex in one of the assumed points, the degree of this cone will be m — 1, and the number of its generating lines suf- ficient for its complete determination is the same as that of the number of points necessary to determine a plane curve of the 1*^ t . m (m + 1) , m — 1 1*" degree, viz. — —- 1. The greatest value of Tc for which such a cone can be con- structed is — ^- — —\ this is therefore a superior limit, although other lower limits to the number h may be obtained in general from other considerations. Thus, a curve of the third degree cannot be made to pass through more than six arbitrarily chosen points. 268. If ^{n) —2 points he given, all surfaces of the w* de- gree which can he drawn through these points, will pass through w° — (^ (w) + 2 more fixed jpoints. Let M = 0, -0 = 0, w = be the equations of three surfaces of the w*"* degree which pass through ^ (m) — 2 points, and which have not a common curve of intersection, they will pass through m' common points, and \m + /i« + I'M* = is th& equation of another surface of the m"" degree, which passes through the same points, and by giving different values to A. : /* : v we can obtain all sur- faces which pass through these points. Any surface will be particularized when two points are given, which do not lie on all three of the surfaces, or both on the same two : and all such surfaces will contain w' - <^ («) - 2 common points besides the given points. Thus, all conicoids which pass through seven points will pass through a fixed eighth, as is easily seen if each conicoid be two parallel planes, the seven points being angular points of a parallelopiped. . A surface of the third degi-ee, drawn through 17 points, passes through 10 others. 202 INTERSECTIONS OF SDEFACES. 269. All surfaces of the m*'' degree which pass through a certain nwniber of given points in the curve of intersection of two surfaces of inferior degrees, q and s, that number being {n — q)+(}){n — s) + 2. II. Let nZ.q + s- In ttis case, since H'l^n-tl^a + VW^W, = fl {V^ + pVW, 0„^_s) V^ in which pn^_, = contains 4>{n-q-s) + l constants, and will diminish the number to be determined in v,^ = 0, and w^=0, by so many. Hence, q + s — i, it is easily shewn that the num- ber of new points is t, +^) which is independent of the degree of the surface m = 0. If s = 1 and qS Pi. r." a" «," S" cos (p, p') = ?^ - + /,' e. + ^' I. + p' e_ Px P2 i'a p. 4 _a'a" JS-^' al^" + oi'^ = — f H T" + • • • cos UJJ — ... Pi A PiPi which gives the required angle. 277. The class of a surface is the number the planes of which can be drawn through a given straight line so as to touch the surface. If (a, )8, 7, S) and (a', /S', 7', S) be two planes, the co-ordi- nates of any plane passing through their line of intersection will be \a + /*«', X^ + /i^', ; \ : fi being any arbitrary ratio, and the particular planes which touch a surface, whose equation is F (a, j8, 7, S) = 0, supposed a homogeneous alge- braical equation of the w* degree, will be determined by the values of X : /t which satisfy the equation F{\a + fia', )=0; the number of values of the ratio is n, and this is therefore the class of the surface, and corresponds to the degree of the surface in the plane co-ordinate system. 278. It is easy to express in the language of four-point co-ordinates the results of this chapter. Thus, a surface of the w*'' class is determined if

= (»»?«+ «2o) («-««)+ and, if the line be a tangent line, two values of s are equal to z,; .: l=mp„ + nq„, and, by eliminating m and n from this equation and the equa- tions of the line, we obtain the locus of all the tangent lines through (ajj, y„, s„), whose equation will be which is therefore the equation of the tangent plane where such exists, i.e. unless ^^ and q„ assume the indeterminate form - . This equation is immediately deducible from the equation of Art. (282), by means of the equations p^'{z„)+F-{x,)=0, and ?„i^'K) + i^'(y,) = 0. p2 212 INTERSECTIONS OP SUEFACES 286. Geometrical explanation of the nature of the mterseor tion of a surface with its tangent plane at any point. ETeiy plane intersects a surface of the m'" degree in a curve which is of the same degree; hence a tangent plane at any point intersects the sitrface in a curve of the w"" degree, passing through the point of contact. Now when a tangent plane exists, since it is the locus of the tangent lines at the point of contact, and each of these tan- gent lines contains two points, which coincide in the point of contact, it follows, that any line, drawn through the point of contact in the tangent plane, meets the curve of intersection in two points at the point of contact. The point of contact is, therefore, a singular point in the curve of intersection. This singular point may be either a conjugate point, as in the- case of contact with an ellipsoid; or a multiple point, as in the case of a hyperboloid of one sheet ; or a point through which two coincident lines pass, as in the case of a cylinder. If the surface is of the second degree the curve of inter- section is of the second degree, and, since it must contain a singular point, the only admissible lines of intersection are either an indefinitely small circle or ellipse, or else two straight lines which cross one another, or are coincident. 287. If a plane intersect a surface in a curve which contains a singular point, the plane is generally a tangent plane to the surface at that singular point. For a straight line drawn in any direction in the plane, through a singular point, meets the surface in two points which ultimately coincide, and therefore generally satisfies the con- dition of being a tangent line to the surface. If the point which is a singular point in the curve of inter- section is also a singular point in the surface, the condition of passing through two coincident points is not sufficient to define a tangent line. Thus, if at any point of a surface there be a conical tangent, there may be a multiple point in the curve of intersection of a WITH TANGEJJT PLANES. 213 plane intersecting the conical tangent, which will not make the cutting plane a tangent plane at the multiple point. 288. The form of the curve of intersection of a surface with the tangent plane at any point may he illustrated by taking the case of an anchor ring, supposed to be generated by the revo- lution of a circle about an axis in its plane not intersecting the circle. The figure represents the ring, with the generating circle in different positions as it revolves about the axis Oz, The plane U is drawn through the axis Oz, intersecting the surface in the circles GHc, DLd. Suppose this plane to move, parallel to itself, towards the position V, the closed curves in which it intersects the surface become elongated until they meet one another in the point A^ 214 INTERSECTIONS OF SURFACES forming for the position F of the plane a figure of eight, vi«. EPAqFQ which has a double point at A. Here we observe that the concavities of the circles AKa and A GBD, which are sections by planes perpendicular to V and to each other, lie in opposite directions with regard to the plane V, and that the ■tangent lines at A lie in that plane, which is therefore the tan- gent plane at A; and it is a tangent plane at no other point of the curve of intersection. The sections by planes through A perpendicular to V change the directions of their concavities as they pass from the position AKa to A GBD, when they cross the tangents to the branches pA Q, PAq at the multiple point. If the plane move past V to the position W the curve of intersection gradually assumes an oval form, which degenerates into a conjugate point at d. It is clear also that a plane may meet the ring in the circle GHKL, in which case it is a tangent plane at every point of the curve in which it meets the surface, which is composed of two coincident circles, as may be seen by moving the plane inwards parallel to itself. 289. To find the equations of the tangent line to the curve of intersection of a surface with its tangent plane at any points Let the equation of the surface be F{x,y,z)=0, and that of the tangent plane at (a;„, y„, aj {x - X,) F' {x,) + {y- y,) F' {y^ + {z- z,) F' {z,) = 0. Let the equation of the tangent line to the curve of inter- section at the point {x^, y^, z^ be x-x^ ^ y-y, ^ z-z, __^^ The points in which it meets the curve are given by the values of r, which satisfy the equations F{xi + \r, y^ + fM; «, + vr) = 0, d . d , d • (A or e /«+''^+0 jr(a,^, y^, aj = 0, WITH TANGINT PLANES. 215 and {x, + \r -x,) F\x,) + {y^ + fir- y„) F\y^ And since the tangent line meets the curve at two points gene- rally, and in more than two at a singular point, we have for the general tangent line 'S-F'{x,)+,iF'{y,) + vF'{z,)=.Q, ■>^F'{x,)+^iF'iy,)+vF'{z,) = 0; which equations determine, the ratios X : fi : v, except in the cases in which more than one system of values of X : /i : v satisfy the equations ; and this will happen, I. wheii F'{x^) = 0, F'{y^) = 0, and F'{z„) = simulta- neously, which is the case when there is a tangent cone at the point (aj„, y„ »„) ; II. when F'(x^)=0, F'{y^ = 0, F'{z^==Q, which is the case when {x^, y^, «J is a singular point in the surface ; III. when -p^ = -pAgj = _A_ij, ^^ich will happen when (ajj, y^^ z^ is the point of contact of the tangent plane. In the last case, the tangent line meets the curve in more than two coincident points, and the condition for this is which, combined with the equation \F{x,)+,LF'{y,) + vF'{z,) = 0, give two systems of values of the ratios X : (l : v, unless the differential coefficients of F{x^, y^, «„) of the second order vanish, in which case all lines drawn in the tangent plane through (a;„, y^, z^ meet the curve in three coincident points, and the multiple point is of a higher order of multiplicity, and if the s* differential coefficients are the first, which do not all vanish, A tt . r + ^ H'm . r' + i -D'm . »•' = 0. 2 D Now the four constants in the equations of a straight line may be chosen so as to satisfy the equations m = 0, Du = 0, U^u = 0, Z>'m = 0, and all straight lines having such constants lie entirely in the surface, since the above equation is then satisfied for all values of r. If a plane be drawn through such a straight line its line of intersection with the surface will be, in the general position, composed of that straight line and a conic forming a group of the third degree. Now there are five positions of the plane for which the conic breaks up into two straight lines. For the equations of any surface of the third degree which contains the axis of x as one of its straight lines is of the form F, {y, s) + xF, {y, z) + x'F, {y, z) = 0, Fg {y, z) denoting a function of ^ and z of the third degree, and F^ {y, z), F^ (y, z) of the second and first degrees. If s =Xy be the equation of a plane containing the axis of x the projection of the curve of intersection on the plane of xy has for its equation * Cambridge and Dublin Mathematical Jowrnal, Vol. iv. p. ii8. SINGULAR POINT^OF SUKFACES. 217 where M^) = %f + 2a,y + a,, the subscripts of the letters a, h, c being the degree to which X rises in the respective coefficients. The curve of intersection consists therefore of the axis of x and a conic the equation of the projection of which is a,/ + ib^ + c^a? + 2a,y + 25^03 + a, = 0, and the conic breaks up into two straight lines for values of \, which satisfy the condition a^ + Ci< + afi^ - a^c^a^ - 2\aJ)^ = ; which, being of the fifth degree in X, shews that there are five positions of the plane, for which the conic becomes two straight lines. In the general position of a plane through a straight line the plane is a tangent plane at the two points of intersection of the conic and straight line (Art. 279); in the five particular positions, the plane, intersecting the surface in three straight lines which form three double points, touches the surface at these three points, and it is therefore a triple tangent plane. Through each of the three straight lines can be drawn, be- sides the plane in question, four other triple tangent planes, giving rise to 12 new triple tangent planes, and 24 new straight lines situated on the surface, making in all 27. These are the only such straight lines which can be drawn on the surface, for any straight line on the surface must meet one of the three straight lines in any triple plane, since these three straight lines form the complete intersection of the plane with the surface ; and the plane passing through such straight, line and the line which it intersects must be one of the triple tangent planes containing that line, since it intersects the surface., in two and therefore three straight lines. Each triple tangent plane contains three lines, and five can be drawn through each of the 27 lines, therefore the whole 5*27 number of triple tangent planes is —^ = 45. 218 RULED SURFACES. Ruled Surfaces. 291. The student is already familiar with certain surfaces which are capable of being generated by straight lines, or through every point of which some straight line may be drawn which coincides, throughout its length, with the Surface. For example, a plane, a cone, a cylinder, an hyperboloid of one sheet, an hyperbolic paraboloid. Among these surfaces he is aware that any portion of a conical or cylindrical surface, if supposed perfectly flexible, might be developed into a plane without tearing or rumpling. We shall now give some account of the general character of surfaces which have this property, distinguishing them from those which, although capable of being generated by the motion of a straight line, are incapable of development into a plane. 292. Def. a Buled Surface is a surface which is capable of generation by the motion of a straight line; or a surface through every point of which a straight line can be drawn, which lies entirely in the surface. Def. If a ruled surface be such that each generating line intersects that which is next consecutive, the surface is called a Developahle Sarface. Def. If a ruled surface be such that consecutive positions of the generating line do not intersect, the surface is called a Shew Surface, Developahle Surfaces. 293. Eoeplanation of the development of developable surfaces into a plane. Let Aa, Bb, (7c, ... be a series of straight lines taken in order, according to any proposed law, so as to satisfy the condition that each intersects the preceding, viz. in the points a,l, c, ... Since Aa, Bb intersect in a, they lie in the same plane, similarly, the successive pairs of lines Bb and Co, Oc and Dd, &c. lie in one plane; thus, a polygonal surface is formed by the successive plane elements AaB, BbO, &c. EULED St(gPACES. 219 This surface might be developed into one plane by turning the face AaB about £b, until it formed a continuation of the plane BbG, and again turning the two so forming one face about Oc until the three AaB, BhG, CcD were in one plane, and so on ; the whole surface might, therefore, be developed into one plane without tearing or rumpling. The same is true however near the lines -4a, Bh, ... are taken, hence, in the limit, we arrive at the property from which this 220 RULED SUEPACES. class of surfaces derives Its name, which as we have seen is derivable from the fact that two consecutive positions of the generating line always intersect. Edge of Regression, 294. The polygon abed, ... whose sides are in the direction of the lines Eb, Cc,... becomes in the limit a curve which is generally of double curvature and is called the Edge of Egression, from the fact that the surface bends back at this curve so as to be of a cuspidal form, and every generating line of the system is a tangent to the edge of regression, which is therefore the envelope of all the generating lines. In the case of a cylinder, the edge of regression is at an infinite distance. 295. To find the, general nature of the intersection of a tan- gent plane to a developable surface with the surface. The plane containing the element BdH of the smrface repre- sented by the figure evidently becomes in the limit a tangent plane to the developable surface at any point D in the generat- -ing line J)d, since it contains the two tangent lines JDd, and the limiting positions of lines joining such points as D and E, which ultimately coincide ; and again, supposing DdJS in the plane of the paper, Ff meets this plane in e, Ggf meets it in some point/', BTig in g', &c., and similarly for Cc, Bb, ... on the other side. The complete intersection of the surface and tangent plane is therefore the double line formed by the coincidence of Dd, Ee, and the limit of the polygon a'b'c'def'g ...\i]iic\i is a curve touching the double line Dd at the edge of regression. COE. To find the nature of the contact of the edge of re- gression and the tangent plane. The plane containing the generating lines Dd, Ee contains the three angular points c, d, e of the polygon in the limit, therefore the tangent plane contains two consecutive elements of the curve edge of regression, and is what is called the osculating plane at that point. [To face page 321.] EULED SURFACES. 221 296. The shortest line which joins two points on a develop- able surface is that curve wliose osculating plane contains the normal to the surface at every point. If the surface be developed into a plane, the shortest line must he developed into the straight line joining the two points. If on the polygonal surface in the figure of page 219, ABGD...K be the polygon, which in the limit becomes the shortest line joining A and K, since on development this becomes a straight line, two consecutive sides EF, FO must be inclined at equal angles to the line Ff. Hence a straight line drawn through F, perpendicular to the line Ff va. the plane bisecting the angle between the planes EFf GFf -mH. evidently lie in the plane EFQ, and bisect the angle EFG. This line will be in the limit the normal to the surface, and the plane EFQ will be the osculating plane. Therefore the shortest line is the curve whose normal at every point lies in the osculating plane at that point. Such a line is called a geodesic line of the surface, and it will be hereafter shewn that the property enunciated for de- velopable surfaces is true for geodesic lines on all surfaces. If the geodesic line, joining two given points, be drawn on a right circular cone, the equation of the projection upon the base can be shewn to be - sin (7 sin a)=r sin {6 sin a) + - sin {{{a)y + 's^{oi), a being the parameter, and (a),'\/r(a) functions of that para- meter, given in any particular case. In skew surfaces, the equation of the tangent plane at any point will involve the parameter of the generating straight -line passing through the point, but not containing the consecutive straight line, will involve sonie other parameter which fixes the tangent plane among all the planes containing that straight line. We may also arrive at the conclusion that the equation of the tangent plane to a developable surface can only involve one parameter, from the consideration that if it involved two, we should by varying them infinitesimally, obtain the equations of three planes, which woijld ultimately intersect in a definite point, instead of in one straight line, so that the plane could in general have only one point of contact with the surface which it touched ; and the surface would therefore not be developable. 300. To find the form of the curve of intersection of a de- velopable surface with a tangent plane. Let the equation of a plane be given in the form s = ate + ^ (a) y + V^ (a) (1) containing one parameter a, by the variation of which the plane assumes different positions. The equation of the plane in its next consecutive position is « = (a + c?3i) a;,+ <^ (« + dot) y + -^{p. + drt),. 224 EULED SURFACES. and the line of intersection has for its equations = x + '{a)i/ + ■^'(a) and a = aaj + ^ (a) y + V^ (a). If we eliminate a between these equations we obtain a surface which is the locus of all such lines. If the equation of the plane involved two arbitrary para- meters, the plane would not move in such a manner as to give with a consecutive position some definite line the locus of which would be a surface. Assuming therefore (1) as the- equation of a plane of the system, let the plane of xy correspond to the value a = 0, let the axis of y be the line of intersection with the next consecutive, and the next to this pass through the origin ; .-. ^ (0) = 0, ^' (0) = 0, -^ (0) = 0, V^' (0) = 0, ^jr" (0) = 0. Also in order that y may be determinate ^ttt^ must vanish. Hence we may express ^ (a), ■^ (a), as follows : ^(a)=aa'"*'(l+6), ^(a) = Ja"«(l+e'), e, e' vanishing simultaneously with a, and m being < w + 1. The intersection of the surface with the plane of xy will be given by the elimination of a between = a {a; + aoT (1 + e) ^ + &a""(l + e')} and = a!+ (»»+ 1) aoT (1 + »?) 2^ + (« + 2) 5a"*'(H- V), a = corresponds to Oy, and, for the curve of intersection, in the neighbourhood of 0, = maoT'y + (« + !) Ja"*S and = mx— {n + l — tnjha'*^; f mx ['^'-'" _ f may I'" •'• \{n + l-m)b] ~\ {n+l)bl ' .-. y^=Gaf*^-^, or the curve is parabolic touching Oy at 0. SINGULAR TANGENT PLANES. 225 301. If a series of straight lines, generating a surface, he described according to a law siich that the shortest distance be- tween two consecutive lines is of an order superior to the first, it will he at least of the third. Since the four parameters, entering the equation of a line, must be capable of being eliminated, there must be three rela- tions between those parameters besides the two equations of the line ; hence if the equations 'h&x = mz + a, y = ns + h, m, n, a, h must be functions of one parameter, which by its variation gives rise to different positions of the generating line. The shortest distance between two consecutive lines of the system is A?w A5 — An Aa Am]^ + Aw]^ + {mAn - nAmf and Am = dm+-d^m + -d^m,+ and similarly for Are, Aa, and A5 ; .*. Am A5 — Aw Aa = dm dh — dn da, + - {dm d^h + dh dSn — dn d'^a — da d^n), + terms of the following order higher than the third, the denominator is of 'the first order, and if dm dh — dn da be not zero, the numerator is of the second order, but if dm dh — dnda = always, we have also dm d^h + dh d/'m — dn d^a — da d\ = ; or the numerator is of the fourth or higher order. Hence the trutli of the proposition which is due to M. Bouquet. Tangent planes touching along a curve line. 302. We have seen (Art. 287) that, when a plane intersects a surface, at every point of the curve of intersection, through which Q 226 SINGULAR TANGENT PLANES. an arbitrary line drawn in the plane passes through two coinci- dent points, the plane is a tangent plane to the surface, or such a point is a multiple point on the surface. If the curve of inter- section consist of two or more coincident lines, this will occur at every point of such curves, hence, either the plane will be a tangent plane to the surface at every point of such multiple curve, or will contain a multiple line on the surface. Conversely, if a tangent plane touch along a curve line such a curve line will be a multiple line on the tangent plane. Thus in the case of the anchor ring, the plane which touches the anchor ring at every point has for its curve of intersection the two circles coincident in LKH; also the tangent plane to a cone contains two gene- rating lines which ultimately coincide, and is therefore a tangent plane at every point of the generating lines which it contains. Similarly, a surface of the fourth degree admits of the case of a double conic, or of a quadruple straight line, as in the case of two cones touching along a generating line. A surface of the fifth degree might be composed of one of the third, and one of the second degree, in which case it is possible that a tangent plane might meet the former in a triple, and the latter in a double straight line. 303. To find the conditions that a tangent plane may touch the surface at every point in which it meets it. Let the tangent plane at the point [x„, y^, «„) in the surface whose equation is <: = f{^,l/) (1) have the property. Sufficient conditions are that p =p„ and q=qo throughout the curve common to the surface and tangent plane. Since the curve of contact is a compound curve contain- ing two coincident curves at least, therefore if {x, y, z) and {x + dx,y + dy, z + dz) be two points common to the surface and tangent plane, two values of dx : dy must be coincident at least. The equation of the tangent plane is SINGULAR TANGENT PLANES, 227 .'. p„ dx + jd dy —p dx + g[dy +- {rdoe\ +-2sdxdy + tdy]^) + &c. Only one value of the ratio dx : dy exists ultimately, unless and in this case the two values are given by roJa; I + Isdxdy + tdy\ = ; and since the roots are equal, because at every point two tan- gent lines coincide, we have the necessary condition, rt — s^ = 0. Or, since the tangent plane remains constant for all points common to it and the surface, p and q are constant when x and y receive small increments ; /. dp = = rdx + s dy, dq = = sdx+ t dy, whence rt — s'=0 (2). It is easily seen that the condition rt — 8' = ia not sufficient, although necessary, since for the curve common to (1) and (2) ^T + ^T =(»•+<) (J"^? + ^sdxdy + idy\), which is not necessarily = 0, for values of dx : dy obtained from ^ these equations. It will be seen hereafter that rt — s^—0 may be true for other points in the surface which are not in the curve of contact. Thus, in the case of a developable surface always touched by a plane whose equation is z = ax + 6 {(x) y + '<^ {a) , s = r^'{p), and< = s^'(^). Therefore rt — s^ = Q at every point of the surface, and we Tiave shewn that the tangent plane does not necessarily touch at every point in which it meets the surface. Q2 228 CURVE OP GREATEST SLOPE, 304. To find the curve of greatest inclination to a given plane tchich can he traced on a surface. Let the equation of the surface be F{x, y, z) = 0, Z, »i, n the direction cosines of the given plane. The tangent plane at any point of such a cui-ve is {x - x^ F\x,) + {y -y,) F'{y,) + {z- z,) F'{z^) = 0, and the direction cosine of the line of intersection with the given plane are proportional to mF\z,)-nF'{y,), nF'{x,) - lF'{z,), and lF'{y,) -mF'{x,). The direction of the curve line having the property proposed, is perpendicular to this line ; therefore the differential equations of the curve are {mF'{z) - nF'iy)} dx + {nF'{x) - lF'(z)] dy ^\lF'{:y)-mF'{x)]dz = % and F'{x) dx + F'i^) dy + F'(z) dz = 0. The first of these equations with the equation of the surface and any point chosen on the surface through which the curve shall pass, are sufficient completely to determine the curve. If the plane be that oi ocy, l=m = 0, and the equation becomes F'{y)dx-F'{x)dy = 0, which with F{x, y, z)=0 is sufficient. Cor. If the equations for obtaining dx: dy : dz become iden- tical, the direction of the line of greatest inclination will be indeterminate ; in this case mF'(z) - nF\y) _ nF\x) - IF'jz) _ IF'jy) - mF'jz) F\x) F'(i,) F\z) lF'{x)+mF'{y)+nF'{z) Therefore F'{x) : F'{y) : F'{z) = 1 : m : n &t every point of the surface, which can only happen when the surface is a plane parallel to the given plane. TANGENT PI^NES. 229 Tetrdhedral Co-ordinates. 305. To find the equation of the tangent plane at a given point in a given surface referred to tetrahedral co-ordinates. Let 4> = F{(x, B, 7, S) =0 (1) be the equation of the given surface, (a„, /S,, %, 8„) the given point P, and h+mB+ny + rS=0 (2) the equation of the tangent plane required. Since the tangent plane contains all the tangent lines to the surface, it must be satisfied by the co-ordinates of points taken in any direction on the surface, which ultimately coincide with the proposed point. Hence (1) and (2) are satisfied by the co-ordinates of P and' also by a^ + da„, j8„ + (?y8„ . . . when da^, dfi^, ... are indefinitely diminished ; . •• MiXa + md^o + m<^o + '"^^o = 0, F{a,) da,+ F' (fi,) dB, + F (7„) dy, + F (S„) dS, = 0, and ddo + dfi^ + (a„,/3„,7„,8,)=0, which is of the w"' degree, determine generally n.{n — ly points, and the same number of tangent planes, hence the surface given by an equation of the w**" degree is of the n.n — lY class. It will be shewn hereafter how this number is diminished when there are multiple points and lines on the surface. Four-Point Co-ordinates. 309. The proposition in the system of four-point co-ordi- nates, which corresponds to that in plane co-ordiiiate systems of finding the equation of the tangent plane at a given point of a surface, is to find the equation of the point of contact of any plane whose co-ordinates satisfy the tangential equations of the surface. 232 TANGENTIAL EQUATION. The property which, we shall employ, is that the point of contact of a tangent plane is also a point in a contiguous tan- gent plane which moyes up to and ultimately coincides with the former. 310. To find the equation of the point of contact of a tan- gent plane to a given surface determined hy an equation in four- point co-ordinates. Let = F{a, /3, 7, 8) = (1) be the equation of the surface, a„, /8„, 7„, \ the co-ordinates of the given tangent plane. Assume Za + jw/8 + «7 + rS = (2) to be the equation of the point of contact. Then, a„+ 2 Pb P* and similarly for /i, v, p. Art. (108). If therefore we assume A and B so that I = AF (a.) + Bk, and m = AF' (fi,) + /S/t, TANGENTIAL I^UATION. 233 we obtain in consequence of the indeterminateness of (^„ : dS^ n = AF' (7„) + Bp, and r = AF' (S„) + Bp. Multiplying these equations by a„, /3„, 7„, S„, and observing that Z«. + m^„+ ... =0, and «„i?"(a„) +^„i?"(/3„) + ... =0, and also we obtain B=0, and we derive la + m^ + ... =Xl(nF' {a„) + ...}. Hence the equation of the point of contact is ai^'(«o) + ^F' (;8„) + ryF' (7j + BF' (S„) = 0. 311. To find the tangential equation of the curve line in which the tangent plane to a surface given hy four-point co-ordi- nates touches along a curve line. In the case of a singular tangent plane, for which there is not a single point but a curve of contact, i?"(«„)-0, i^'(/3„) = =0 (1). If we take a plane («„ + {?»„, ^^ + d^^, ...) indefinitely near to (cf„, /8(,, 7„, S„) the equation of the surface gives the relation ('^«o|:+'^^o|/-r^«=« (^). and if (a, j8, 7, S) be any plane passing through the line of inter- section of these planes, we may obtain as in Art. (275) or directly by geometrical considerations, if 9 be the inclination of this plane to the first of the above planes and the angle between these two planes, a sin ^ = («(, + da^ sin (^ + ^) - a„ sin 6 = ofj cos ^ sin ^ + da^ sin 0, ultimately ; " a-a„coad yS-jS^cosfl 234 POLAR EQUATION. therefore, having regard to equations (1), we obtain from (2), (a ;J + /3 ^ + ...)Vo = 0, as in Art. 307, the relation which holds for all planes which touch the curve of contact. This is therefore the tangential equation of the curve. 312. To find the degree of a tangential surface of the w* class. Let the equation of the surface be F{a, jS, 7, S) = 0, and (a, /8', 7', S') (a", /3", 7", h") two tangent planes intersecting in a given line. Any point in the given straight line and surface is in the given tangent planes, and if («„, yS„, 7„, S„) be the tangent planes of which this point is the point of contact, oiF\oi,)+^F'{fi,) +... = 0, .-. a'i^'(«o)+^^'(/3„) + ...=0, a"i^'(«o)+i8"i^'(^o) + -=0. These equations, which are of the n — if", and the equation of the surface, which is of the w*'' degree, determine generally fi . n — \\ tangent planes, and the same number of points of contact. Hence, a surface given by a tangential equation of the n* degree is of the n .re — ij degree. Polar Oo-ordinates, 313. To find the polar equation of the tangent plane to a surface at a given point. Let the equation of the surface be- = M=/(^, ^), and Mj, 0g, g co-ordinates of the point of contact of the tangent plane. POLAK E^JJATION. 235 The equation of the tangent plane is of the form pu = cos a cos ^ + sin a sin 6 cos (^ - |S), Art. (79) , and the constants p, a, and jS are to be determined from the con- sideration that the tangent plane contains not only the point of contact but adjacent points which have moved up to and ulti- mately coincided with that point. Hence the values of -t» and tt at the point of contact are do d^ ^ the same for both tangent plane and sui-face, let v„, w^ be those values ; pv^ = — cos a sin 6^ + sin a cos 6^ cos (0„ — jS), pw^ = - sin a sin 6^ sin („ - ^) ; •■• P ("o sin ^0 + "o cos e^ = sin a. cos („ - ^S), p (m„ cos 6^ — ?;„ sin 6^ = cos a ; the last three of these equations give readily the values of the constants : and the equation of the tangent plane is M = (m„ cos 6^ — v„ sin 0^ cos 6 + (m„ sin 0^ + v^ cos 0^) cos (^ - <^„) cos + w„ cosec 0^ sin ((f> — <^„) sin 0. This equation can also be written in the form -^ = ^ K (sin ^0 cos 5 - cos 0„ sin ^ cos (^ - ^„)}] - -j-ifi cosec sin ^ sin {6 — (b). 314. To find the perpendicular distance from the pole upon the tangent plane. This may be obtained from the first three equations of the last article by squaring and adding, whence / (V + «„'+«'o' cosec' ^) = 1, 236 or. POLAR EQUATION 315. The following method serves to shew the geometrical signification of the partial differential coefficients, and may be useful as an exercise. Let P be the point of contact, PB a tangent line passing through OZ, and PQ a tangent line in the plane through OP perpendicular to the plane POZ, take JR and Q points very near toP; in OQ, OB take Op= 0P= Op', then ^ = r sin ^i^ and Pj>' = rd0 ultimately, and Qp,—Bp' are respectively the values of dr due to changes of 6 and ^ considering the other constant, .■.*-f-c.OPS, and -J^ = -^ =- cot OPQ. r sm Od^ Pp Draw OF perpendicular to the tangent plane QPB, and on a sphere, whose centre is P, let aSjS be a spherical triangle, with its angular points in PQ, PO, PB, join S7, 7 being the intersection of PY and ayS, then &y is perpendicular to a.^, and aSyS is a right angle. Hence cot ah = cot 87 cos ahy, and cot ;8S = cot Sy sin a87 ; POLAR PLAN^ AND POLES. 237 cot' aS + cot" /8S = cot' Sy = ?^^^ ; 1 11 fdrV 1 fdrV Fohxr Planes and Poles. 316. To find the locus of the points of contact of tangent lines, drawn to a given surf ace from a given point. Let ^ = F{x,y, z) =0 be the equation of the given surface, and a, /8, 7 the co-ordinates of the given point. Then, if (x, y, z) be one of the points of contact, the tangent plane to the surface at {x, y, z) must pass through (a, P, 7). This gives the condition which, combined with the equation of the surface, determines the required locus, or the curve of contact. It has been shewn (Art. 283) that for points on the surface x-^ + y-^ + z-^isoi the degree n — 1, or lower by unity than the degree of the given surface ; therefore by combining the above equation with that of the surface we obtain an equation of the (« — 1)* degree. The curve of contact for any conicoid is therefore a plane curve, and it is obvious that the equation of this plane is always real, whether the points of contact be real or imaginary. 317. Dep. The polar plane of a given point with respect to a given conicoid is the plane on which lie the points of contact, real or imaginary, of the tangent lines drawn from the point to the conicoid; and the point from which the tangent lines are drawn is called the pole of the plane. 318. To find the equation of the polar plane of a given point, with respect to a given conicoid, in three-plane co-ordinates. 238 POLAR PLANES AND POLES. Let ao^ + i/ + cs' + 2a'ya + iVzx + Idxy + 2a"a! + ^V'y + 2o"s + «? = be the equation of the conicoid, (a, ;8, 7) the given point. Then, by the last article, the equation of the polar plane is a {ax + cy + Vz + a") + ... + a"a! + Zi"2^ + c"^ + d(f> d d d da _d$ _dy _ dS I m n r ' which determine the ratios of the co-ordinates. The equation of the pole of a given plane (o , ^ dther plane cwrve. The curve of intersection of two conicoids is met by an arbi- trary plane in four, real or imaginary, points. Two of these will lie on the plane curve which is, by hypothesis, the partial intersection of the conicoids. Their remaining partial intersec- ENVELOPIN* CONES. 241 tion will therefore be met by an arbitrary plane in two points, and will therefore be a plane curve. It follows from this that if a proper conicoid can be described containing two plane curves of the second degree, an infinite number can be so described. For the two planes may be con- sidered as one conicoid, and the two curves are therefore the intersection of two conicoids, whence, by Art. (258), an infinite number of conicoids can be drawn containing them. Making the two move up to coincidence, we see that an in- finite number of conicoids can be described touching another along a given plane curve. The equation of any of these coni- coids, containing only one parameter, (Art. 258), will be deter- mined if we make it pass through a fixed point. Hence if we take a point, and its polar plane with respect to a conicoid, the only conicoid which can be drawn, passing through the point and touching the conicoid along the curve of intersection with the polar, will be the corresponding enveloping cone. Enveloping Cone. 325. To find the equation of the enveloping cone of a given conicoid, whose vertex is at a given point. I. Three Plane Co-ordinates. Let F{x,y,z)=F^{x,y,s) + '2F^{x,y,s)+F, = Q be the equation of the conicoid, (a, ^, y) the given point. Then the equation of the polar of the given point will be aF' {x) + ^F'{y) + yF'{z) + 2 {F, {x, y, z) + F,] = 0, Art. (318). The general equation of a conicoid touching the conicoid along the curve of intersection will be Fi?!, y,z)=h [aF'ix) + ^F'{y) + ryF'{z) + 2F, {x, y, z) + 2F,}\ If this be the enveloping cone, the point (a, ^, 7) must be a point on it, that is F{a, /3, 7) = * {2i^. («> A 7) + 4^. («, ^, 7) + 2F,Y = ih{Fioi,^,y)r, 242 ENVELOPING CONES. whence the equation of the cone is iF{a, ^, y) F{x, y, z) =. [,xJF'{x) + ^F\y) + yF'{z) + 2F, {x, y, z) + ^F,\\ II. Tetrahedral, or Quadriplanar, Co-ordinates. Let F{a, jS, ly, 8) = be the equation of the conicoid, («„, /So, 7„, So) the given point. The equation of the cone will then be F{a., /3, 7, S) = ;fc {aF'(«») + /3i^' W + V^^'W + Si?" (S.)}', i being determined as before. Hence i^(a., A, 7., «o) = /t {«, 7o. So) F{a, A 7, «) = {aF'ia,) + ^F'ifi,) + yF'iy,) + BF'{S,)}\ ASYMPTOTIC LINIB AND PLANES. 243 Asymptotes. 326. Def. a straight line is an asymptote to a surface when it meets the surface in two points at least at an infinite distance, while the line itself remains at a finite distance. Def. a plane is an asymptotic plane, when in it an infinite numher of asymptotic lines can he drawn to the surface. Def. a plane is a singular asymptotic plane when all straight lines drawn in it are asymptotic lines. Def. An asymptotic surface is the surface which is en- veloped by all asymptotic planes to the surface. 327. To find the asymptotic lines, planes, and surface to a given surface. Let = Fi^, y,z) =0 be the equation of a surface, (a?„ , y„, «„) a point in an asymptotic line, 5 = 3. — ^o = . " == r, its equations ; and let F{\, /*, jf) be arranged in a series of homogeneous"*- function^ of degrees n,n—l, ... so th^t F{\, fi,v) = 4>„+^„.i + .-- 4>r- The points in which the line meets the surface are given by ■the equation Fix^ + Xr, y„ + iir, z„+vr') = Q, d d d or, if D denote the operation ^^o ^ + ^o jr + ^'o ^ : r".^„ + r"-' (I><^„ + <^^,) + r"-^ (-D'n.. + ^ .^^J + . . . = 0, and, since two of tie roots are in,finiie, (^„ = 0, and ^,^, + Z»sE>„ = 0, E2 244 ASYMPTOTIC LINES AND PLANES. ^„ = denotes that the line is parallel to a generating line of the cone F„ {x, y, z) = 0, and ■^^»+^-=('«o|; + yo|^ + «o|)«^» + <^«-x = o...(i), is the equation of a plane in which the asymptotic lines in any direction must be drawn, since the condition shews that the line lies entirely in the plane. This is therefore the asymptotic plane containing the lines in the direction corresponding to any solution of ^„ = 0. Since the plane (1) is a function of \, fi, v, which are con- nected by the relations <^„=0, and \*+ /*'+ v^= 1, the asymptotic surface may be found by eliminating X, fi, v, and dX, dfi, dv between the equations (1), ^„ = 0, and the diflferentials of the three equations. If (»„, y„, «„) be an arbitrary point, the equations ^„ = 0, and (1), determine n{n — l) directions in which asymptotic straight lines can be drawn through the point. 328. To find the asymptotes of a central conicoid. Let the equation of the conicoid be aa? + hy^ + cz^ = 1, the directions of the asymptotic lines are given by a\' + hfi''+ci/'=0 (1). The equation of the asymptotic plane containing lines whose directions are (X, /i, v) is d\x^ + bfiy^ + cvz^ = (2). To obtain the asymptotic surface, we have the above equations and the equations in the differentials, ax^ d\ + hy^ dfi + cz^ dv = 0, a\ d\ + b/i dfi + cv dv = 0, X (?\ + /i d/i + V dv = 0. ASYMPTOTIC LINHB AND PLANES. 245 Eliminating by indeterminate multipliers ax^ + AclK + ^ = 0, hy^^■Ah^L^rB^L = (), cSj + Acv + Bv = 0. .: by (1) and (2), 5=0, and ^ = ^=?°; X fJb V .'. by (1) , axo" + %o' + c^o' = 0. whicb is the asymptotic cone, every tangent plane to which is an asymptotic plane, lines in which parallel to the line of contact meet the surface in two points at infinity, viz. the points in which it meets the parallel generating lines, which are the lines of intersection of the surface and the plane (2). . 329. To find the asymptotic lines of closest contact in any asymptotic plane. If the surface be of a higher degree than the second, we can determine generally lines which meet the surface in three points at an infinite distance, the conditions that this should be the case are <^„ = (1), ■O^« + <^«_i = (2), and i)V„ + 2i?<^^, + 2(^^, = (3). For any set of values of \ : /i : v, which satisfy the equation (1), the intersection of the surface (3) with the plane (2) is the locus of the points through which the corresponding asymptotic lines have closer contact with the surface. Since, if we write a3„ + \r for Xq, &c., and let ^^^dK + f'dj. + 'dii' J)'„ becomes (Z> + rAy„ = D'„ + 2rA (i)(/)„) + »-=A'(^„ = D^„ + 2r . {n-1) D<}>„ + r\n . {n-1) ^„, and D^i becomes {D + rA) ^^^ = D^^ + r . (n - 1) ^,^i , 246 ASYMPTOTIC LINES AND PLANES. the equations (3) and (2) are simultaneously satisfied for all values of r, i. e. they intersect in two straight lines which are the asymptotic lines required, 330. If we applied this to the case of the conicoid the equation (3) would he identical with the equation of the surface, and the two particular asymptotic lines would be the generating lines in which the plane cuts the surface, since a line which meets a conicoid in these points lies entirely in the surface. 331. If we take as an example the surface x* — ifi? — 2a^yB = 0, the equations become \*-/tV = (1), 2x^^-IJ.v{nZa + vyQ)=(i (2), 6a;oV-2yo3o/*»'- (A'«o + »'yo)'' = 2aV (3), or, \' = ±fiv, by (1); .-. 2xq\ T (jjlZo + vi/o) = 0. Hence for the intersections of (2) and (3), 3QiSio+ vyoY - ^yo^o/*" - 2 0[i»o + vyoY = ^aV" ; ,'. fiZg —vi/q=± 2a\, if X" = /tw, and (3) is evidently reducible to 12aroV + (/*»o - ^^o)' - 3 (/i2« + vi/oY = ± 4aV, which represents a hyperboloid of one or two sheets. If we take \' = - fiv, we obtain imaginary asymptotic lines. 332. Mr Walton* has defined an asymptotic plane as a plane touching a surface at an infinite distance and passing within a finite distance from the origin of co-ordinates. If the asymptotic plane correspond to an ordinary tangent plane, the tangent lines drawn from different points in the plane to the point of contact become parallel when 4;his point moves to an infinite distance. Hence the ordinary asymptotic plane is • Cambridge and Dublin Mathematical Journal, Vol. iii. p. 28. ASYMPTOTIC SURFACES. 247 one in which all lines drawn in a certain direction meet th? sur- face in two points at an infinite distance. As in the tangent plane there are generally two directions, real or imaginary, in which the tangent lines pass through three consecutive points, so among the parallel asymptotic lines there are generally two lines which possess the property of Art. (329). A singular asymptotic plane is one which touches along a line at infinity, if considered as a limit of a tangent plane ; and, if considered as the locus of asymptotic lines, it is a plane such that lines drawn in any direction in it meet the surface in two points at an infinite distance. The analytical condition is ohtained by considering that for some solutions of 0„ = the equation Z>^„ + ^^^ = must give an equation which is independent of the values \, fi, v. Thus for the surface xyz = a', X/iv = 0, and x^v + y^\ + z^fj. = 0, we have three singular asymptotic planes x^ = 0, y„ = 0, and z^ = 0, which are the only asymptotic planes. 333. Asymptotic surfaces which have a contact qf a higher degree than the second. If any relation which satisfies ^„ = 0, makes D^^ + jt^^ = independently of any relation between x^, y^, «„, we may find a locus of straight lines drawn in the direction corresponding to that relation which shall pass through three points at an infinite distance : the equation of the locus is which, the reasoning of Art. (329) shews, is the equation of a cylindrical surface or of two planes. The existence of such asymptotic surfaces shews that there is a singular point at an infinite distance. Thus in the surface whose equation is z{x + yY-a{x'-f)^Vz = 0, 248 ASYMPTOTIC SURFACES. we have the equations .;(\ + ,.)= = (1), {2v{x, + i/,)+eoQ^ + (')-a{X-/i)]{\ + fi)=0 (2), 2^0 K+2'o)(>»'+M)+''(a'o+yo)'-2a (a!„\-2^^) + &V=0...(3), r = gives the asymptotic plane «o('^ + /*)-«('^-/*)=0, and the particular lines one of which is at an infinite distance. \+ fi = satisfies (2) identically, and the asymptotic surface is V [x, + y,T - 2aX (a;„ + y,) + Vv = 0, which gives two planes if a'X, = or > J", and it is easily seen, that the straight lines in which these planesmtersect the plane lie entirely in the surface. 334. If the equation Z)^„ + ^^j = be of such a form that the terms involving x^,y„, z^ disappear, this indicates that the tangent lines for those particular directions are at an infinite distance ; in this case we can find an asymptotic surface. For let V, /*', v be values of \, /*, v which are near those which make ^„ and ^^^ = 0, and reduce i)^„ to zero identically, and suppose ^/ to be the factor of ^„ which gives rise to a factor ^, in Z)^„ ; then at the points in which 05 _ y _ a _ \— 7 — —7— -7 " JU V meets the surface we have, ^' being the corresponding value of 0, ,^„'r" + f^,r"- + ...=0, or <^'„^.^;V»+f,^,r»^+... = 0; therefore at an infinite distance. N0EM4f.S. 249 ■which reduces- to form ■which is the equation of the asymptotic surface. Thus z{x + yf - as» + Ix^ = 0, being the surface, v (V + ^y r" + av'V + 5XV = 0, when X', /jf make V + fi nearly = ; .-. v'i\' + fi,'yr'-av"r = 0, and the equation of the asymptotic surface is i/ {x + i/Y - av'z = 0, or (a; + yf — az = 0, a parabolic cylinder. « Normals. 335. Def. The normal to a surface at an ordinary point of a surface is the straight line drawn through the point perpen- dicular to the tangent plane at that point. Def. The normal cone at a singular point of a surface is the locus of the normals to the tangent planes to the conical tangent at that point. 336. To find the equations of the normal at any point of a surface. Let the equation of the surface be F {x, y, a) = 0. The equation of the tangent plane at any point, which is not a singular point, is {x - x„) F' {x,) + {y-y„)F' (2,„) + (a - a„) F'{z,) = 0. The direction cosines of the normal are proportional to F'ix,), F'{y„), F'{z,). Therefore its equations are x-x„ y-yo _ g-go riwJ-F'iy,) F\zy 250 NORMAL CONE. which is a determinate line, except in the cases in which F\x:) = 0, F'iy,) = 0, and F' ^ = 0. 337. If the surface be given by the equation z =f{x, y), the equation of the tangent plane is and the equations of the normal are «'-'^o + Po(»-»o) = 0, and y-y, + q,{z-z^=0. 338. To find the equation of the normal cone at a singular point of a surface. Let ^ = F{x, y, z) = be the equation of the surface, and let u,v,w,u,v, and w be the values of -j^, -j-^, -yi , -^-V) dor ay dz dydz d'd) , d'A ,^, . , . , d^ ^°^ d^y ^* *^^ '"^^^"^ P°'^* (^o' y-* ^«)- Any of the sides of the normal cone is perpendicular to each of two consecutive tangent lines to the surface at the singular point, or the normal cone is reciprocal to the tangent cone. Let X, ft, V be the direction cosines of a side of the normal cone, and I, m, n those of a tangent line at («„, y^, z^ ; .-. fu + m*« + w'w + lumn + 2«'wZ + iw'lm = 0, ?+m' + »'=l, and ?X + ?»/* + mi* = (1), and similar equations are true for the next consecutive tangent line; .*. ^u-^-mw ■\-nv)dl-^ =0, ldl-\- =0, \JZ+ = 0, hence, employing the arbitrary multipliers A and jB, we obtain lu-\-mw-\- nu'4- ^? + 5\ = 0, Iw -\- mv-\- nu + Am + B/i= 0, and Iv + mu + nw + An + Bv = ; NOEM^S. 251 therefore multiplying by I, m, n and adding, the equations (1) give ^ = 0, hence, writing vw - «'" =p, v'w - uu' =p', &c., I {uvw + ^u'v'w' - uu"' - vv'^ - ww'^) +A{\p+ iir' + yg') = 0, also Z\ + m/i + Mi/=0; .-. \> 4- /** 2 + v^r + 2[Lvp' + 2v\g' + 2X/ir' = 0, and the equation of the normal cone is therefore p{x-x,y+ + 2/(y-y,)(a-0„) + ...=O. 339. CoE. The condition that the normal cone should re- duce to two planes is =p {qr-f"") + r {pg^ - rr) + c^ {p'r' - q^) = 0, but qr —p'^ = (uw — v") {uv — w'^) — {v'w' — uu'Y = u {uvw + 2u'v'w' — mm" — v'w' — w'w") = uT suppose. Similarly, p'q—rr' = w'T, a,nd. p'r —qq' = v'T; .-. N= {pu + r'w + q'v) T= T ; therefore when N= 0, T= 0, or the tangent cone degenerates into two planes, as it ought to be, from the nature of the nor- mal cone. 340. To Jind the normal to a surface given hy the tangential equation. Let the equation of the surface be F{a, jS, 7, S) = 0, and let (a„, jSj, 7„, SJ be a tangent plane, also (a, /S, 7, S) a plane contain- ing the normal ; this is perpendicular to the tangent plane, and by Art, (276) the condition of perpendicularity is 252 NOEMALS. where K P^ P> P, and similarly for /ti„, j;^, p^. Since this plane also contains the point of contact aF'ioi,) + fiF'ifi,) + 7i?"(7o) + ^F'{S,) = (2). The equations (1) and (2) determine the normal. The equation (1) gives the direction of the normal since it represents a point at infinity in the direction perpendicular to the tangent plane, see Arts. (107), (110). 341. To find the ntmiber of normals which can be drawn from a given 'point to a surface of the w"" degree. Let F{x, y, e)=(i he the surface. The number of normals will be the same from whatever point they be drawn, the num- ber will therefore be found by investigating the number of nor- mals which can be drawn from a point at an infinite distance, which we may assume in Ox produced. The number will therefore be equal to the number of nor- mals parallel to Ox, together with the number of normals to a plane section at an infinite distance. If (as„, y^, e^ be the foot of a normal parallel to Ox, F'{y^) = 0, F'{z^) =0, which combined with the equation F{x^, y„ s„) = gives n.{n — 1)* solutions. Again, any plane section of the surface will be of the w* degree, and the number of normals drawn to any curve/ (a;, y) = of the n**" degree is, in like manner, the number of normals par- allel to Ox, together with the normals which can be drawn at points at an infinite distance, whose number is n; now, the number of normals parallel to Ox are given by the number of solutions of/' (2^„)=0, and/(fl5„, y„) = 0, which are n.{n-l), hence, the number of normals to the plane section at an infinite distance is n\ Therefore, the number of normals which can be drawn to the surface from any point = n . n-lf + n^ = n''-n^ + n. TANGENT PLANE%AND NORMALS. 253 342. Tangent plane of a central comcoid. The equation being ax" + by^ + cz' = 1, that of the tangent plane at {x„, y„ z^) is ax^x + ly^y + ca„a = 1, and if Z, m, n be direction cosines of the normal to this plane I m n V( !l. + '^ + ^ \ a b c therefore the equation of the tangent plane may be written in the form //r m" n^ h> + my + nz=^i^- + ^ + -j. 1. For the ellipsoid x" «" a" , ^2 + J, + ^2 1, ^0 ,, 3/0. ■ ^0 _■! T 78 "T 2 -l. The tangent plane a^ + j» + c» ~ meets the ellipsoid when jx-xf (y-yf {z-z,T _ a' "*■ 6^ ■*■ c» "" ' or in the single point (a;„, y„, aj. 2. For the hyperboloid of one sheet 2 3 B ^8 + ja ^2-1, a' + jii c' ~ The tangent plane o» "^ 6« c' ~ 254 TANGENT PLANES AND NORMALS. meets the hyperboloid in points which satisfy the equation " (a^)-=(i^)'; or in two straight lines determined by the equations xy,-yx, _ ^ z-z„ ab ~ c ' and ^«+l^_£?o = i 343. Tanffent plane and normal to the paraboloid. The equation being that of the tangent plane is {x-x^x, , {y-y„)yo _ g-g„ those of the normal are g' {x - X,) ^ y (y ~y„) ^ cje- x,) Xo ±y» -1 {x - a!„) X, + [y - y„) .y„ + (g - g„) 2g„ a" - 5" c therefore, the equations of the normal may be written a'(a!-a;o) _. 6'(.y-y,) _Q x<, yo ' and a!„ (a; - x^ iyAy" ^o) + 2«o (« - go) = 0. TANGENT PLANES ^ND NORMALS. 255 The last equation shews that the normal to a paraboloid at any point lies in the tangent plane to a surface whose equa- tion is drawn through the same point. 344. Tangent plane and normal to the Jielicoid. The helicoid is generated by the motion of a straight line, subjected to pass through a given axis to which it is perpen- dicular, and about which it twists with an angular velocity pro- portional to the velocity of the point of intersection. The equation of the helicoid is s = ctan~ — , if the axis of z be the line to which the generating line is per- pendicular, cif _ ex The equation of the tangent plane at (»„, y„, »J is i^o + y^) (« - «o) = c (- ^Vo +.^a;o). and the equations of the normal are x-x^ _ y — y<, _ c (g - gp) -y<,~ a-o x.^ + y^'' The tangent plane cuts the surface in the points which satisfy the equation {a>« + y^ (g - go) = c»(,a! (^tan 2. - tan ^ j , or in an infinite number of straight lines, one of which coincides with the generating line. The normal is a tangent line to a circular cylinder whose axis is that of the surface and which passes through the point at which the normal is drawn. 345. Tar>gent plane to the anchor-ring. Iiet the plane containing the centers of the generating circles be taken for the plane of xy and the axis of rotation for -fee axis 256 TANGENT PLANES AND NORMALS. of z, and p be the distance of any point {x, y, z) from the axis, c that of the center of the generating circle, a its radius, fi'=x^ + y\ and a' + (p - c)' = a\ The equation of the anchor-ring is (a;'+y» + s' + c=- a)»- 4c' (x' + 3/') = 0. The equation of the tangent plane at (a;„, y,, z^ is K (35 - a!o) +yo (2^ -^o)} (Po - c) + 2„ (a - »„) /»„ = ; or, (Po - c) {x^ + y„y) + /)„«„3 = p/ (p„ - c) + p„V To find the form of the curre ^^^in figure, page 213, let Xf^ = c—a, yo = 0, Zq = Q, pQ—c — a, then the equation of the tangent plane is x = c—a, and the form of the curve of intersec- tion is given by the equation \f + z' + 2c{c-a)Y=iu'\ wu>v'\ MU> w", (Art. 225). Example. The surface whose equation is will be found to have a conjugate line in the plane ofys. 258 CONE WHOSE BASE IS A FOCAL CURVE. 342. If a cone be descrihed with any point of a conicoid as its vertex, and a focal as its base the normal to the conicoid at the vertex is an axis of the cone. Let the equation of the conicoid be S 3 2 a' b" c' ' and let x'y's' be the co-ordinates of P the vertex ; let a tangent line through P meet the dirigent cylinder in E, E', and let F, F' be the foci corresponding to the directrices through E, E'\ and Q, Q the feet of these directrices. Then, since EE' passes through a fixed point P, QQ will pass through a fixed point {x'y') the projection of P, the tangents at Q, Q' to the dirigent conic will intersect on the polar of x'y' with respect to the dirigent conic, that is, on the straight line whose equation is 2 2 12 2 a —c , h' — & , , . — ^fl;a; +— ^^y =1 (1), Now, since F, F' are the poles, with respect to the trace of the surface on xy, of the tangents at Q, Q, FF' will pass through the pole of (1) with jespect to that trace, or through a point whose co-ordinates are But, the equations of the normal at P being ' x — x' y—y' g — g' ^ y g|~' ^ a" V c" it appears that FF' always passes through the point where the normal meets xy. Also, by Art. (207), EPE' makes equal angles with FP, F'P, hence the normal at P, being perpendicular to EPE', makes equal angles with FP, F'P. That is, any plane through the normal will cut the cone in two straight lines making equal angles with the normal, or the normal is an axis of the cone. The two other axes of the cone will be normals to two other conicoidfl through the point confocal with the given one, and the PROBIiiMS. 259 axes of the cone being mutually at right angles, the confocal surface will also cut each other at right angles. Hence, through any point may he drawn three surfaces having a given focal curve, and these surfaces will cut each other at right angles at all points of intersection. These properties may readily be proved otherwise. Changing the signs of a*, h^, ^ we obtain the theorem for all central conicoids, and the non-central surfaces being the limits of these, the proposition will be equally true for them. XV. (1) The tangent planes to an ellipsoid at points lying on a plane section -will intersect any fixed plane in straight lines which touch a conic section. (2) The locus of the intersection of two tangent planes to the cone a h c which are at right angles, is the cone (6 + c) a;' + (c + a) y' + (a + 6) «^ (3) Find the equation of the tangent plane upon the principle that no other plane can pass between it and the surface in the neighbourhood of the point through which it is drawn. (4) If two planes be drawn at right angles to each other touch- ing the central conicoid av? + hi^ + c^ = 1, and having their line of intersection in a given direction (Z, m, w) ; shew that the locus of their line of intersection is the right circular cylinder a;" + y" + «r = («« + my + wa) + + — v — H • (5) If the non-central conicoid a be taken, the locus is 2,l(}x + my + nz) - 2x = a{n'' + r) + h (P + m'). S2 260 CONE WHOSE BASE IS A FOCAL CURVE. (6) The locus of the intersection of three tangent planes to the conicoid oa;' + ty* + ca* = 1, which are mutually at right angles, is . , , 1 1 1 a c and to the conicoid «'«'„. a + b - +-r =2x, IS X= a — . a b 2 (7) The locus of the intersection of three tangent lines to the eUipsoid x' y' a? , a'^b'^ c'~ ' mutually at right angles, is (6" + c') a;' + (c" + a") y' + {a' +b')i^ = J V + c V + a'b'. (8) If p, p' be the perpendiculars from the center on parallel tangent planes to two confocal conicoids, p'-p" is a constant quantity. (9) If three conicoids be drawn, through a given point (suV*'), confocal -with the ellipsoid x' 7/" ^ , L ^ 1 — 1 a' b' c' ' the locus of the intersection of three tangent planes to them, mutually at light angles, is the sphere x' + y' + sl' = a/' + y" + s/'. (10) If three planes be drawn, mutually at right angles, and each passing through a tangent line of a plane curve of the second degree, the locus of their intersection is a sphere. (11) The tangent plane to the surface xyz = a? cuts off a tetra- hedron of constant volume from the co-ordinate planes. (12) If two surfaces of the second degree have two common generating lines of the same system, they will have two other common generating lines, and touch each other in four points. (13) The tangent plane to the surface whose equation referred to tetrahedral co-ordinates is iPy + mr/a + mo)8 + ToS -I- m'fiZ + w'yS = 0, at the point .4 is l'h+n^ + mr/=0. PROBLEMS. 261 (14) If the tangent planes at A, B, G, D form a tetrahedron cAcd, find the equations of the lines Aa, &c., and shew that they ■will meet in a point if W = rrnn' = nn'. (15) If ^a, JBb intersect, then also Cc, Ddvill intersect. (16) Shew that the surface, whose equation is mn/ly + nlya + lma/3 + IraZ + Tur/SS + nryS = 0, satisfying the conditions of (14), can never be a ruled surface ; and that it will be an elliptic paraboloid, if 1111111111 -5r+— i+-5+-5= + -j+ r- +T-+ + . t m n r mn m Itn Ir mr nr (17) The straight lines in which the tangent planes at A, B, C, D to the surface in (16) meet the opposite faces of the fundamental tetrahedron will lie on the plane, la + m/3 + ny + rS = 0. (18) If Aa, Bb, Cc, Dd meet the above surface again in the points a', b', o', d', the tangent planes at A, a' and the plane BCD intersect in the same straight line, and the four straight lines so determined lie on the plane ^a + m/S + ny + rh = Q. (19) In the surface la^^myB which passes through the edges BC, CA, AB, BB, find the points in AG, BB, and in BG, AB, at which the tangent planes are parallel, and thence shew that the center of the surface Hes on the line joining the middle points of AB, CB. (20) This surface will be a paraboloid if l = m. (21) If the straight line joining the middle points of AB, GB meet this surface in P, Q, ibhe tangent planes at P, Q are parallel to AB and GB. (22) The surface la' + mP' + ny'.+ rB' -0, will be a paraboloid, if -H 1 H_ = 0, and will be elliptic, or hyperbolic, according I m n r as Imnr is negative or positive. (23) If this condition be satisfied, and if a, b, c be the middle points o{BA, BB, BG, a', V, d oi BG, GA, AB, this surface will touch the planes b'c'a, c'a'b, a'b'c, ahc ; and also the three planes bch'd, coda', aha'b' ; and the points of contact of the former are the angular points of a tetrahedron whose faces intersect the corresponding faces of ABCB in four straight lines lying in one plane; and this plane passes through the points of contact of the latter, and is parallel to the axis of the paraboloid. (24) The surface lay + maS + n^y + r^SS = 0, of which the edges AB, GB are generating lines, will be a paraboloid^. 262 CONE WHOSE BASE IS A FOCAL CURVE. i£l + r = m + n; and i£' l + m + n + r = 0, the straight line joining the middle points of AB, GB will lie on the surface. (25) If two arbitrary points be taken on each of four straight lines meeting in a point, the only conicoids which can be described through the eight points are cones, or combinations of planes. (26) Investigate the condition that the general equation of the second degree may represent a cone from the consideration that every plane will have the same pole with respect to it. (27) is a fixed point, P a point such that the polar planes of 0, P with respect to a given conicoid are at right angles, shew that the locus of 1 is the plane diametral to all chords of the conicoid perpendicular to the polar plane of 0. (28) In any conicoid passing through the sides of a quadrilateral ABGD in space, the polar plane of the center of gravity of the tetrahedron ABCD will be parallel to AG and BD. (29) The polar plane of any point on a directrix will pass through the corresponding focus, and the line joining the point to the focus will be at right angles to the polar plane. (30) If be a fixed point on a conicoid, OP, OQ, OR any three chords mutually at right angles, the pole of the plane PQP wiU lie on a fixed plane. (31) The surface, whose equation is — h^ + — +^=0, has a a , p y o tangent cone at each of the angular points of the fundamental tetra^ hedron. Any two of these cones have a common tangent plane, and a common plane section containing the edge of the tetrahedron oppo- site to the common generating line : also the six planes of these common sections meet in a point. (32) The points on a conicoid, the normals at which intersect the normal at a fixed poiat, lie on a cone of the second degree, having its- vertex at the fixed point. 33 V (33) From different points of the straight line - = t,«=0, . a^ y" s!" asymptotic straight lines are drawn to the hyperboloid ~s + « + t = 1 J shew that they wiU all lie in the planes r = =*= — J2. a c (34) Find the tangent planes to the two surfaces (1) {x' + y' + ^'=ia'{a^ + y^; (2) b'z={x' + yy-a'{x' + i/^i ■which touch them along a curved line. PROBLEMS. 263 (35) Find the tangent cone at the origin to the surface (x' + 1/^ + axy-{c' -a'){x' + sr) = 0; and shew that as a diminishes and ultimately vanishes, the tan- gent cone contracts, and ultimately becomes a straight line, and as a increases up to c, it expands and finally becomes a plane. (36) Shew that the asymptotic planes to the surface z{x' + y') - atxi' - by'= 0, are parallel to the plane of xi/, and that the locus of straight lines in these planes having contact of the second order at infinity is » = a; ora = 6; and that the axis of s is an evanescent asymptotic cylinder. (37) If a globe be placed upon a table, the breadth of the elliptic shadow cast by a fixed luminous point is independent of the position of the globe. (38) If an ellipsoid having its least axis vertical, be substituted for the globe, determine the condition of the shadow of the globe being circular. It may be shewn that the locus of the luminous point must be an hyperbola, and that the radius of the circular shadow is independent of the mean axis of the ellipsoid. CHAPTER XV. METHOD OP RECIPROCAL POLARS, 343. If we take any plane passing through a given point, the polar line of the point with respect to the section of a coni- coid by this plane will be the intersection of the polar plane and the plane of section; for the points of contact of all tangent lines, real or impossible, from the point to the conicoid, lie on the polar plane, and therefore the points of contact of the two tan- gent lines drawn from the point of the section lie on the inter- section of the polar plane and the plane of section, which is therefore the polar line. Hence, any straight line through a point will be harmonically divided by the conicoid and the polar plane of the point. If P be any point, and Q any point on the polar plane of P, then, taking any plane section through PQ, since Q is on the polar line of P, P is on the polar line of Q, and therefore on the polar plane of Q. Conversely, any plane passing through P will have its pole on the polar plane of P. Hence, any plane passing through PQ will have its pole on the line of intersection of the polar planes of P and Q, and the polar plane of any point lying on PQ will pass through the same line of intersection. The relation between PQ and this line of intersection being thus reciprocal, the straight lines are said to be reciprocal to each other with respect to the conicoid. It is manifest that reciprocal straight lines cannot intersect unless both are tangent lines to the conicoid. If a cone be described, with vertex P, meeting the conicoid in one and therefore (Art. 324) in two plane sections, these planes will intersect in a straight line lying in the polar plane of P; for any plane through P intersects the two planes in two straight lines which meet in a point on the polar line of P, and therefore on the polar plane of P. EECIPEOOAL POLAES. 265 Let Q be the pole of the plane through P and the line of intersection of the two planes, then by taking plane sections through PQ, it is obvious that a cone can be described with vertex Q meeting the conicoid in the same plane sections. If therefore we take any two plane sections of a conicoid, there will generally be two points lying on the straight line reciprocal to the intersection of the planes, with which, as ver- tices, two cones may be described containing the plane sections. If the two planes be parallel, their line of intersection being at infinity, the reciprocal straight line will pass through the center. Of two, reciprocal straight lines, one will always meet the conicoid in real, and the other in impossible points, the tangent planes at the two real points passing through the reciprocal sCraight line. These reciprocal properties are particular cases of the general method of reciprocal polars, to which we now proceed. 344. Reciprocal points and planes. Suppose a system A of planes such that, corresponding to every plane P of the system, is determined one, and only one, point p' of a system A' ; then, if (a',)S', 7', S') be the co-ordinates of p, the equation of P must be of the form (a^a' + Jj/S' + cy + d,Z') a + («/ + . . .) ^ + (a^a' + ...) 7 + Ka' + •••) ^ = ; a„ Jj, Cj, , then since all the tangent planes to 8 pass through p, all the points of 8' will lie on the plane P', and since any tangent to 8 has an infinite number of points of contact, there will be at each point of /8" an infinite number of tangent planes. 8' will therefore in this case be a plane curve, every point of 8' corresponding to a tangent plane to ;S^, and every tangent line to 8' corresponding to a generating line of 8. If 8 be any developable surface, every tangent plane to 8 will have an infinite number of points of contact lying in a straight line, and accordingly at every point of 8' will be an infinite number of tangent planes intersecting in one straight line. 8' will therefore in this case be a curve of double curva- ture, a tangent line to 8' corresponding to a generating line of 8. Also, it is immediately seen that the developable generated by the tangent lines to 8' corresponds to the edge of regression of/S. 345. The equation found in Art. (319) for the polar plane of a given point with respect to a given conieoid shews that the relation between a point and its polar plane is a particular case of the general relation between a plane and its correspond- ing point described in the preceding article. If the point be limited by lying on a surface of the n^ degree, which is met by an arbitrary straight line in n points, the reciprocal surface BECIPKOCAL POLAES. 267 • will be of the n* class, since it will have n tangent planes, passing through the straight line which is the reciprocal of any- arbitrary straight line, and which is therefore itself arbitrary. The relation between these two surfaces is reciprocal by the properties already discussed. Such surfaces are said to be polar reciprocals, -the one of the other, with respect to the conicoid, which is called the auxiliary conicoid. If the auxiliary conicoid is not specified it is always supposed to be a sphere, and any change in the radius of the sphere not altering the species of the reciprocal surface, but only its di- mensions, one surface is in this case said to be the polar reci- procal of the other with respect to the point which is the center of the sphere. The polar reciprocal of a conicoid is always a conicoid, since a conicoid is both of the second degree, and the second class. Also, since the equation of the auxiliary conicoid involves nine constants, which will enter into the equation of the polar re- ciprocal, we may, by properly choosing the auxiliary conicoid, make the polar reciprocal coincide with any assigned conicoid, so that from any proposition which is generally true for a conicoid, we may form one true for all conicoids which are the reciprocal polars of the former, that is, for all conicoids whatever. If we restrict the auxiliary conicoid to real surfaces, this will not be strictly true, for, in that case, the polar reciprocal of every ruled surface must be a ruled surface, since to a straight line every point of which lies in the one surface, corresponds a straight line, any plane passing through which will be a tan- gent plane to the reciprocal surface, and which must therefore be wholly on the surface. Thus the polar reciprocal of an ellip- soid may be any ellipsoid, elliptic paraboloid, or hyperboloid of two sheets ; and of a hyperboloid of one sheet, any hyperboloid of one sheet, or hyperboloic paraboloid. The polar reciprocals of cones and cylinders will of course be plane cm-ves, and in the latter case the plane will pass through the center of the auxiliary conicoid. 268 EECIPKOCAL POLAES. 346. On the species of polar reciprocals of conicoids. The polar reciprocal of an umbilical surface, namely, an ellipsoid, hyperboloid of two sheets, or elliptic paraboloid, will be an ellipsoid, an elliptic paraboloid, or a hyperboloid of two sheets, according as the center of the auxiliary conic lies within, upon, or without the surface. For to any tangent plane to the surface {A) passing through the center of the auxiliary conicoid (5,), corresponds a point on the reciprocal surface {A'), its pole with respect to {B), at an infinite distance. Hence if the center of [B) lie without {A), there will be a plane section of {A!) at infinity, con-esponding to the enveloping cone from the center of {B). If the center of B lie upon A, the plane at infinity, which is the polar plane of the center of B, will touch A'. If the center of B be within A, the enveloping one becomes impossible, and the plane at infinity will not meet A' in real points. Hence, since the polar reciprocal cannot be a ruled surface, the results will be as stated. The polar reciprocal of either of the skew surfaces will be an hyperboloid of one sheet, or an hyperbolic paraboloid, according as the center of the auxiliary conic does not, or does, lie upon the surface, the reasoning being precisely similar to that for the umbilical surfaces. 347. On the reciprocal polar of a conicoid with respect to a given point. If be the given point, and a sphere be described with center 0, and radius k, then if OP be drawn perpendicular to any plane, the pole of the plane with respect to the sphere will be a point Q on OP, such that OQ . 0P= A"; since we take any plane through OP, we shall get a circle and straight line, and Q will be the pole of the straight line with respect to the circle. Hence the construction for the reciprocal polar of a conicoid with respect to a given point is as follows. Through the given point let fall a perpendicular on any tangent plane to the conicoid, and on this perpendicular take a point such that the rectangle under the whole perpendicular and the part of it intercepted between this point and the given point is constant; the locus of the points so determined is the reciprocal polar with respect to the point. We might of course equally take the reciprocal KECIPROCAfc POLAES. 269 construction, namely, draw a plane perpendicular to any radius vector of the conicoid from the given point, and at a distance from the given point such that the rectangle under this distance and the radius vector is constant ; the planes so determined will touch the reciprocal pole. The reciprocal polar of a sphere with respect to a point will be a prolate surface of revolution of which the point is the focus, and the line joining the point and the center of the sphere the axis ; for, taking any plane through this line, the section of the reciprocal polar by this plane will be a conic of which the point is the focus and the line before mentioned the major axis. The reciprocal of the polar of the point, with respect to the sphere, will be the center, and the reciprocal of the center, the directrix plane of the surface of revolution,, exactly as in two dimensions. Properties of conicoids of revolution having a common focus may be immediately obtained in this manner. These are, however, generally at once deducible from the corresponding properties of plane curves. It is shewn (Art. 210), that the enveloping cone from a point on a focal curve of a conicoid is a righ<^^!7 If we take then the polar reciprocal of the conicoid ^iA. re- spect to this point, the tangent planes to the asymptotil cone*!^?], the reciprocal surface will be perpendicular to generatin^Jiaes of the enveloping cone of the conicoid, and the asympt cone will therefore also be a right cone, or the surface will be' one of revolution. This result is of course true whether the asymptotic cone employed in the proof be real or impossible. Conversely, the reciprocal polar of a surface of revolution with respect to a point will be a conicoid of which the point is a focus. Hence, from a sphere may be obtained, by successive reci- procations, any of the umbilical conicoids, but, as before shewn, the ruled surfaces cannot be obtained in this manner. 348. 2b find the equation of the reciprocal polar of a coni- coid with respect to any point on a focal curve. Let the equation of the conicoid be a c 270 EECIPROCAL POLAES. let the given point be (a, )8) lying on the focal curve, and let the radius of the auxiliary sphere be h. Then if we take a tangent plane to the conicoid at the point (a;') y', s') the corresponding point in the reciprocal polar will be given by the equations x — a ,y — /3 z I? px' py' ~ pz'~ Tn^ a' 'W '7' p being the perpendicular from the center, and w from the point (a, ^) on the tangent plane. TT ■ •or otx' By' Mence, smce - = 1 ^--i C^ , the equations for the point on the reciprocal become x — a y— /8 a —k' V x y z" The equation of the reciprocal polar is therefore a'{x-ay + Iy'(^-0y + cV-{a{x-a)+^{y-^)+}^Y = O. 349. The reciprocal of a conicoid, with respect to a point on a modular focal curve, is a conicoid of revolution capable of generation by the modular method ; and, with respect to a point on an umbilical focal curve, is one capable of generation by the umbilical method. Removing the origin to its center, the equation will become {a'-a^7?-2aPxy+{V-^f + ^z': 1— ^. K we again transform to the principal axes, the equation will become EECIPEOCAl* POLAKS. 271 A', P being the roots of the equation a* S^ and since -j ^ + p — j = 1, we see that c" is one root, and the fit — c 0^0 other will therefore be a' + b^ — d'— (a" + /S"). Now a' + /S' is less than a^—c^, and greater than J" — c"; the second root is therefore > J' < a", and the reciprocal surface is an oblate spheroid for points lying on the modular conic. Similarly, for points on the umbilical conic, the reciprocal surface is a prolate conicoid of revolution, and will be either a spheroid, paraboloid, or hyperboloid of two sheets according as the point lies within, upon, or without the ellipsoid. Corresponding results may be deduced for the other umbilical conicoids. Hence, the reciprocal polar of an umbilical conicoid is, an oblate spheroid for a point on the modular focal, and a prolate conicoid of revolution for a point on the umbilical focal. For the hyperboloid of one sheet, we shall have, changing the sig-n of c\ as the equation of the reciprocal siu-face in its siniplest form, since o, ^ are connected by the equations -^+-^-1 Now o? + ^''>h^ + c'' = 0, i>"-'^ = 0, or by the equivalent equations i)"'-'^ = 0, Z>'"-'f = 0, ...... Z>'f = 0, are called the 1st, 2nd, . . . n - 1 1"" Polars of the surface ^ = with respect to the point (a', j8', y, S'), which is called the Pole, The particular Polar i)''"'^=0, or I>'^' = is also called the Polar Plane, and Z)"-''^ = 0, or D"'^' = 0, the Polar Conicoid of the surface. 358. When the equation of the surface is given in the common co-ordinates, as f{x, y, s) = 0, it may be reduced to ' the homogeneous form by writing - , ^ , - for x, y, z, and the 282 GENERAL THEOEY OP POLAES AND TANGENT LINES, equation of the polars will be obtained by means of the operation and in the equations D(p = 0, so obtained making < = 1 = <'. The equation t = Q will then represent the plane at infinity. If the function/(a!, y, z) be arranged in homogeneous functions of ascending degrees, as m„ + m^ + m^ + = 0, this equation reduces to the homogeneous form u/' + u^f- + u/''^ + = 0. Ifa;' = 0, y' = Q, a' = 0, iy^ = n. (n-1) ... (w-r+1) uft-'+{n-\) ... (n-r) uff^^ + ... Hence, the equation of the r**" polar of the origin is n-r , (w-r) (m-r-1) " n '■ n.{n—\) " that of the polar plane of the origin is MMj + Mj = 0, and that of the polar conicoid, - W . (W - 1) Mo + (n - 1) Mj + Mjj = 0. If n = 2, the result of Art. (318) for the polar plane of the origin is obtained with respect to a conicoid, to which the polar conicoid reduces. 359. Geometrical properties of Polars. If p, p„ be the distances of F and B^ from P', Mm ■ ^m •• P~Pm '• PmJ I)"'6' = is the locus of a point P, such that, if ^, ^, Aj \ be the n values of ^ corresponding to the intersections R^, B^... of a line P'P with the surface ^ = 0, Hence, the (w - r)^ polar is the locus of P, such that \\P PJ \P pj \P Pr))~ GENEEAL THEORY OP POLAgS AND TANGENT LINES. 283 Thus, the polar plane with respect to F is the locus of R, such that S( 1=0, \P pJ nil 1 or -=- + -+ + _ P Pi P„ Pn which gives the well-known harmonic property of the polar of a conicoid with respect to a point. 360. Connexion between Diameters and Polar Surfaces. When the point, with respect to which the polars are taken, is at an infinite distance, the condition for a polar plane becomes "Z {p — pj) = 0, or !S {PR) = ; therefore, the polar plane is a polar diametral plane corresponding to a system of parallel chords drawn in the direction of the infinitely distant point. The condition for the polar conicoid becomes tip- p,) (p - p,) =0, ovt [PR, . PR,) = 0, and the conicoid is a polar diametral conicoid for a system of parallel chords. And generally, the polar surface of any degree with resnetS to a point at an infinite distance in a given direction if the polar diametral surface of the same degree correspondii^ to a system of chords in that direction. 361. If tangent planes be drawn to a surface at the points in which a straight line meets it, and from any point Q in this line any other straight line he drawn meeting the surface in P, , Pj . . . and the tangent planes inp^,p^ ^(ip)=^(i) If three straight lines be drawn through a point Q, meeting two surfaces in the same points R^, R^, 8^, S,, and Tj, T^, R, 8, T the corresponding points in the polar planes with respect to Q are the same for the two surfaces; hence, the polar planes for these surfaces are the same. This is true when the three straight lines become ultimately coincident, in which case the surfaces touch one another at iZ^, R^ Suppose now a straight line QR to meet a surface in ^j , ^^ » • • • and at these points tangent planes to be drawn, the surface and 284 6ENEBAL THEOET OF POLAES AND TANGENT LINES. the system of tangent planes form two surfaces which have the same polar plane. Hence, if any other line through Q meet the surface in Pj, Pj and the tangent planes inp^, p^ Hhy-H^y Properties of Polar s. 862. Every polar of a surface, with respect to a given pole, is a polar, with respect to the same pole, of every polar of a higher degree than its own. For D^',j> = P'{D'^). 363. Every line,- drawn through a pole to a point in the curve of intersection of the first polar with the surface, meets the surface in two coincident ^points. For the equation Ve ^ = has two values of fi equal to zero if ^ = 0, and D^ = simultaneously. 364. If a surface have a multiple point of the m'^ degree, that point will he a multiple point of them — 1 1* degree on the first polar, with respect to any point not on the surface. Let F be the pole, B the multiple point, P any point in P'B; m values of /x : \, in the equation X'ef^ = 0, corre- sponding to the multiple point B, are equal ; hence, the equation dfju haswi-1 equal values of/i : \; i. e. the first polar of i^ = has a multiple point of the m — ll"" degree at B. This is also obvious from the property of the polar given in Art. (359), for, if a straight line through the pole meet the surface in a multiple point of the m^ degree, whose distance is p^, the points in which it meets the first polar are given by the equation :0, \P PJ\P Pm+J {--- \P Pt GENERAL THEOEY OP POLAES AND TANGENT LINES. 285 wiicli has m — 1 roots equal to p^, hence the multiple point in the first polar is of the m — l]"* degree. The r* polar has a multiple point at B of the m — r\ degree. 365. If a tangent cone on a double point of a surface he- comes two non-coincident tangent planes, the first polar touches the line of intersection at the double point. In this case, if P' be the pole, P any point in the plane through P' and the line of intersection, there are two coincident positions of P'P, such that for each position \''e \^ = Q gives two equal values of the ratio /i : \ ; and therefore one of these values satisfies the equation \''"'eA 2)^ = for each of two coincident positions of P'P; that is, two coincident points in the line of intersection on the douhle point lie in the first polar, or the line of intersection is a tangent to the first polar. Or, if /3j be the distance of the double point from the pole, since there are two equal values of p^ , one value of p in the first polar is p^, and, if the double point has two non-coinci- dent tangent planes, the line of intersection is a tangent line ; therefore, for two coincident directions the same is true, and p = Pj for two points, coincident with the double point, in the line of intersection, which therefore touches the first polar. 366. If the two tangent planes at a double point are coin- eident, the first polar has a tangent plane at that point coincident with them. For P'P intersects the surface in two coincident points for any direction indefinitely near the multiple point; hence, the first polar has a point in the plane coincident with them, not only at the multiple point, but at the adjacent points; the plane is therefore a tangent plane to the first polar. 367. If r generating lines of a conical tangent coincide, r — 1 of the conical tangent of the first polar will also coincide. For, P'P, passing near the multiple generating line through r ultimately coincident points, will pass through r — 1 ultimately coincident points of the first polar. 286 GENEEAL THEORY OF POLAES AND TANGENT LINES. 368. If a surface have a multiple line of the «i"' degree, the first polar contains the same line as a multiple line of the jn — l]"" degree. For if P be any point on the multiple line, PP has m equal values for the surface, and therefore m — 1 for the first polar. 369. The propositions of Articles 364, 365, and 366,. can be shewn directly as follows. Let the angle D of the tetrahedron be taken for the multiple point and the angle A for the point Q with respect to which the polar is taken. The equation of the surface will be dx ^' da^^' da.' /. •^^ = 0, %j = is a tangent line to the first polar. III. If the planes be coincident ^^ = •y^^, da - ^^' d^ ' .'. ■(^j = is a tangent plane to the first polar. 370. If the pole he on the surface, the polar plane will he a tangent plane, at the pole, to the swrface, and also to all the corresponding polars. For B'^ = is the equation of the tangent plane at («', /S', 7', S'). GENERAL THEORY OF POLARS AND TANGENT LINES. 287 Also, since the polar plane is the polar plane for the r*'' polar, in which the pole evidently lies, the polar plane is also a tangent plane at the pole, to the r"' polar as well as to the surface. This is easily seen also from the equation of the surface, M1 + M2+ =0, in which the origin, a point on the surface, is taken for the pole, since the equation of the r"" polar is ('-s)«.+('-3('-^)«.+ -»■ Mj = 0, the equation of the polar plane, is also the equation of the tangent plane to the surface and »•"' polar. 371. The locus of poles, whose polar planes pass throwfh a given point, is the first polar with respect to that point. The polar plane of F is i)'(^' = 0. If this plane pass through Pj , we have the equation ( aj -=-7 + . . . . j i^' = 0, therefore P' must lie in the surface, whose equation is [ a^ -^+ .... j ^ = 0, which is the first polar with respect to P^ . 372. Every plane is a polar plane corresponding to {n-\l)^ poles. Take three arbitrary points Pj, P^, Pg, in the plane, the first polars of these points are of the w — l]"' degree. The first polar of Pj is the locus of all points which are poles of planes through P„ and therefore contains all poles of the given plane ; the three surfaces which are first polars of P„ Pj, Pg, each con- tain the poles of the given plane, and, therefore, since every common point is the pole of the plane containing P„ P^, Pg, there are n — lf such poles. 373. 7%e first, polars of all points in a straight line have a common curve of intersection. The n- l~f poles of any plane through two of the points lie on the curve of intersection of the polars of the two points, and this curve must therefore he the locus of the poles of all such planes ; any point in the line of intersection of the planes 288 REDUCTION OF CLASS. must therefore have its first polar passing through the curve of intersection of the first polars of the two points taken. Such a curve is a Polar Curve corresponding to the line. Cor. 1. -• If two lines intersect, their polar curves lie on the first polar of the point of intersection. Cor. 2. If any number of planes pass through a point, their poles lie on the first polar of the point. Cor. 3. A tangent line to the surface touches its polar curve on the point of contact with the surface. On the Degree of a Reciprocal Surface, 374. The properties of the polars of a point with respect to a surface have been employed by Salmon, in the Oamhridge and Dublin Journal, Vol. Ii., to explain the reduction of the class of a surface or the degree of its reciprocal, in the case of certain singularities in the surface; and we give some of the theorems relating to this reduction in order to introduce the student to some method of dealing with the subject. 375. To estimate the ^ect of an ordinary double point of a surface upon the class of the surface, or the degree of its reci- procals. The number of tangent planes which can be drawn through a given line may be found by constructing the polar cm-ve of the line, which is the intersection of the first polars correspond- ing to any two points in the line ; the intersections of this curve with the surface gives n .n — l]" points, and a plane drawn through anyone of these points' and the given line will gene- rally be a tangent plane to the surface, since all the lines drawn from the point in that plane will be generally tangent lines. But (Art. 364), it is seen that, if there be an ordinary double point on the surface, the first polars of any kind pass through the double point, and therefore the polar curve of the line passes through the point ; hence, the lines drawn from the double point in the plane containing it and the given line, although they meet the surface in two coincident points, are not generally tangent lines. EEDUCTIOK OF CLASS. 289 Two of the planes, therefore, corresponding to the two points in which the polar curve meets the surface at the double point, are not tangent planes to the surface. The number of tangent planes is therefore dinjinished by two, for each ordinary double point of the surface. 376. To estimate the effect' on the class when the conical tangent redibces to two planes. If the tangent planes at a double point be not coincident, the first polars touch their line of intersection ; hence, to the number of coincident points of intersection of the three surfaces in the ordinary case is added one, since the intersection of each polar surface with the given surface touches the line of intersection. If the tangent planes at a double point coincide, each tangent plane contains three coincident points in the three surfaces, and the whole number, by which the class is reduced, is therefore six. 377. The surface of the third degree, whose equation is « + | + £+| = o, has four double points, one at each angle of the fundamental tetrahedron. Hence the class of surfaces of the third degree, which is in general 3 . 2"= 12, is reduced for this surface by two for each double point : the surface is therefore of the fourth class. If we reciprocate the surface with reference to a'' + /8= + 7= + S'=0, let (a', /3', 7', S') be a point in the reciprocal surface ; .-. a'a + j8'y3 + 77 + S'S = is the equation of a tangent plane to the given surface, and is therefore identical with where — , +-oir+-77 + w='J; a p 7 o 290 EEDUCTION OF CLASS. "a b ~ c ~ d ' .-. (««')» + (&/3')» + (c7')» + (rfS')» = is the equation of the reciprocal surface, which is of the fourth degree, which is therefore the class of the surface as reduced above. 378. The surface Za^Sy = 8' has double points at A, B, 0, and the tangent surface at A reduces to two tangent planes, ^ = and 7 = 0, the class is therefore reduced by these for each double point, and the degree of the reciprocal will be 3.2''-3.3 = 3. 379. The wave surface, whose equation is (aV + Vy + Zz") {a? + f + z^)- a' (J" + c')a?- V (c' + a') f has four double points, real or imaginary, in each principal plane, x' v' and if we write — for x^, — for y, &c. the symmetrical form shews that there are also four in the plane at infinity ; hence there are sixteen double points, and the degree of the reciprocal surface will be 4 . 3" - 16 . 2 = 4, 380. To estimate the effect of a dovhle straight line in a surface. The polars contain the line singly, and the number of points which correspond to the multiple line which is common to the three surfaces is 5m — 8 (Art, 272). Now if in tetrahedral co-ordinates CD be taken for the mul- tiple line, the equation of the surface may be written in the form Pa' +2Qa^ + B^=0,m which P, Q, B are of the m^V degree. There will therefore be certain points at which the tangent planes will be coincident, which will be determined by the intersection of the surface PB = (^ with the straight line, the number of points being 2 (w — 2). Now at each of these points there will be an additional point common to the surfaces, and the whole number by which the class of the surface will be diminished will be 7w — 12, REDUCTION OF CLASS. 291 381. To estimate the effect of a multiple line of the r*"" degree of multijplicity in the surface. The polars contain the line, each in the r — if^ degree of multiplicity : and the number of points which correspond to the common multiple lines is (Art. 272) (r-l)^w4-2»-. (r-1) (m-l)-2r.y-lf = (»--l) {(3»--l)w-2r'}. Now, if the line be taken for one of the edges, as OD, of the fundamental tetrahedron, the equation of the surface may be written in the form i?;(a, ^8) = Pa'H- ^a^'^/S + . . . = 0, in which the coefl&cients are of the (m — rf^ degree. The equation of the tangent planes to the surface at any point (0, 0, 7', S') will be P'a'+ QoT^^^... = 0, where P', Q... are the values of P, Q... when 0, 0, 7', S have been substi- tuted for a, j8, 7, S, Now the points in CD at which there are two coincident tangent planes wiU be obtained by eliminating a, /8 between the equations ^i?;(a,^)=0, and^^,(«,^) = 0, and the eliminant will be of the degree »• — 1 in each of the coefficients. The degree of the resulting equation in 7', S', will therefore be 2 (r-1) (w-r). And since the polars touch the line at each of these points, 2 (r — 1) (w — »•) additional points will lie in the multiple line. Hence the total number of points corresponding to the line, each of which is a point which gives an improper tangent plane, is (r-1) {3(r-l)ra-2r=4-2(n-r)} = (r-1) {(3»- + l)w-2r(r+l)}. This is therefore the number by which the degree n . w — l]* is reduced in the reciprocal surface. Thus, if a surface contain a multiple line of the w — ] ]* degree, which must be a straight line, the degree of the reciprocal surface will be U2 292 EEDUCTION OF CLASS. n.n-lY-{n-2){{Sn-2)n-2{n-l)n} = n {n— ll"— (n — 2) n} = n, or the reciprocal surface is of the same degree as the original surface. 382. As an example of such a reduction, we will take the surface whose equation is a/3"~' — 7S""' = 0, in which a straight line is of the n - l"!* degree of multiplicity, and the reciprocal surface will he found to be of the w"" degree. 383. To estimate the effect of a line of the r* degree of multiplicity, the line being the intersection of two surfaces of the ¥^ and Z"* degrees, K=0, and L=0. The number of points which correspond to the multiple line is (Art. 273) lk{nr-lf+2{n-l)r.{r-l) - {k+l)r.V^^} The number of coiacident tangent planes is obtained from the equation F^ {K, L) = 0, whose coefficients are of the degree necessary to make each term of the w**" degree. The eliminant of ^K{K,L) = Q, and^j;(^,i) = 0, is of the degree (re — Icr) (r — 1) + (w — Ir) {r — 1) = {r-l)[2n-{k-^T)r], which gives a surface meeting the line in points whose number is {r - 1) {2»i - (A + Z) r} U. The degree of the reciprocal is therefore reduced by kl{r -l)\n.{r -I) + 2{n-\)r + 2n- (Je + I) [r {r-l)+ r}] = iZ (r - 1) {(3r + 1) » -2r + (^ + Z) 7^}. On the Relation of Straight Lines to Surfaces. 384. Since the methods employed in this chapter aflford peculiar facilities in the examination of the positions of straight lines satisfying certain conditions relative to a given surface and THEORY OF TANGENT LINES, 293 its singular points, we shall here follow Salmon in his applica- tion of them to the contact of straight lines with surfaces, in the Quarterly Journal, Vol. I. page 329, repeating some propositions which have been already discussed. 385. To find the condition that a straight line may touch a surface at a given point. Let P', (a, /3', 7', 6') be the given point, P, (a, /8, 7, S) any point in space, audi?, (yt^) •••) any point in PP, and ^ = the equation of the surface. The values of X : /i for the positions of R in which PP' meets Kpr the surface are given by the equation yu,"e 1^ = 0, and if PP' meet the surface in two points which coincide with P', two values of X are zero, or <^' = 0, and D'^' = 0. These are necessary conditions that PP' should touch at P', and unless Z>'^' = is satisfied for all values of a, /8, 7, B, i.e. unless -S-, ,. 5§r, ... are all zero, the locus of P such that PP' ig ; aa. ap ^-^ ,. a tangent line is f a t-, + ... j <^' = 0, which is the equation of tJf^\? tangent plane at P'. \y' ,- D'' = is not a sufficient condition for tangency if the differ- ential coefficients are zero, for in that case PP' meets the surface in two coincident points for all positions of P, or P' is a multi- ple point, in this case P may be determined so that PP' meets the surface in three coincident points if its co-ordinates satisfy the equation D'^^' = 0, and unless this equation be satisfied for all values of a, /3, 7, S or -7^ , .... _, ,j„, , ... are all zero, da. eta. dp (,^ + ...)V-o is the locus of such positions of P', and is the equation of the tangent cone at the double point. The argument is easily continued in the case of triple ... r""' singular points. 294 THEORY OP TANGENT LINES; 386. To find the locus of the points of contaAt of all tangent lines which can he drawn from a given point to a surface. Let P' be the given point, P the point of contact in a tangent line to the surface, drawn through P ; in this case two positions of B at least coincide in P, and the equation X"eT ^ = has two values of /j,:X = 0, i.e. = and D^=0, the intersec- tion of the two surfaces represented by these equations is the locus of the points of contact including the singular points of the sur- face which may lie in the curve of intersection for which P'P is not a proper tangent. Hence, the locus is the intersection of the surface and its first polar with respect to the given point. 387. To find the tangent lines at an ordinary point of a surface, which meet the surface in three coincident points on the point of contact. Let P' be the point of contact, P any point in such a tangent line, /i."e i^ ' = must have three values of \ = 0, hence ' = 0, J)'^' = 0, and Z>"0' = 0. The intersections of the surface iJ'^^' = 0, with the tangent plane D'^' = 0, give the positions of the two tangent lines required, which are obviously the tangents to the section made by the tangent plane. 388. If the surface be of a higher degree than the second and if I)''^+i/i=Z>''^=0 and {X + fiD)DD'y.- 391. To find the number of tangents which can he drawn from a given point, to meet a surface in three consecutive points. Let P' be the given point, then, if three positions of B coin- cide in P, \"e a ^ = must have three values of the ratio /i : X = ; the conditions for tlMLare >■ ' + . . . + X"^'^ = 0, I »■+ 1 the eliminant F of tt = and t- = is of the same degree as OK ail (D"*'^')»"^<^""'"^ or of the (to + »• + 1) (w - »• - 2)'" degree, or of the [n (w - 1) - (r+ 1) (r + 2)}"' degree. The intersection of 7=0 with D"^ = Q gives all such tan- gent lines. ^ 394. To find the locus of tangents which can he dravmfrom a singular point to a surface. Let F be the singular point, suppose of the r* degree, r of the values of X : /t in /t-e c ^' = are equal to zero, .-. <^'=0, i)'f = 0, ... Z)"^' = o, and the equation which gives the remaining values is u=^ )it"-^i) V + . . . + X»-<^ = 0, and, if the line PP is a tangent, two of the roots are equal, and _ = — =0, have a common root, and the eliminant F is d\ dfi. of the degree of (jD'y)""^' ^""^', which is (w + r) (w-r- 1). THEORY OF TANGENT LINES. 297 V— is tlie equation of a surface containing all tangent lines through P' to the surface. 395. To find the, numher of double tangent lines which can be drawn from a fixed point not in the surface. Take P' the fixed point, P one of the points of contact ; then two positions of R coincide in P, and two other positions coincide, therefore from the equation \"e k {n) — l given arbitrary points, have a common curve of intersection. 2. The polars of any order r, of all surfaces of the w* degree passing through <^(ra) — 2 points, have (n — rf common points. 3. Prove that the surface reciprocal to the surface whose equa- tion is (a? + y' + si')' = a' {k? + if) is of the fourth degree, and explain the reduction of class. 298 PEOBLEMS. 4. Prove that the reciprocal stirfiice of the surface whose equa- tion is I m n r ^ is of the degree n m + ll"- 6. If P, Q be two points, -P„_i, Q„_, their first polars with re- spect to S, prove that the first polars of jP with respect to Q^_^ and of Q with respect to P„_^ are the same surface. 6. Prove also, that the as*'' polars of F with respect to Qy and the y* of Q with respect to F^ are the same surface. 7. If F^ Has a double point Q, 6„_,+, has a double point F. 8. If the polar conicoid of I 971 n r - with respect to F, resolve into two planes, there are four positions of F given by equations similar to ^ _^_y_S — Imnr' The corresponding conicoid is the planes a = 0, and ■^ + I+- = 0, and the plane polar is I 7n n r ' 9. The conditions that the first polar of -+-+-+- = 0, a p y o with respect to (a', j8', -/, 8'), may be a sphere, are a" mn+.a" lr = hUn + b"' mr = c'lm + (/'mr = p ; and the locus of all such polfes, corresponding to all surfaces of this form, is the curve a'Py + a"aB = b'ya + 6"/88 = c'a^ + c"yS. For a particular sur&ce 7n n r rrmr CHAPTEE XVII. FUNCTIONAL AND DIFFEEENTIAL EQUATIONS OF FAMILIES OF SURFACES. 396. In finding the equation of a surface generated by the motion of a curve of given specios, it is obvious that the curve must satisfy such a number of conditions as will enable us, on expressing them analytically, to eliminate from the equations all the constants which distinguish the curve in any particular posi- tion, and to obtain a final equation involving only the current co-ordinates, and constants which are the same for all positions of the curve. Thus, if the equations of the curve involve n pa- rameters, which vary with the position of the cui-ve, we shall require (w— 1) fixed and independent conditions, to be satisfied by each curve, in order that the locus of the curve may be^a surface. Then (n — 1) equations, expressing these conditians,]- together with the equations of the curve, will give (n + 1) in^- pendent equations satisfied by the co-ordinates of any point o£. the curve, from which the n parameters may be eliminated, and a final equation obtained, which is the equation of the locus of the curve. 397. If, however, a less number of conditions be given, although there is in this case no determinate locus, we may find a functional, or a partial difierential equation, which must be satisfied by all the diflferent surfaces generated by curves satisfy- ing these conditions. Thus, if (m — 2) conditions be given, we may eliminate any (n — 1) of the n parameters, obtaining an equation of the form m = a, where m is a determinate function of X, y, z, and such constants as do not depend on the position of the curve, and a is one of the parameters. Similarly eliminat- ing other (« — 1), we may obtain an equation t) = /8. The equations M = a, « = /3, 300 FUNCTIONAL AND DIPPEKENTIAL EQUATIONS may then be considered the equations of the generating curve, and, in order that this may generate a determinate surface, another condition will be necessary, which will give a relation between the parameters of the form /3 =/(a). Hence, the equation of any surface so generated must be of the form V =f{u), where u, v are determinate functions of x, y, z, and constants, and / denotes an arbitrary function. Eliminating the arbitrary function, we obtain a linear partial differential equation of the first, order ; and either of these equa- tions may be considered as the general equation of a family of surfaces, generated by a curve of given species, fulfilling a number of conditions insufficient to determine a locus, but such that if any new condition be imposed, a particular surface be- longing to this family may be found. Again, if only (re — 3) be given, we may obtain two final equations of the form /3 =/i (a;, y, e,a); 7 =A (^> 2^' «> «) 5 /j, f^ denoting known functions, and a, /S, 7 being parameters. For a determinate surface, two new independent conditions will be necessary, which will supply two equations of the form ^ = ^{a), y = f{o.). The general equation of this family of surfaces will then be found, if we eliminate a from the equations ^ (a) =/i i^^ y^ «*»«); "^ (a) =/» (»> V' ^> «)• This is not generally possible, ^, ■^ being, for the family, arbitrary, and determinate only for a particular surface. In cer- tain cases, however, it happens, that on eliminating {n — 2) of the constants, one of the remaining two also disappears, leaving an equation of the form u = a, as in the former case. When this occurs, we may also obtain an equation of the form where / is determinate. The general equation of the family of surfaces will then be /fa;, y, 2, ^(m), V"(m)) = 0, OP FAMILIES OF SUEFACES. 301 • inToIving two arbitrary functions, from which may be obtained a partial differential equation not involving the functions. This differential equation will generally be of the third order, but occasionally of the second. Similarly, if (w — 4) conditions be given, and if the equation be such that it is possible to deduce one equation of the form M = a, we may obtain a general equation for the family of sur- faces so generated which will involve three arbitrary functions, and so on for fewer given conditions. 398. On the functional equations of families of Ruled Surfaces, The equations of a straight line involve four parameters ; hence, if a straight line moves so as to satisfy three conditions, each condition being such as to give rise to one independent relation among the parameters, a determinate locus will be gene- rated. Intersection with a given curve, or tangency of a given surface, are examples of such conditions. If a straight line move so as to satisfy only two such con- ditions, we shall be able to obtain a functional or differential equation, which will include the whole family of surfaces so generated. Of this kind are conical surfaces with a given vertex, or cylindrical surfaces with a given direction of generating line ; the condition that a straight line may pass through a given point, at a finite or infinite distance, giving rise to two relations among the parameters. If a straight line move so as to satisfy only one such con- dition, the general equation of the family of surfaces generated cannot usually be obtained. If however the one condition be that it move parallel to a fixed plane, the exceptional case mentioned in the last Article arises, since this condition gives rise at once to an equation of the form m = a, m = being the equation of the given plane. 399. To find the general functional equation of cylindrical surfaces having their generating lines parallel to a given straight Let I, m, n be proportional to the direction-cosines of the given straight line. 302 FUNCTIONAli AND DIFFERENTIAL EQUATIONS Then the equations of the generating line may Tbe written I m n ' in which /3, 7 are the two parameters. Hence, ip=ly — mx, ly = lz — nx, and the general functional equation will be ly — mx =f{h — nx), or F{ly — mx, lz—nx)=0. 400. To find the general functional equation of conical surfaces having a given vertex. Let (a, h, c) be the given vertex, then the equations of the generating line may be written X — a _y — h z —c I m n ' in which the ratios l:m:n are the parameters. The functional equation will then be X —a _ j,fx — a\ nfx—a x—a\ - or F[ =, =0. \y — o z—cj This shews that if an equation m = represent a cone, the first member of the equation may, by a change of origin, be reduced to a homogeneous function of x, y, z. 401. To find the general functional equation of conoidal surfaces, having a given axis and a given directing plane. Def. a conoidal surface, or conoid, is any surface generated by a straight line moving so as to intersect a given straight line, the axis, and to remain parallel to a given plane, the directing plane. If the axis be perpendicular to the directing plane, the surface is called a right conoid. Let the equations of the axis be x — a y — h z — c y ■ == ^ == r, I m n OP FAMILIES OP SUEPACES. 303 and (r, »i', n') the direction-cosines of the directing plane : also, let the equations of the generating line be x — a — lr y — h — mr z — c — nr X fj, V ' The equations will then involve the three parameters r, and the ratios X: fi'. v, connected by the equation l'X-\-m'iJL + nv = 0. We shaU then have the equations V {x — a)-r m' {y — V)+ n' (» — c) = (K' + mm' + nn) r, ■, in [x — cl) — l[y — h) _ mX — Ifi n{x — a) — l{z — c) nX—lv' The general functional equation will accordingly be r(af-«)H-m'(y-5)+n'(.-o)=i^ f i"-"j-;;^-? l. ^ ' ^"^ ' ^ ' [n{x — aj — l{z — c)) If the axis be taken as the axis of x, and the directing plane as that of yz, this equation reduces to The general- equation of a right conoid is of the form 7\ / N T-rfw* (cB — a) — Z(v — 6)1 li—a)+m{y-i)+n{.-c)=F)^ ^^^_^>_^'^_^^ '^. 402. To find the general junctional equation of surfaces generated hy a straight line moving parallel to a fixed plane. Let the equations of the generating line in any position x — a._ y — p _ z — y X /i V ' and let I, m, n be the direction-cosines of the fixed plane, We shall then have IX + mji + ni' = = 0, and therefore lx + my + ns = la, + '. (w/3 + ny, for any point on the line. 304 FUNCTIONAL AND DIFFERENTIAL EQUATIONS Also we may write one of the equations of the line in the form which shews that the general functional equation is z = x(f> {Ix + my + nz) + ^^ [Ix + my + nz). It may also be taken as z=y^i {Ix + my + nz) +1/^^ [Ix + my + nz). These equations will be found to lead to the same differ- ential equation, which we shall hereafter find in a different manner. They may also be seen to coincide as follows. Taking the former z = x^{u)+-\jr{u), lz={u — my — nz)(j){u)+l-\lr{u), * z {l + n(j}{u)} =u(j){u) + lylr{u) —my^{u), my.», will also be satisfied. The equation (I) is therefore the differ- ential equation required. We may also write it in the form jdz , dz n = l-jT +7111 ^ . dx dy 406. To find the differential equation of conical mirfaces, having a given vertex. If (a, j3, 7) be the co-(»dinateS of the given vertex, we shall have a-ir _ /3 — y _ 7-g \ ~ /t V ' and we must have, at every point of the surface, ^dF ,a .dF , .dF . the differential equation required, whence it follows, that the operations being performed on F alone. The differential equation may be written DIFFERENTIAL EQUATIONS OP FAMILIES OP SUEPAOES. 307 407. To find the general differential equation of conoidal surfaces, having a given axis, and a given directing plane. Let the equations of the axis he — j — = ^ ~ = ^-^^ , and I m n let V, m', n' he the direction-cosines of the directing-plane, \, fi, v those of any generating line. "We shall then have, at any point of this generating line, {x-a) {mv - nfj.) + {y-l) {nX - Iv) + {s-c){lfM- mX) = ; and since ?\ + w'/i + wV = 0, we obtain, putting p = U + mm + nn'. p =1' {x — a) + m! (y — 5) + n' {z — c), X _ fi _ V p{x — a) — Ip p {y — i) — mp' p{z — c)— np ' therefore the differential equation is {p {x-a)-lp']-^+ {p {y-h)-mp']-^ + {p (a-c)-V}-^ =0. The coefficients being linear functions of x, y, z, it follows, from this equation, that the other equations of the system (A) will hold. If the axis be the axis of z, and the directing plane the plane of xy, this equation reduces to dF dF „ dz dz ^ 408. To find the general differential . equation of surfaces generated by a straight line moving parallel to a fixed plane. Let I, m, n be the direction-cosines of the fixed plane, X, /i, v those of any generating line ; we shall then have IX + mfi + nv = 0, and the differential equation satisfied by such surfaces will be found by eliminating X, p,, v, from this equation, and dx dy dz ' ^■f+-+^''"|l+-=»- x2 308 DIPFEEENTIAL EQUATIONS OP FAMILIES OF SURFACES. The equation will accordingly he I dF dFyd^F .^(jdF dF\f dF dFY d'F , which may "be written / , dsVd's „/ , ds\/', dz\ d^z dxdy If the fixed plane be taken as that of ocy, these equations reduce to dii /diy £F_ 2 dFdF dW_ fdiy^_ \dy J da? dx dy dxdy \ dxj dy^~ ' and l'— V ^ _ 2 — — ^'^ (^§lI \dy) da? dx dy dxdy \dx) dy^ ~ The remaining equations of the system (A) may be shewn to be satisfied in this case, but with the general form of the equation the work is tedious. In the case when the equation of the surface is z=f{x, y), and the fixed plane becomes that of xy, we shall have whence X fi (dsVd'z dzdz d'^z /dzY d'z_^ dz~ dz'' \dy^J da? dx dy dxdy \dx} dy^ dy dx Differentiating the last equation with respect to x and y successively, multiplying the results by t- and ^ respectively, and adding, we obtain (dzVd'z r,(dzydz d\ dz /dzV d'z fdzY d^^^ \dy) ds^ \dy) dx da?dy dy \.dx/ dxdy' \dx/ df ' DIFFERENTIAL EQUATIONS OF FAMILIES OF SUEFACES. 309 whence and, proceeding similarly, the whole of the equations (A) will be seen to be satisfied. 409. To find the general differential equation of devehpahle surfaces. In this case, the two directions in which the tangent plane to the surface at any point meets it must coincide, or the values of \ : /* : K giren by the equations ^dF dF dF^ dx dy dz ' d d . d^^ \ dx dy dz) dy must coincide. This gives the equation [dx) [dy'' dz" [dydzj]^'" dFdF {d"F d'F d'F d'Fl dy dz [dzdx dxdy dy? dydz) + ...=0. If we takei^(a;,y, s) = s-/(a;,^), this reduces to d^zd'z (d'zV^^ da^ dy" \dxdyl Equation (A) may be shewn to hold as follows. The equa- tion ' dx dy) has equal roots. We shall then have , d'z d^z ^ ^ d'z ^ d'z_^ ^^^fd^y^^' ^d^y + f''^'-^' simultaneously. • 310 DIFFEBENTIAL EQUATIONS OP FAMILIES OP SUEFACES. Differentiate each of these equations with respect to x and y successively, multiply the four resulting equations by X", X/i, \/4, and /i", and add : the result will he which reduces to by the former relations. Similarly, we may proceed to shew that the higher equations are also satisfied. We may also obtain this equation from the general equation of a developable surface given in Art. 300. It is there shewn that the equation obtained by eliminating a from the equations z = ax + y {a) + ^fr (a), = x+y^' (a) + i/r' (a), represents a developable surface. Now at any point of this surface I = <^ (a) + J {^+# W + 1' («)} = («)• Hence £ = K^)' and eliminating the function by differentiation, we obtain the equation dx' dy^ \dx dy) . 410. To find the general differeniial equation of surfaces of revolution about a given axis. The general functional equation is F{x, y,z) =f [{x-ay+ {y-hf-V {z - c)'} ~f [h:+my^-nz) = 0. DIFFEEENTIAL EQUATIONS 05*^AMILIES OE SUEFACES. 311 f=2(3/-5)/;-m/;, whence [m{a-c)-n{y- 1)] -7- + [n{x-a)-l{z-c)] -5- or {m (a - c) -TO («/ -6)} ^ + {«(«;- a) - Z (a - c)} ^ = Z (y — S) — TO (aj — a). 411. -ip^Zim«io»j of the differmtml equations of families of surfaces to conicoids. Take Fix, y, s) = ckc* + hf + cs' + ^dyz + 25'aa; + 2c'^ + 2a"a! + 25"y + 2c"» + (Z = ; then, by the equation of Art. (405), in order that this mayS^^ cylinder, we must have at every point I [ax + c'y + I's + a") + m {c'x + hy + a'z + h") + n{h'x + a'y + cs + c") = 0, for fixed values ail : m ', n. This cannot be the case at every point, unless al + c'm + h'n = 0, c'l + hm + an = 0, h'l + a'm-\- cn = 0, a"l+i"m + c"n = 0; from the first three of these equations we obtain aa'-" + W^ + cd^ - aho - 2a'5'c - (1), and I (JV - ad) = m {dd - W) = n {a'b' - cc') ; 312 DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. whence by the last, we have the further condition y+-jzr h" b'c'-aa'^c'a'-bb'^a'b'-cc' 7+T7I7- ■J = (2). The conditions (1) and (2) must hold among the coefficients in order that the surface may he a cylinder. Compare Art. (233). Similarly, if the equation F {x, y, z)=Q represent a cone, we must have the equation (a; -a) {cae + c'y + b'z 4 a") + (^ - /3) {c'x + by + dz + &") + (« - 7) (^'35 + «V + ca + c") = 0, satisfied at every point of it. This cannot be the case unless the following four equations be simultaneously true : aa + c'/S + &'7+o"=0, c'a+ &/S +a'7+&" = 0» &'a + a'|8+ C7+c"=0, a"a + J"/3 + c"7+tZ=0; and, the condition that the equation of the second degTee may represent a cone, is the determinant = 0. This is the condition of Art. (231). The application of the condition for a conoid leads to some rather tedious work, if we take the most general form of the equation. The condition finally requires the relation among the coeflScients „ , „, , „ ad"" + bV + od"- - abc - 2db'c = 0, with the further relation for a right conoid We may obtain these by taking the axis of z as the axis, and the ditecting plane -parallel to the axis of a; ; in which case Z = 0, «i = 0, ?' = 0, a, c, y, a" t b, a; b" b\ a, c, c" a". b ,c , d CIPFEEENTIAL EQUATIONS OP%AMILIES OP SUEPACES. 313 and the differential equation reduces to dF dF m dF ^ dx ^ dy n ^ dz Hence the equation X {ax + c'y + S'a + a") + y {c'x -Vly-^- az+l") -^y {h'x + ay + ca + c") = 0, or aa? + /, a'rn'\ s , / , cm'\ + V.zx + xyUo' - ^) + d'x + (b"-^) y = 0, must be true at every point for which F{x, y, s) = 0. Subtracting, we shall obtain the following equation, satisfied at every point of the surface, am' J , 2 , / , , cm"\ ,, b'm' „ /,„ c"m'\ „ „ , _ + b'sx + —r- + a"x+lb" + — ~] y + 2c"z + d = 0. These equations must then coincide, and we have „ , 2a'm' „ , JW „ „ , „• a = 0, 1= — —, c = 0, c=—r, c =0, d = Q. n n and the equation of the surface will be 6/ + a'yz + h'zx + c'xy + a"x + h"y = 0, the coefficients being connected by the equation bb' = 2a'c'. Hence the condition aa'" + bb" + cc" - abc - 2a'b'c' = 0, must be satisfied when these axes are taken, and therefore when any other axes are taken, since the left-hand jnember of the equation is unaltered by transformation of co-ordinates. 314 J)IFPEBENTIAL EQUATIONS OF FAMILIES OP SURFACES. If the conoid be a right conoid, we shall have m' = 0, and therefore b" = 0, c' = 0, and the equation becomes a'yz + h'zv \ ci'a^ + J"y ^ 0, in which the condition a + 5 + c = 0is satisfied. This condition must therefore he satisfied for every right conoid of the second degree. The only conoid of the second degree is then a hyperbolic paraboloid, and for a right conoid, the two principal sections must be equal parabolas. The application of the condition for developable surfaces leads to the equation (6c-a")(aa; + 6> + J'3 + a7+ +2 (6V- affl') (c'a; + §3^ + a'«! + J") (5'a; + a'y + ca+ c") + • • • = 0, to be satisfied at every point of the surface. On examination of the coefficients, this will be found to be «»'+ S/+ cz^+ 2a!yz + 2b'zx + 2c'xy + '2a"x+2b"y + 2c"z - A= 0, A having the same meaning as in Art. (231). The condition for a developable surface is then A + c?=0, which shews that the only developable surfaces of the second degree are cones, of which cylinders are a particular case. For surfaces of revolution, we obtain the equation, assuming the co-ordinates of the center of the sphere all zero, which does not aiiect the generality, {ms —xy) {ax + c'y + b'z + a") + (wic - li) {I'x +by + a'z + b") + (Jy - nx) (t>x + a'y + cz + c") = 0, which must either coincide with the original equation, or be identically true. We have accordingly, using an undetermined multiplier Je, c'n — b'm = ha, a'l — c'n = Jcb, b'm — a'b^Tcc,. (1), c'm-6'«-Z(J-c) = 2^a', (2), b"n-d'm = 'iTcb", , (3). From the equations (1) we obtain k{a + h + c)=0, J?EOBLfks. 315 and taking A; = 0, we have aj'l=h'm = c'n, and the system (2) gives us V o' ~ a" ^' a'b' , c'a' h'c' or c--r = h--jT-=^a---r c a I • The third system will then give a a =00 =cc I which equations will however not be necessary in the case when the axis does not pass through the origin. The factor a + b + c may possibly be shewn to be foreign to the, investigation. XVIII. (1) Knd the general functional equation of a family of surfaces such that the tangent plane at any point (oa, y, z) of one of them intercepts on the axis of « a length — jp . Determine the arbitrary function so that the intercepts on the axes of X and y may be in the ratio as ; y. (2) Shew that the differential equation of aU surfaces -which are generated by a circle, whose plane is parallel to the plane of yz, and which passes through the axis of x, is <^-'-)S-(«-|){-(S)"}=»- (3) A surface is generated by a straight line always passiug through the two fixed straight lines y = mas, « = c j y=~ ttuo, z=~c; prove that the equation to the surface generated is of the form mcx — s-ya _ „ (m%x - cy \ also that its differential equation is (cy — mux) -7- + jw (mcx — y (4) The two branches of the curve of intersection of a surface {cy-mzx)j^ + m{mcx~yz)-r^+{c -»=) — = 0. 316 PROBLEMS. and its tangent plane will bo at right angles to each other at every point, if the equation of the surface satisfy the condition /dFV /d'F ^ d'F\ ^ ^dFdF d'F \dx) \d^ ds? J '" dy dzdydz'"~ (5) Shew that the only surface of revolution in ■wliich the two branches of the curve of iatersection with the tangent plane are at right angles to each other at every point, is the surface generated by the revolution of a catenaiy about its directrix. (6) Shew that the only conoid possessing this property is a right conoid, and that its equation may be reduced to the form y = x tan wi«. (7) Apply the condition of question (4), to determine at what points of the surface ax' + ly' + ci^ + ^a'ye + 2h'zx + 2c'xy + 2a"x + 2h"y + 2d'z + d=0, the generating lines are at right angles to each other. (8) The functional equation of surfaces generated by a straight line intersecting the axis of «, and meeting the plane of xy on the circle a^ + y' = a'', is Ja^ + y'-a = zf Find also the differential equation. (9) Find the functional equation to a family of surfeces gene- rated by a straight line of constant length c sliding between the co-ordinate planes of yz, zx, and remaining parallel to the plane of xy. Shew that the differential equation is / & dzy (/dzV , /dzy\ , fdzy (dzV {''Tx-'^d-y) iU) ^KTyJr' Kd-x) {.d^J ■ (10) Shew that a certain differential equation of the third order must be satisfied at every point of any niled surfe,ce whatever. (11) Shew that every right conoid of the m* degree will be cut by any plane perpendiculai- to the axis in a number of straight lines not exceeding (w-1). (12) In a right conoid of the third degree, iu which only one generating line passes through any point of the axis, shew that the Action made by any plane through the axis will consist of the axis, and two generating lines, the sum of whose distances from any fixed point on the axis is constant. (13) The general functional equation of ruled eurfaces whose generating lines pass through the given straight line PROBLEMS. 317 x—a y—h z—G I ~ m n ' is st = xf{u) + y^{u), n{pa — a) — l{z—c) and the general differential equation is V ^ dy) da? \ ^ dy) \ dx) dxdy A dzVd'z -• ^(^ + ''^)^= = ^' X, )«., V, being proportional to m{z — c)—n(y — h), &c. What do these equations become respectively ■when the given straight line is the axis oi zl (14) Shew that all developable surfaces of the third degree are cones or cylinders. CHAPTER XVIII. PROPERTIES OF CONICOIDS SATISFYING GIVEN CONDITIONS. FORMS OF THE EQUATION OF THE SPHERE. 412. To find the form of the general eguatton of a conicotd passing through eight given points. If M = 0, V = te the e(iuations of two particular conicoids satisfying the given conditions, then u + 1cv = will be satisfied when u and v simultaneously vanish, and will therefore represent a conicoid passing through the eight given points, and since it involves one arbitrary constant, it is the general equation re- quired. 413. If a conicoid pass through eight given points, ike polar plane of any other given point will pass through a fixed straight The polar plane of the point A of the fundamental tetra- hedron is da, da. which passes through the straight line determined by the equa- tions du dv — = — = 0. da ' da. There are moreover certain points whose polar planes are altogether fixed. For if (a', ^, y', B') be a point such that du' du' du' du dv ~ d7 dv dv' ' d^ W H dS • CONICOIDS THEOUGH GIYEN POINTS. 319 its polar plane will be independent of k ; or will be fixed. This number will afterwards be found to be four. 414. If a conicoid pass through eight given points, the pole of any given plane will lie on the curve of intersection of two conicoids. The equations determining the pole of the plane BGD are du J, ^^ _(\ <^** h '^^ —n ^^ A.l.'^'" —n whence the locus of the pole is the curve given by the equations du dv du dv _ du dv du dv _ d^'d^~~d^'dB~ d^'M~dE'dY~' which are both of the second degree. There will be four planes whose poles are altogether fixed, namely, the polar planes of the points determined in the last article. The locus of the center of a conicoid passing through eight fixed points is a curve of this species, since the center is the pole of the plane at infinity. 415. Reciprocating these propositions, we obtain the fol- lowing. If a conicoid touch eight given planes, the pole of any other given plane will lie on a fixed straight line. Hence the locus of its center will be a straight line. The polar plane of any given point will envelope a develop- able surface of the fourth class. In this case also there exist four polar planes whose corre- sponding poles are altogether fixed. 416. Ihur cones of the second degree can in general be described passing through eight given points. The equations for determining the center of the conicoid u + kv = 0, are du . dv _du I, dv _du ■, dv ^du j dv di'^^di^d^'^'^Wd^ dy'dS'^'^M- Ji the conicoid be a cone, its center lies on the surface, and each member of these equations du -, f dv \ - a + i8 + 7+8 = 0, 320 CONICOIDS THROUGH GIVEN POINTS. Now if we take four of the given points as the angular points of the fandamental tetrahedrons, we may write u=lfiy + mya. + na^ + I'aZ + m'^h + n'yS, v = Xfiy+ amJ the condition for a cone will become 0, n + ku, m+kfi, I' + kK' n + kv, 0, l + k\, m'+kfi m + kfi, l+kXf 0, n' + kv r + k\', m' + kfi', n' + kv', which, being a biquadratic in k, shews that there are in general four cones of the second degree passing through eight arbitrary points. ' The vertices of these cones are the points whose polar planes are fixed with respect to any conicoid passing through the eight points. For if, in Art. (413), 'the value of the ratio du' dv' , . \ will be a root of this equation, and the point («', ff, y', S') will be the vertex of the cone u + k^v=0. Let P, Q be two of these points, then since the polar plane of P is the same for all conicoids passing through the eight points, it will be the polar plane of P with respect to the cone whose vertex is Q, it will therefore pass through Q, and simi- larly through the other two points. Hence of these four points, each is the pole of the plane passing through the other three, with respect to any conicoid passing through the eight points. If we take these points as the angular points of the fandar mental tetrahedron, the general equation of the conicoid will be I, m, n, r each involving an arbitrary constant h in the first degree. It follows from this result, that the equations of any two conicoids may be obtained in the form U + m^ + n-f + r^^Q, l'o!' + m'l3^+n'y'+r'S' = 0. p> n — h, m, V n — k, q, I, m' m, I, r, n m\ n', s CONICOIDS UNDER G^EN CONDITIONS. 321 417. These results will fail if the eight points lie on two planes, for in that case, taking the two planes as a = 0, /3 = 0, the equation of any conicoid passing through the eight points may be written m — 2A a/3 = 0, in which u=^a? + c[^ + ry' + s^ + 2l^y+...+2l'aS+... and the equation giving the values of k, for which the conicoid becomes a cone, is = 0, which is only a quadratic in k, so that only two proper cones can be described through the eight points, agreeably to Art. (343). 418. Three conieoids can he described passing through eight given points, and touching a given plane. Let the equation of any conicoid passing through the eight given points be {I + W) a= + (m + km!) /S" + (« + kn') 7' + (r + hr') ^ = 0, and let a', /3', 7', S* be the perpendiculars from the angular points on the given plane. Then if the point of contact be (a", /S", 7", h"), we shall have (l + kl')a" jm + km')^" ^ {n + kn')y" ^ {r + kr')B" a' i S' whence the condition that the given plane may touch the conicoid is l + kl m + km n + kn r+kr' ' a cubic equation for k, provided a', |8', 7', S' are all finite. If a' = 0, one solution isl + kV = 0, which reduces the required conicoid to one of the four cones of the second degree, passing through the eight points, which is not properly a solution, as the plane will not in general be a proper tangent plane to the cone, but, passing through its vertex, of course will satisfy the analy- 322 CONICOIDS UNDER GIVEN CONDITIONS. tical conditioii of tangency. So if the given plane pass through two of the angular points, there will be only one proper solution, and if through three, no conicoid can be described as required. Eeciprocally, we can in general find three conicoids, touching eight given planes, and passing through a given point. Cor- responding cases of exception arise, when the given point lies on one, two, or three of four planes, fixed with respect to the eight given planes. 419. To find the general form of the equation of a conicoid passing through seven given points. Take m = 0, v=0, w = 0, tSe equations of three particular conicoids, not having a common curve of intersection, and satis- fying the required conditions ; then the equation lu + mv + nw = will be the general equation required. For it is satisfied when- ever u, V, and w simultaneously va,nish, and it involves two arbitrary constants, the ratios I : m : n, hy means of which it may be made to pass through any two other given points. It can therefore be made to represent any conicoid passing through the seven points. Since the equations u = 0, v=0, w=0 determine eight points, we see that any conicoid which passes through seven fixed points will necessarily pass through an eighth fixed point, whose position may be determined from that of the seven. 420. J^a conicoid pass through seven given points, the polar plane of any other given point will pass through a fixed point. For the polar plane of the point A is - du dv , dw . Z -=- + m -J- + M-T-= 0, da, da. da. which passes through a fixed point determined by the equations ^*^ = ^ = — = 0. da, 'da. 'da. If the fixed point be a vertex of any cone of the second degree passing through the seven points, the polar plane will pass through a fixed straight line. For, take m = as the CONICOIDS UNDER GIVEN CONDITIONS. 323 equation of a cone whose vertex is A, passing through the seven points. Then t- = 0, and the polar plane of A will be dv dm m ^--h n -r— = 0, da da. which passes through the fixed straight line -^ = 0, -^ = 0. In general, no polar plane can be altogether fixed, for, in order that the polar plane of A may be independent of ? : m : n, du ^ dv „ dw „ , , ■ 1 . . 1 ^ = 0, T- = 0, j-^ = Oj must be equivalent to only one equation, whence each of the equations m = 0, « = 0, to = 0, must be of the form OM^ + Mj = 0, aw, + w^ = 0, aWj + Wj = 0, where m, is of the first degree, and u^, v.^, w^ do not involve a. Hence through the seven points can be described two cones of the second degree u^ — v^ = 0, v^ — w^= 0, having a common vertex A, and the seven points ifiust lie on four fixed straight lines passing through A. 421. When a conicoid passes through seven given points, to find the locus of the pole of a given plane. /^ If the equation of the conicoid be lu-^mv+nw = 0, andfif -^ j the given plane be that oi BCD, the equations determining tlte / pole are , du dv dw ^^ + '"^ + '*^ = <^' , du dv dw I j- + »J-7-+«-j- = 0, a7 drf ay , du dv , dw ^ Hence the locus of the pole is the surface of the third degree du du du m' ^' 5S dv dv dv d^' d^' dS dw dw dw d^' H^' dS Y2 = 0. 324 CONICOIDS UNDER GIVEN CONDITIONS. The locus of the center, which is the pole of the plane at in- finity, is obtained by eliminating I, m, n from the equations , du dv l-j-+m j- + ( da. da. dw da , du _jdu dy ,du and is therefore = 0. du du dv dv dw dw d^~da' d^~di> d^~da du du dv dv dw Sw dy da. ' dy da.' dy da. du du dv dv dw dw dS da. ' dS da' dS da 422. To find how many conicoids can he described passing through seven given points, and touching two given planes. Let the given planes he a = 0, /3=0; then at the point of contact with the first we must have ^ T du dv dw . .du ^ -0 Z — + ..-0, f^g + = 0, and eliminating /3 : 7 : S from these, the eliminant is of the third degree in I, m, n. Similarly the condition of touching ;S = will lead to a relation of the third degree in I, m, n, and the final equation for determining I : m will be of the ninth degree. There are therefore, generally, nine conicoids satisfying the given conditions. 423. To find a general form of the equation of a conicoid passing through seven given points, six of which lie ly threes on two non-intersecting straight lines. The two straight lines must lie altogether on the conicoid, and if through the seventh point we draw a straight line inter- secting the other two, three points on this line, and therefore the whole line, will lie on the conicoid. Take these as the edges AB, BG, CD of the fundamental tetrahedron, then, since they are contained by the three pairs of planes a7 = 0, aS=0, y3S=0, CONICOIDS UNDER GIVEN CONDITIONS. 325 the general equation is Za7 + maS + n^B = 0. If we take two such conicoids {I, m, n), and {U, m', n'), their fourth line of intersection will be found to he a (Jm! — I'm) = /8 (mn — m'n), 7 {nV — n'l) = S [mn' — m'n) . 424. In the conicoid of the last article, the pole of a given plane lies in a fixed plane. Let La + iIf/3 + -Wy + -RS = be the equation of the given plane, let (a', /3', 7', S') be its pole with respect to the conicoid larf + maS + to^SS = 0, we shall then have V + mZ' _ wS' _ la _ ma + n^' a" W+mS') + ^' {nS') - 7' (^g') - 8' (ma' + n^') La' + Ml3'-Ny'-B8' Hence the locus of the pole is the plane La' + Ml3' = Ny'+Ii8'. The locus of the center is therefore a' + /3' = 7' + 8', which is a plane passing through the center of gravity of the tetrahedron, and parallel to the edges AB, CD ; or the plane bisecting the edges A 0, AD, BG, BD. The polar plane of the point (a', /S', 7', S') may similarly be shewn to pass through the point a_j8__7__ 8 a'~/S'~ i~ 8" which may be determined as follows. Take a plape through AB, and the pole, meeting CD in P, and a plane through CD and 0, meeting AB in Q. Then if R be taken on PQ such that OPQR is a harmonic range, the polar plane of will pass through K Compare Art. (343). 425. To find a general equation of a conicoid passing through seven given points such that four straight lines can he drawn through them and through each of four given points. 326 CONICOIDS UNDEE GIVEN CONDITIONS. Let the four given points be the vertices of the tetrahedron ; then any seven of the eight points represented Ij the equa- tions I m n r will satisfy the required conditions, and the eighth point will lie on any conicoid passing through the other seven. Since the points lie on the pairs of planes I TO ' TO w ' n ~ r ' the general equation required will be \i ml \m nj \n rj ' which we may write La^-^-M^ + Nrf + R^^O, L, M, N, B being connected by the equation Ll + Mm-\-Nn + Br = 0. In this conicoid each angular point of the fundamental tetra- hedron is the pole of the opposite face. The polar plane of a given point (a', /3', 7', S') will pass through the point I TO n r ' and the pole of a given plane aa' 4-/8/3'+ 77' + S8' = will lie on the surface la! TO/3' 717' rS' 1- -3- ■\ H -5- = "• a /a 7 The locus of the center is therefore the surface I m n r - a /3 7 The centers of the eight spheres which touch the faces of the fundamental tetrahedron are a particular case of these eight points, since they are given by the equations CONICOIDS UNDER GIVEN CONDITIONS. 327 The locus of the center of any surface passing through them is therefore JLj-JL JL J_-n Pi a Pi^ Kl Vl^ which equation may be interpreted geometrically to mean that the feet of the perpendiculars from any point of the surface on the faces of the fundamental tetrahedron lie in one plane. 426. To find the, number of conicoids passing through six given points, and touching three given planes. If we take four of the given points as the angular points of the fundamental tetrahedron, we may write the equation of the conicoid Z/87 + mva + na.^ +l'aZ-\- m'^B + n'yB = 0, and I, m, n, V, m', n' will he connected by two linear equations expressing the conditions of passing through the two other given points. The condition that this conicoid may touch a fixed plane L^a. + M^0 + N,y + B,8 = 0, may be found by eliminating (a, /8, 7, S) from the equation I'a + w/3 + my _nct + m'/3 +ly _ ma + 1^ + n'y _ I'a. + jw'yS + n'y and the eliminant is of the third degree. Hence I, m, n, V, m', n will be connected by two homogeneous equations of the first degree, and three of the third, and the number of solutions will therefore be twenty-seven. 427. To find the number of conicoids passing through five given points, and touching four given planes. Proceeding as in the last article, we shall have 7, m, n, V, m, n' connected by one homogeneous equation of the first de- gree, and four of the third degree, and the number of solutions will be eighty-one. 428. Hence by reciprocating the results of Arts. (428) , (427) , and (423), we see that eighty-one conicoids can be described pass- 328 CONICOIDS UNDER GIVEN CONDITIONS. ing through four given points, and touching five given planes ; twenty-seven conicoids through three given points, and touching six given planes ; and nine through two given points, and touch- ing seven given planes. 429. To find the general equation of a conicoid containmg two non-intersecting straight lines. If these straight lines be taken for the edges a = 0, S=0; /3 = 0, 7 = 0; of the fundamental tetrahedron, we see that the terms in the general equation involving a*, ^, y, 8", aS, /Sy, must vanish, and the resulting equation may he written m.'^a. 4- ma/3 + ra.'/3S + w'yS = 0, 430. To find the condition that the conicoid of the last article may he a paraboloid. In order that this conicoid may be a paraboloid, it must have contact with the plane at infinity ; we must therefore have, at the point of contact, my+n^ = na + m'S = ma +n'S = m'^ + n'j; a + ^+y+h = 0. These give a. _ jS _ 7 _ S m' —n' m — ri m' — n m — n' whence the required condition is m + m' = n + n' ; which will give at the point of contact a + S = 0, |S + 7 = a. Hence the axis of the paraboloid and the two generating lines are parallel to the same plane. The polar plane of any point («', j8', 7', B') may be readily shewn to pass through the fixed point a ^ 7 _^ a p 76- which may be determined geometrically as before (Art. 425). 431 . If a conicoid contain two given non-intersecting straight h'7ies, the pole of a given plane will lie on a fixed plane. CONICOIDS UNDER GIVEN CONDITIONS. 329 Let the given plane be its pole will be given by the equations m, andZ=0, m = 0, n = 0, l' = 0, m' = 0, because the substitutions a = A/3, a = — A;/3 must lead to the same equation, namely, that of the cone whose vertex is A, and which contains the two conies ; and similarly for /3 = t , /8 = — ^ . The general equation of any conicoid containing these two given conies is therefore \ {poi' + ql3' + 2n'yB) +/j,{o?- J ® ^1'^ ^'^ rectangles of the segments of chords of the sphere passing through A, B, G, D respectively. Hence, since the left-hand memher of the equation is always proportional to the rectangle of the segments of any chord drawn through the point (a, ^, 7, 8), Art. (137), it must be equal to that rectangle. 437. To find the radius of the circumscribing sphere in terms of the lengths of the edges. Its equation heing a'^iS^ + 5'V + c"a/3 + o'aS + P^S + c^ =/(«> P, 7, S) = 0, the equations of its center are /'(«)=/'(/3)=/'(7)=/'(8) = '^- Hence a/' (a) + /S/'(;8) + -if'{i) + S/(S) = (a + /3 + 7 + S) \ = \, or 2/(a,A7,S) = >" But —/(a, /8, 7, S) is the rectangle of the segments of any chord through (a, /3, 7, 8), and therefore is equal to — i?^, if i? be the radius : or \ = 2i?^. The equations determining the center are then c'';8 + &"'7 + a''S=2ii:^ c'=a+a"7+&'S = 2E^ &'^a + a''^+ c'S = 2R\ a+/S + 7 + S = l, which give, on eliminating a, ^, 7, S between these equations, 4^B^bVa''+cVb''+a'h'c''+a''bV-a'a'\b'+b''+(^+c''-a'-a'')-...} = aV + ¥b" + c*c" - 2b'b"cV - 2cV^ a'a" - 2aWJ'^ Now, if V be the volume of the tetrahedron, 6 V will be the volume of a paralklopiped whose edges are J) A, DB, DC, or 6 V= ahc X ^/l-cos^BB C'-cos" OJJA-cos'ADB-2cobBD Ocos CD A cosDAB, 144 r=4aW - a" {V + c^ - a J - V (c» + a' - V^ - c\d^V-dy _ (J' + c*" - a'^) (c^ + a^ - V) [a' + V - c") =bVa''+cVb''+aVc''+a''b'Y'-a'a''{b'+b''+c'+c''-a'-a'')-... TANGENTIAL EQlfixiON OF SPHERE. 335 whence AR'xU4:V'=lGs {s-aa!) {s-lV) [s-cd), where 2ssaa'+&5'+cc', we have then, finally, P_ Vs (s — aa) (s — Ih') {s — cc') ^ w • 438. To jind the general tangential equation of a sphere. Let (a', ^', 7', h') be the tetrahedral co-ordinates of the center of the sphere, p its radius, then we shall have for any tangent plane, whose co-ordinates are (a, j8, 7, 8), a'a + jg'yS + 77 + 8'S = ± /3, Art. (109), and the homogeneous equation of the sphere will therefore be («'a+^'/3+7'7+S'S)==/>=K + f. + ^. + ^. - 2 ^cos(^i))-...l . Ifi Jf'a ra Jfi Jest's J 439. To find tJie tangential equations of the spheres touching the faces of the fundamental tetrahedron. In order that the surface may be touched by the plane BCD, the tangential equation must be satisfied by the plane /S = 0, 7 = 0, 8 = 0; hence the coefficient of a? must vanish, and hence, for the spheres touching all thtf faces of the tetrahedron, we must have j,,V=_p/^-=i,3V=i./S'^ = p^ The tangential equation of the inscribed sphere will therefore be ' /S7 ^AD , ^-f- cos^— -+ ... = 0, and of the remaining seven spheres touching the faces, ^87 ^AD , 7a ,BI> a^ .en -^-i- cos^ —— -f- -^— cos'' -— + -'— cos'' -— F,Ps 2 p^, 2 p,p, 2 aS . ,B0 ^B . ,GA 78 . ,AB „ sm" -— - -^-- sin' '— sm"-— =0, PiP, 2 p,p, 2 p,p^ 2 with three similar equations, and j3y ^AD a8 ^BG ja . ^BB ^-^ cos' -^ + cos' '— sm' -— - P^z 2 p,p, 2 p,p^ 2 0/3 . ,GB ^S . ,GA 78 . ,AB ^ ^— sm' — ^^^ sm' — sm' —— = 0, FiP, _ 2 p^p^ 2 p^p^ 2 with two similar equations. (- 336 CENTERS OP SIMILARITY. If a, ^, 7, S be the co-ordinates of any tangent plane to one of these spheres, we shall have ± — + -±- + - = 1, Pi P2 Po~P4 the signs being all positive for the inscribed sphere, three posi- tive and one negative for the second set of spheres; and two positive, two negative, for the last set. 440. To find the centers of similarity of the inscribed, and the four escribed, spheres of the fundamental tetrahedron. If any point be taken on the line joining the center of the inscribed sphere, and that of the escribed sphere opposite A, dividing the distance between them in the ratio jj, : \, the equa- tion of this point is + ^ + 7^i) -«+^H-^ + l -+-+-+- +-+-+- Pi P2 Pz Pi Pi Pi Pi Pa but if this point be a center of similarity, the ratio /i* : \ must be the ratio of the radii of the two spheres, or V Py Fs Fa PJ \Pi Pi Ps Pi) and the equations of the two centers of similarity are Pi Ps Pi with similar equations for the centers of similarity of the in- scribed sphere, and any other of the escribed spheres. In a similar manner for the two escribed spheres opposite A and I), we shall' find the centers of similarity to be «=1; ^ + 1=0; Pi Pi Pi Pi the former being external, the latter internal. The centers of similarity of the escribed spheres are therefore the points in which the planes, bisecting the internal and external angles between the faces, meet the opposite edges. In tetrahedral co-ordinates these results are, (1) the center of CENTERS OF SIMILARITY. 337 similarity of the inscribed sphere, and the escribed sphere oppo- site A, are given by the equations /3 = 0, 7 = 0, S = 0; a = 0, ^^=^37=j?,S; (2) the centers of similarity of the inscribed spheres opposite A and I) are given by the equations ^ = 0, 7=0, i?,a+p,S = 0; a = 0, B == 0, p,^ = p,y. The twelve centers of similarity of the inscribed sphere and the three escribed spheres opposite A, B, and 0, will lie by sixes on the eight planes S = 0; a = 0, /3 = 0, 7 = 0, ^,a +p^ +i?37 - 2^,S = ; pfi +Psy - Pi^ = 0, i?s7 +^1" -Pi^ = 0: ?!« +p^ +Pi^ = ; the first containing the six external centers ; each of the next four containing three external and three internal ; and each of the last three containing two external and four internal poles. The twelve centers of similarity of the four escribed spheres will lie in the same manner on the eight planes p^CL+pfi+p^rf+jp^B=0; a = 0, /3 = 0, 7=0, S = 0; Pi"- + pfi -p^y -p3 = 0, PiO. -pfi -pa + P4S = 0, Px^-pfi+P{i-p3 = ^- 441. To find the distance between the centers of the msortbed and circimiscribed spheres. The tetrahedral co-ordinates of the center of the inscribed sphere are — , — , — , — , if »• be its radius : hence the rect- ^ Pi P^ P, P* angle of the segments of any chord of the circumscribed sphere passing through this point PiP^ PlPi PzPx PiPi P1P2 P^P. or, if A be the distance between the centers, t^ = n^-r^[ + + 4- + + - \P^Pi PtPi PsPx PiPi PlP^ PiP^- a result due to Mr Salmon. Quarterly Journal, No. 15, 33& PROBLEMS. XIX. 1. If A, B, C, D be the vertices of the four cones of the second degree, whi.ch can be described through the curve of intersection of two coniooidsj the triangle BCD will be a conjugate- triad of the sec- tion made by its plane of the cone whose vertex is A. 2. If of eight given points six. lie by threes on two nonrintei-- secting straight lines, shew that no cones can be described through the eight points ; but that there is an infinite number of points, lying on two straight lines, which have their polar planes, with respect to any conicoid containing the eight points, fixed 3. If of seven given points six lie by threes on two non-inter- secting straight lines, shew that the remaining line of intersection, of any two paraboloids passing through the seven points will be a fixed straight line at infinity. 4. If a conicoid be described containing the edges AB, BC, CD of a tetrahedron, the pole of the plane bisecting the edges AB, CD, AG, BD will lie on the plane bisecting the edges AB,, CD,, AD, BC. 5. Shew that only one conicoid can be described containing two given non-intersecting straight lines, and touching three given planes. 6. Shew that eight conicoids can in general be described passing through four given points, touching three given planes, amd having the intersections of the tangent planes at the four given points, witli the corresponding planes containing the points, lying in one plane. 7. Shew that four conicoids can in general be described passing through five points in one plane, two other giveia points not in that plane, and touching two given planes. 8. Shew that a sphere can be described touching the edges of a tetrahedron, whose lengths are a, a'; b, b' ; e, c' ; if a ± a' = 6 ± 6' = c ± c', the ambiguities being independent. 9. If a tetrahedron be self-conjugate with respect to a sphere, shew that the opposite edges aire, two and two, at right angles ; and that all the plane angles containing one of the solid angles must be obtuse. Shew that this angular point wUl lie within the sphei'e, and the three others without, and determine the radius of the sphere. 10. The number of paraboloids, which can be drawn through eight given points, is, in general, three. 11. Prove that the equations of two conicoids cannot both be obtained in the form of Art. (416), if they have a common generating line. 12. Four cones of the second degree can be drawn, each contain- ing the locus of the center of a conicoid passing through eight given points ; and having their vertices coincident with the vertices of the cones on which lie the eight given points^ F£OBp:MS. 339 13. Prove that, in general, one conicoid can be described, which is self-conjugate with respect to one given tetrahedron, and also with respect to another of which three angular points are given. Shew also that the fourth angular point of the second tetrahedron will be fixed. 14. If two conicoids be so related that a tetrahedron can be drawn, whose faces touch one of the conicoids, and two pairs of whose opposite edges lie on the other, an infinite number of tetrahe- drons can be so drawn. 15. If the conicoids la' + m^ + ny' + rB' = 0, I'a' + m'^ + n'y' + r'8' = 0, be so related, prove that a similar relation will exist between the conicoids l'^ m'' n'' r'^ la'+mB' + n'-Z + r'^^O and ■ra? + — B'+—-/ + - ?l' = 0. 1 6. Prove that two tetrahedrons may be iascribed in the coni- coid having their faces pai'allel to the faces of the fundamental tetrahe- dron, provided that (l + m + n + r) 7 + — + - + -)< 4. 17. Prove that, in order that the two conicoids in (15) may be so related, one of the following conditions must hold : V 1 + m n = r' I' I + n' n = m + r' r' I' r Tlflf n' I + r = m + n ' Z2 CHAPTER XIX. CURVES. 442. We have already considered curves as the complete or partial intersection of surfaces ; hut in the investigation of the properties of curves we may also have to consider curves as the locus of points each of which satisfies given laws, the algehraical statement of which assumes the form of equations between the co-ordinates of any point in the curve, and variable parameters, the number of equations being two more than the number of parameters. Instances of this mode of representation of a curve occur in dynamical problems in which the curve is defined by equations between the co-ordinates of the position of a particle and the time of its arrival at that position. If from such equations the parameters were eliminated, the result would be two final equations, that is, the equations of two surfaces whose complete or partial intersections are the curve in question. As an example of a curve in space considered in this point of view, we may take the Helix, which is generated by the uniform motion of a point along a generating line of a right cylinder as the generating line revolves with uniform angular velocity about the axis of the cylinder. If we take the axis for the axis of z, and the axis of x through the generating point at any initial time, Q the angle through which the generating line has revolved when the point has moved through a space z on the generating line, we have, for the co- ordinates of the point, a being the radius of the cylinder, a; = a cos ^, y = a sin 0, z= naff ; here is the variable parameter, and the curve is the intersection of the surfaces US' + y^= a^, and y = x tan — . TANGENT LINE. 341 Tangent line to a Curve. 443. To find the equations of the tangent line at a given point in a curve. Let {x, y, z) be the given point P, (»+ Aa;, y+ Ay, z + Az) an adjacent point Q, the equations of the line PQ are Ax Ay Az If Q move up to and ultimately coincides with P, the limiting position of PQ is the tangent at P, if, therefore, dx : dy : dz be the ultimate value of Ax: Ay : Az, the equations of the tangent at P are ^-x ^ 7}-y ^ ^-z dx dy dz (1) Let the equations of the curve be given in terms of a variable parameter d, in the form f=.^(^), v=t{0), r=%w, dx:dy:dz = '{^):^lr'{^):x'{^), and the equations of the tangent at a point corresponding to 6 are ^ — X _7] — y ^ — z wwrwwrYWY (2) Let the equations be those of surfaces containing the curve, P(f, 7), = 0, and O (?, v, ?) = 0. Then on any point P of the curve, F'{x) dx + F'{y) dy + F'{z) dz = 0, and G' {x) dx + G' [y] dy + G'{z) dz^O; whence the equations of the tangent PQ may be written F'{x){^-x)+F'{jj){'n-y) + F'{z){K-z) = 0, and G\x) {i-x) + G'{y) {v-y) + G' {z) (?- z) = 0, which equations represent analytically the fact that the tangent to the curve lies in each of the tangent planes to the surfaces at the common point P. (3) If the surfaces, of which the intersection gives the curve, be cylindrical surfaces, whose sides are parallel to the two 342 CURVES, axes of « and y, and their equations be v=fi^)} ?=^(l)) the equations of the tangent will he ri-y=-f{x) i^-x), ^-z = '{x){l-x). Those equations are the analytical representation of the fact that the projections of the tangent to the curve are the tangents to the projections of the curye on the co-ordinate planes of xy, zx ; which is obviously true, since the projections of P and Q have their ultimate coincidence simultaneously with that of JP and Q. 4A4:. If the point P be a multiple point on the curve, there being more than one tangent line, the equations given in (2) of the last article will be satisjSed by more than one system of values of dx : dy : ds; therefore, either F'{x) = F'{j/) = F'{is) = (1), or G'{x)^a'iy)=G'iz)=0 (2), or both sets of equations hold simultaneously, or else a\x) - a'ijj) - G\z) ^^^' The case (I) occurs when the first surface has a multiple point at P, in which case the tangent lines at the multiple point of the curve are the intersections of the conical tangent to the first surface with the tangent plane to the second. The case of (1) and (2) holding simultaneously is that of both surfaces having multiple points at P, and the tangent lines are the intersections of the conical tangents. The case (3) occurs when the surfaces have a common tan- gent plane at P. 445, To find the directions of the tranches of the cwve of intersection of two swrfaces, at a multiple point. The equations of the surfaces being ^(?,^,?)=0, and ailv,^ = % and {x, y, z) being a multiple point P on the curve, let L:±=tZ^ = t^=r (1) I m n CUKVES. 343 he the equations of a line through P, th^ points in which this line meets the surfaces are given by the equations F{x + lr, y + mr, a + nr) = Ol and G{x + lr, i/ + nir, s + nr)=0) ^ there are an infinite number of directions which give two values of r equal to zero, since the curve has a multiple point at P: therefore the two equations must be one or both identically satisfied, or else they must not be independent equations, 1. If (3), be identically satisfied, then of the values of r common to the equations, two become zero for the directions given by equation (4), and and these determine the directions of the tangent lines at the multiple point. 2. If (3) and (4) be both identically satisfied, the directions of the tangent lines are given by the two equations f4 + -| + -|)>(-.3/,«)=0 dx dy ds 3. If neither be identically satisfied, let (3) x \ + (4) x /i = be an identical equation, and by (2),- XF{x + lr, y + mr, z-^nr)->riiG{x + lr, y + mr, z + nr)=0; therefore the directions of the line (1) which give two values of r common' to (2), each equal to zero, are given by the equations (3) or (4), and (l^ + ,n^j + n^J{XF(,x,y,z)+fiG{x,y,^)]^0....i5)/ 344 CURVES; If (5) combined with (3) or (4) give equal values of l:m:n, the two tangents coincide and the point P is in this case a cusp on the curve. We can proceed in a similar manner for higher orders of multiplicity, 446. To find the differential coefficient of the arc of a curve. If X, y, z and x + Asc, y + Ay, » + As be the co-ordinates of P, and an adjacent point on the curve, As the length of the arc between them, pe==A^VA^' + AJl°; PO therefore, since limit -^ = 1, dividing by (As)' and proceeding io the limit we obtain If the position of P be determined by the variable t, \dt) ~ \dt) "•" \dt) "*" \dt) ' and ■=-, -f- . T- are the direction-cosines of the tangent. as as ' as 447. To find the differential coefficients of the arc referred to polar co-ordinates. Transforming to polar co-ordinates x = r sin 6 cos ^ = p cos ^, y = rsiii.6am^ = p sin -|^'- + (^ + ^.J-2-J+''r^^ + l3? + ''J¥} + *0, • '' where e, e', ... vanish wben t is Indefinitely diminished, there- fore ultimately .dx dy dz ^d^x , d-'y ^ d^z . and '^df+''d^ + 'df='^' ^^e'^^® 'dydh_dz_iY^^d^ ^A" ~3i df dt df dt df ■*■ dt d^ therefore the equation of the osculating plane is idyd:'z d^dSj\.^_.ld^d2x_d^^.. , fdx d^y dy d%\ .y . _ ^ + Kdt-dF-rtw}^^-'^-'^' OSCULATING PLANE. 347 452. The osculating plane may te considered also as the plane which has the closest contact with the curve at the given point. Thus, if L (^ -x) +3f{v-y)+N{^-z)=6 be the equation of the osculating plane at {x, y, z) and x + Aaj, y + Ay, s + A^ be the co-ordinates of a point near {x, y, e), the perpendicular distance from this point on the plane is XAa; + M/^y + iVAa (AST + %T + Ail')^' If x, y, z be functions of t, where e vanishes in the limit ; therefore L, M, N must be chosen so as to make ZAa; + JfAy + iVAs the least possible when A* is made as small aa we please ; therefore Ldx + Mdy + Ndz = 0, and Ld^x + Mdy-irNd^z = 0, whence the equation of the osculating plane is found as before. 453. The osculating plane is also the plane containing two tangent lines at consecutive points. Their equations will be ^ — = ■ , " = ^ — , ax dy az , ^ — x — dx_7]—y — dy_^—z — dz dx + d^x dy + d^y dz + d^z If therefore the equation of such a plane be I {^ — x)+m{ri -y) + n (?— s) = 0, we obtain the equatiojas ldx + mdy + ndz =0, and l^x + md^y + nd''z = 0. 454. To find the direction-cosines of the osculating plane. If I, m, nhe the direction cosines, I m n 1 dy d*s — dz d'y ~ dz d^x — dx d'z ~ dx d^y — dy d^x ~ u ' 348 PRINCIPAL NORMAL. where m° = {dy d^z — dz d'yY + = (J^' + %]' + dzj {d^^' + d''^ +^=) - {dx d'x + dy d^y + dz d\)\ and dx^ + dy^ + dz^ = ds^; .'. dx d'x + dy d^y + dz d'z = ds d's ; .-. i^ = dg'{^^' + d^'+d^]'-'^sY; , dy d'z — dz d'^y . .'. I - , , ^ =■ , &c., ds V^^ + d'y]' + d^'-d'^'' in which d^s = if s he the independent variable. 455. The osculating plane is perpendicular to the line of intersection of consecutive normal planes. This is ohvious geometrically, since consecutive normal planes are the limiting positions of planes perpendicular to two adjacent sides of the polygon whose limit is the curve, and their line of intersection is perpendicular to the plane containing the sides which is ultimately the osculating plane. The equation of the normal plane being {^ -x)dx+{ri- y) dy-{-{X- z) dz = 0, and of the consecutive normal plane {f-x-dx) {dx+d^x) +(ri-y-dy) {dy+d^y)+{i-z-dz) {dz+d'z) = 0, at the line of intersection we have {^-x)d'x + {v-y)d'y+{^-z)d'z-d^T-'^J-d^J=0. Hence, the direction-cosines of the line of intersection are pro- portional to dy d'z — dz d^y, &c. The line is therefore perpendicular to the osculating plane. ' Principal Normal. 456. Def. The principal normal is that normal which lies in "the plane of two consecutive tangents or in the osculating plane. 457. To find the equations of the principal normal at any point of a curve. PRINCIPAL NORMAL. 349 The direction-cosines of the tangent at any point P are dx dy dz ds' ds' ds' and if x, y, z be given functions of t, let t + r correspond to a point Q, the direction-cosines of the tangent at Q will he dx (d /dx\ I % dz d^'^_\jt[d^)'^^j'^' ds""" ' ^"^ If X, fi, V be the direction-cosines of a line perpendicular to the principal normal, and the tangents at P and Q, I, m, n those of the principal normal, l\ + mil + nv = 0, ^ dx ^ dy dz . as as as therefore, ultimately, 'i|@+;'|, (f ) + -^ {£) = "■ , d fdz\ , _ dz •■• dx d ds dt ©-■■ ,...= = 0. Also jdx ''Ts' dy , dz ■f m-f--HwT- = as as = 0; • , J5=0; I m n d idx\ d (dy\ d fdz\ dt [dsj Jt \ds) dt \ds] 350 PEINCIPAL NORMAL, 458. If from any "point 'in a curve equal distances he mea- sured along the curve and its tangent, the limiting position of the line joining the extremities of these distances is the principal normal. From the point {x, y, z) let equal distances o- be measured along the curve and the tangent to the points Q, T, The co- ordinates of Q are X and those of T dx ^(d^x ^ W , dx e vanishing in the limit. The equations of the line QT 2ix& ^ — x "n—y K—^ df^^ ds"^^ ds"^^ therefore the limiting position of QT \s the principal normal, being perpendicular to the tangent, since dx d^x dy d^y dz d's _ ds ds^ ds d^ ds da'' ~ Cauchy proposed, as a definition of the Principal Normal at any point, the limiting position of the line joining the points on the curve and tangent, -whose distances from the point of conr tact measured along the curve and tangent respectively are equal; by which means the definition was made independent of the osculating and normal planes. Four-point System. 459. Equation of a curve of double curvatv/re. If a, ^, % B be the four-point co-ordinates of any plane, the equation of a point is h + m^ + ny + rS = (1), and we have seen (Art. 112) that if I, m, n, r involve one variable in the first degree, the locus of all the points, which can be euBVES. 351 obtained by giving to the variable valnes from — oo to + oo , is a straight line. Let I, m, n, r be any functions of a single variable x, we can show that the locus of points corresponding to all values of the variable is a curve line ; for, if the locus be cut by any plane, and the, co-ordinates of the plane be, substituted for a, /S, 7, S in the equation of the poiut, the resulting equation determines a series of values of the variable x, which correspond to the point in which the locus is intersected by the plane ; and, by shifting the plane, we obtain a continuous series of such points which form the different portions of the curve line of which (1) may therefore be considered to be the equation. If I, m, n, r be rational and integral functions of x, not hav- ing a common factor, and one at least being of the w"" degree, any plane determines n values of x, real or imaginary, and there- fore meets the curve in n points, hence, the curve is of the w"* degree. We may observe that, in order to be sure that the curve is of the «*'' degree, it must not be possible to make any substitution of a new variable so as to diminish the degree, while the functions remain rational. 460. If the curve which is the locus of m^+ny + rS = be traced on the fundamental plane BCD, every point in the curve which is the locus of (1) lies on a line joining A with a point of this curve,^ that is, on the srarfece of a cone whose vertex is A and guiding curve mj8 + ny + rS = 0. 461. If l = a^ + a^x + a^+ ni = b^+\x+b^^+ and a^ct + b^ + cjy + d„8 = a'a, the equation of the curve may be written •aV + d'xfi' + c'cffy + dVB' + . . . = 0. This reduced equation exhibits that a curve of the first de- gree is the straight line joining the points a' = 0, jS' = 0. 352 CURVES. Also that a curve of the second degree is a plane curve, the plane containing the three points a' = 0, ^ = 0, 7=0. Again, by the preceding article, a curve of the third degree, ■which is not necessarily a plane curve, lies on two cones whose vertices are A' and D', and whose guiding curves are the conies traced on BCD' and A'B'O' whose equations are V^' + c V + d'x^h' = 0, and a a! + I'x^' + c'a^i = 0, which have a common generating line A'D', 462. To find the equation of tJie tangent to a curve. Let /(») = la. +OT/3 + W7 + rS = be the equation of the curve, and let x^ determine any point P in the curve, a;„ + \ a point Q. adjacent to it, whose equation will be f{x, + \) =f{x,) +\f K) + ^/" {X,) + ... = 0. The straight line whose equation is is the equation of the tangent at P, since, when Q moves up to P and ultimately coincides with it, the straight line ultimately passes through Q. The distance between adjacent points in the tangent and curve is evidently of the order V, generally. If /"(*o) = 0, the distance is of the order V, and the curve, which in ordinary cases lies on the same side of the tangent on each side of the point of contact, in this case lies on opposite sides, or there is a point of inflexion in the osculating plane. The equation /(a;„) +/*/'(»<,) +^/"(a'(,) =0 is the equation of a conic which has a contact of the second order. A double point occurs when I, m, n, r retain the same ratio for two values of x. 463. To find the equation of the osculating pla^ at any point of a curve.. The plane whose equation is /(a!j+/./K)+v/"(ar,) = CUEVES. 353 is the plane wliich passes through the points f{x^ = 0, /' (a;„) = 0, and /"(«„) = 0, and therefore coincides with the limiting position of the. plane which passes through three contiguous points of the curve. The equation is therefore the equation required. The distance of a point f{x„ + \) = from the osculating plane is ultimately cK^f"'{x^, or the curve generally lies on opposite sides of the osculating plane in passing through the point of contact. Singularities of Curves and Developdbles. 464. A curve of double -curvature and the developable sur- face of which it is the edge of regression, may he considered in connexion with one another; and we may expect that singu- larities in one will have corresponding singularities in the other. Cayley in Liouville's Journal, Tom. X., and in the Cambridge and Dublin Mathematical Journal, Vol. v., and Salmon, in the same place in the latter Journal, have investigated equations among the number of such singularities ; and Salmon has proceeded to shew how curves of double curvature may be classified by the con- sideration of the number of apparent double points in the curves. We shall introduce the student to some of the methods em- ployed, and leave him to consult the papers referred to, if he desire to enter more fully into the subject. 465. A curve of double curvature may be considered as the locus of a system of points, or as the envelope of a system of straight lines, and the corresponding developable surface as the locus of the system of lines, or as th* envelope of a system of planes. We may consider the three systems of points, lines, and planes as connected in the following ways. 1, If the system of points be supposed given, each line of the second system joins two consecutive points of the given sys- tem, and each plane contains three consecutive points of the same system. 2. If the system of planes be supposed given, each line of •the second system is the intersection of two consecutive planes of A A 354 CURVES. th'e given system, and each point of tlie first system is the inter- section of three consecutive planes of the given system. 3. If the system of lines be supposed given, each point of the first system is the intersection of two consecutive lines of the given system, and each plane contains two consecutive lines of the same system. 466. The following terms will be employed : A line through two points denotes a line joining any two arbitrary points of the system of points. A line in two planes denotes the line of intersection of any two planes of the system of planes. A point in two lines is the intersection of any two lines of the system of lines which intersect. A plane through two lines is the plane containing any two lines which intersect, A stationary plane is a singular plane which contains four consecutive points, or three consecutive lines, and occiu's when two consecutive planes coincide. A stationary point is a singular point which lies in four con- secutive planes, or in three consecutive lines, and occurs when two consecutive points coincide. Any plane not belonging to the system contains a certain number of lines in two planes. Any point not belonging to the system lies in a certain number of lines through two points. Any plane contains a certain number oi points in two lines. Any point lies in a certain number oi planes through two lines. These numbers will be denoted by I, X, p, and w, respectively, and the numbers of stationary planes and points by s and or. 467. The degree of a curve is the number of points in which it intersects an arbitrary plane. The class of a curve is the number of osculating planes which contain an arbitrary point. The degree of the developable surface of which the curve is the edge of regression is the number of points in which it meets an arbitrary straight line. CUEVES. 355 The class of the surface is the number of tangent planes which can he drawn through an arbitrary point. The rank of the system is the number of planes which can be drawn through an arbitrary straight line so as to contain lines of the system. Hence the rank of the system is the same as the degree of the surface. The class of the curve is the same as the class of the surface. 468. Singularities in curves of dauble curvature are by these considerations made to depend upon the singularities of plane curves. Let a given plane intersect the surface, the plane curve is thus connected with the lines and planes of the system as follows. Every point of the plane curve is in a line of the system, every tangent to the plane curve is the intersection of the cutting plane with a plane of the system. A straight line in the cutting plane meets the surface in r points, if r be the degree of the surface ; therefore the degree of the curve of intersection is r. A point in the cutting plane lies in n tangent planes to the surface if n be the class of the surface, hence, n tangent lines to the curve of intersection can be drawn through the point; therefore the class of the curve is n. When the cutting plane has a ^oint in two lines the plane curve has a double point, since the curve of intersection may be supposed generated by the intersection of lines of the system with the cutting plane, and the generating line in this twice passes through the same point. When the cutting plane has a line in two planes, it may be seen similarly that the plane curve has a double tangent. At the points in which the cutting plane meets the curve of double curvature, two lines of the system meet the plane curve in the same point, this point is therefore a cusp in the plane curve. For every stationary plane, two consecutive planes coincide, AA2 356 CURVES. and therefore two tangents to the plane curve coincide, or there is a point of inflexion. If m be the degree of the curve of double curvature, the plane curve is of the degree r, and of the class n ; it has I double points,^ double tangents, m cusps, and s points of inflexion. Hence, the formula for plane curves give three independent equations among these numbers. These formulae are given by Salmon in the following forms. If /i be the degree of a plane curve, r its class, S the number of double points, k of cusps, t of double tangents, t of points of inflexion, v = fifj,-l — 2B — SK I— K=3 {v — fi), and 2{r-B) = {v-fju){v + /i-9); whence n = r {r-l)-2l-3m, s — m=3{n — r), and 2{p-l) = {n-r){n + r-9). 469. If, instead of considering the system in connexion with a plane, which intersects the developable surface in a curve, we' consider it in connexion with a point, which is made the vertex of a conical surface whose guiding curve is the curve of double curvature, we shall obtain other relations among the number of singularities, which can be connected with those of a plane section of the conical surface. Every plane through the vertex cuts the curve in m points, corresponding to which are m generating lines of the cone; also a plane which cuts the cone meets the curve in m points; therefore the line of intersection contains m points on the curve, the plane section of the curve is therefore of the tm"* degree. Again, r tangent lines meet the straight line joining any point in the cutting plane with the vertex of the cone, and hence r tangent planes to the cone can be drawn through the point in the cutting plane; therefore there are r tangents to the plane section, which can be drawn through the point, that is, the class of the plane section is r. CURVES. 357 The vertex lies in ■ar lines through two points, hence there are CT double generating lines of the cone, or ■m double points on the plane section. The vertex lies in X planes through two lines, hence there are \ double tangent planes to the cone, and therefore \ double tan- gent lines to the plane section. The r planes of the system .which pass through the vertex correspond to three generating lines, therefore there are n points of inflexion of the plane section. For every stationary point, two consecutive points coincide ; therefore the cone has a cuspidal edge, and therefore the number of cusps of the plane section is v. Hence, the plane section of the cone is of the degree m and of the class r ; and it has ot double points, \ double tangents, a cusps, and n points of inflexion. Thus, three more independent equations are obtained, r = jw (m — 1) — 2ot — So-, w — o- = 3 {r — m), and 2 (\ — ot) = (r — m) (r + m — 9). 470. As an exercise, the student may calculate the number of such points, and the order and class of a section made by a tangent plane of the developable surface, and of a conical surface in which the vertex is on the curve of double curvature. He will find that /i = r — 2 for the first, and m — 1 for the second, v=n—l r — 2 S=l — 2r + 8 zT-m + 2 K=m—3 (T T=p-n + 2 \-2r + 8 t=s w — 3 and the results of substitution in the three equations for plane curves lead to the same six equations among the number of singularities. 471. If a cone be described whose guiding curve is a given curve of double curvature, X lines through two points pass through the vertex and determine a double side of the cone. 358 CURVES. The two points through which any line through two points passes may be either distinct or coincident, as in the case of a multiple point of the curve ; to an eye placed at the vertex of the cone two different branches will in both cases appear to intersect, but will actually intersect only in the latter case ; and in the case of actual intersection the intersection will take place for all positions of the vertex. The sum of the apparent and actual double points is \. Salmon has employed the number of these double points to construct a classification of the curves which are the complete or partial intersections of two surfaces of given degrees; in which the distinctions are made according to the number of points in which the surfaces touch, and the nature of the con- stants where they do touch. The student is referred to the article in the Cambridge and Dublin Journal, Vol. v. XX. (1) Tte equations of the tangent to the curve of intersection of the surfaces ax' ■i-b'f +c«*=l and bx" + cy' + as? = 1, x{i-x ) ^ y{v-y) ^ mj^-e) ab — \dx dz dx_ ENVELOPES. Ittt f].r. = 0, <^,^dz^,dF(d^ d^de) dy dz dy d^ \dy dz dy) ' respectively. But, at all the points in question, ^ (x, y, z) = 0, or —F{x,y,z,a)=0, whence we have -yr = 0, and the two pairs of equations coincide. The tangent planes to the two surfaces will consequently be co- incident, or the envelope will touch any one of the surfaces along the curve in which that surface meets its consecutive. The con- tact will he real or unreal, according as the characteristic is real or unreal ; and may he real for one portion of a series of surfaces, and unreal for the remainder. It is this property which gives rise to the name envelope for the locus of the ultimate intersections. 476. That the locus of the ultimate intersections of the surfaces which involve one parameter touches, in general, each of the surfaces, or is the envelope of the surfaces, may he seen by geometrical considerations. For, if Z7j, Z7j, ZJ, be three consecutive surfaces, the two curves of intersection of ?7j, C^and U^, C^ are, ultimately, two consecutive curves which generate the locus of ultimate intersections, and which ultimately coincide ; these consecutive curves lie both on the locus and on the surface U^, therefore, at every point common to the surface U^ and the locus of ultimate intersections, there will be a common tangent plane ; and the same is true of every other surface of the series ; hence the locus of ultimate intersections envelopes every surface of the series along a curve line. As an example of this class of envelopes, we may take a series of spheres having for diametral planes one series of circular sections of an ellipsoid. Let the equation of the ellipsoid be ^ f ^^ _ 1 ^2 + j2 + ^-l. ENVELOPES. 365 and let tlie equation of a circular section be - VZ^^' + - VF^"' = w. (Art. 184.) a c If we take a circular section of the other system a c the equation of the sphere containing the two will he jc= + «'' + s" - 6^ - (ot + w') - V^^" + (ot - ■or') - Vi^^^ + wot' = ; (Art. 186.) c and the condition that the former section may be a diametral plane of the sphere supplies the condition a^ + c"" whence the equation of the sphere may be written »'+«'+ e- h'- -P-, [ax V^36^+ cs VF^"} + iir= ^^ = 0. This gives, as the characteristic, the plane section made by ax V^^^ + cz 'JW^^= ^ (a^ + c'), and for the envelope, the surface whose equation is (a* - c*) (x" + f+z^- V) = {ax V^^^ + cz '/W^\ This equation may be written in the form a^'^h' ^d a%V '■ ^^TT ' which proves that the envelope required is a prolate spheroid, concentric with the ellipsoid, and touching it along a central section. The equations of the axis of this spheroid are :-l — a'/a^-b' c'^b'-c"' the locus of the centres of the circular sections ; and the squares of its semi-axes may readily be found to be a° + c', and 6^ The foci will therefore be two of the umbilici of the ellipsoid. 366 ENVELOPES. The characteristic in this case is given by the equations ax ^/d'-b" + cz Vj^^" = ZT (a' + c"), a? + f + z' = V + ^^'^^, and the ciu-ve will therefore only be real for values of to- satisfy- ing the condition :t>6'+w'' d'(a:'-V)->r6\b'-&) Also, since the characteristics are circles on a series of parallel planes, their ultimate intersections will be impossible and at infinity ; and the edge of the envelope is in this case the two impossible circular points at infinity, which lie in the plane ax ^W^^ + cz ^V-i? = 0. 477. To fivd the, locus of ultimate intersection of a series of surfaces, the general equation of which involves two arbitrary parameters. Let the equation of any such surface be F{x,y,z,a,b)=0, in which a, b are independent arbitrary parameters. The equa- tion of any consecutive surface may be taken to be F{x, y,z, a + ha, b + hb)= 0, and the points of ultimate intersection of the two surfaces will be given by the equations ^=«' '^'i^''fb='-' or, since ha, hb are independent, on the surfaces 7r_0 dF_ dF_ ■^=^' rfa=^' db-^' ENVELOPES. 367 and, by the elimination of a and h from these equations, the equation of the locus of all such ultimate intersections will be determined. 478. To shew that the locus of ultimate intersections is tlve, envelope of a series of surfaces, whose general equation involves two parameters. The locus of ultimate intersections being obtained by the elimination of a and h from the equations j^ n dF dF ^=^' &=^' ^=^' we may assume that a and b are found from the two latter equations in the forms and that, consequently, the equation of the locus is F{x,y, z, d)j, ^ \dx dz dx) dcp^ \dx dz dx) i dy dz dy d(j)^\ dy dz dy) d(j)^ \ dy dz dy) ~ dF dFdz^^ dx dz dx and ,„ ,„ , > • dF dFdz^^i dy dz dy But, at the point in question, we have If \ 1 J / \ dF ^ dF ^ 368 ENVELOPES. whence, also, at this point, dF ^ dF ^ from which equations it follows that the values of ^ , t- are the same in the two surfaces, or that the locus of ultimate inter- sections touches each of the series of surfaces in the points in which that surface meets the consecutive surfaces. The locus of ultimate intersections is therefore the envelope. This contact may, as in the former case, be either real or unreal. 479. To find the envelope of a series of surfaces whose equa- tion involves n parameters, connected ly either n — 1 or n — 2 equations. We have already remarked that no envelope can exist for a series of surfaces, if the general equation involve more than two independent arbitrary parameters. If therefore the equation involve n parameters, we must have either n-1 or ra — 2 equa- tions of condition, by means of whiclf we might eliminate all the parameters but one, or two, respectively, and the envelope might then be obtained as before. A more convenient method, however, may be deduced from the consideration that the equation of the envelope of the surfaces F{x,y,z,a, 6)=0 is ^ipi:,i/,z)=0, where ^ (a;, y, z) is the maximum or minimum value of F{x, y, z, a, h) obtained by variation of a and 6. The envelope of the surfaces, whose general equation is u=.F{x,y,z, a^,a^ a«)=0, in which a„ a^, o„ are connected by the equations <^, = 0, <^, = «^^ = o, will be ■« = 0, if V be the maximum or minimum value of u ob- tained by variation of a„ o„ a„ subject to the re- 1 or w-2 equations of condition. ENVELOPES. 369 The ordinary method of proceeding by undetermined multi- pliers is described and explained in all treatises on the Diffe- rential Calculus. We will only exemplify it in the following case. 480. To find the envelope of a series of planes passing through the centre of an ellipsoid and intersecting it in sections of con- stant area. The equation of the plane may be taken to be u = lx + my + W3 = 0, the parameters I, m, n being connected by the equations F + m!' + n^=l, lW + mV+nV = d^; whence, differentiating, and using undetermined multipliers \, fi, we obtain \x+ fil +aH = 0, \y + fim + ¥m = 0, \z + ij,n + (?n = 0, and multiplying by I, m, n, respectively, and adding \u + fji, + d^ = 0, whence we may obtain the equation .72 -V . ~ 12 _72 1 - . T^ z a^ — d^ — Xu P—d^—Xu c' — d^ — Xu' or, since for the envelope, m = 0, ^^ V" 2' -J 3. I = a^-d^^W-d' ' c^-cP the equation of the envelope, which is a cone whose focal lines are the asymptotes of the focal hyperbola of the ellipsoid. We may observe that since the maximum or minimum value of u is zero for all points of the envelope, we may put it zero at any stage of the operation, which would have given above the equation and the subsequent work would have been correspondingly sim- plified. BB 370 ENVELOPES. 481. To find the envelope of a series of spheres, having jhr diameters a series of parallel chords of an ellipsoid. Take the diametral plane of the ellipsoid bisecting the chords for the plane of xy, its principal axes for those of x and y, and the axis of z perpendicular to this plane. Then, if 2a, 2b be the principal axes of this section, 2c the diameter parallel to the chords, the radius of a sphere, whose center is («„, y^,0), will be given by the equation and the equation of the corresponding sphere will be Hence, for the envelope a'-a'„-Ja;„ = 0, y-yo-^,y, = 0, and the equation of the envelope is cV cy aVx' WcY _ ^' "*" /W _L <.2\2 I * "T /„2 , -3\2 "T /12 . „2\2 '' : The envelope is therefore an ellipsoid, whose focal ellipse is the section of the given ellipsoid made by a plane diametral to the given chords. Also, we see that if a, p, 7 be the semi-axes of the envelope, 27 being the axis perpendicular to the diametral plane, 6? + ^-rf = a^ + V + ia)}+x'{a) {«-%(«)} = 0, which being the normal plane to the curve on which the center lies, shews that the characteristic is the circle which is the intersection of the normal plane through the center of the sphere with the sphere. The edge is given by those equations and the equation i+fRT+7wT=<^"(«) {y-ia)}+xia) {z-x{a)}. If we eliminate a between the last two of the equations, we obtain the developable surface touching the normal planes of the curve of the centers, which joins the edge of the tubular sur- face whenever it is met by the developable surface. 488. To find the envelope of a plane, whose equation is Ix + my + nz = v, the parameters I, m, n, v being connected by the equations P + ni' + n'' = l, "■" «.s W "^ .fi ^' "" We have, for the point of contact, the equations xdl + ydm + zdn = dv (1) , Idl + mdm + ndn = (2), Ml , mdm , ndn ■, { ^ ] ,„\ v—a v—o v—c [{v—a) J ^' Using undetermined multipliers, \, fi, we obtain '^ + '*^+7Z^===0, (4), \y + /*TO+gj— ^, = 0, (5), \a+/*re+^— ^ = 0, (6), ENVELOPES. 375 Multiplying (4), (5), (6) respectively by I, m, n, and adding, we obtain Xv + fi = 0, (8), and, multiplying, them by x, y, s, -adding, and putting -> 3 . , Ix _ my nz „ X(^»_^,») + _^+...=0, (9). Again, from (4), (5), (6) we obtain A, »• = /A + T-2 -jrj + 7-5 Tjr-j + («' - ay ^ {v" - by ^ (v» - cy = \V--by (7) and (8). Therefore \ = -^-^ 57-, /* == - -= j, and substituting in (4), (5), (6), vl y _ vm » vn and, multiplying these by x, y, z, and adding, we obtain the eqiiation of the envelope a;" y g' _ This is the equation of the Wave Surface, and was first ob- tained in this manner by Mr A. Smith. See Gambridge Trans- actions, Vol. VI. 376 PROBLEMS. XXI. 1. A series of similar ellipsoids are described, having a series of sections of a paraboloid, perpendicular to the axis, as principal sections ; prove that their envelope will be a paraboloid, similar to the former. 2. Find the surface always touched by a plane which cuts off a pyramid of constant volume from three given planes. 3. The envelope of the plane Ix + my + nz = a, 1, 7n, n being connected by the equations f + m" + w' = 1, U + fim + vn = 0, is a right circular cylinder, whose equation is (as" + 2/' + 8° - ra') (V + /a' + v") = (Xa; + /ly + vzf. 4. Find the envelope of planes cutting off a constant volume from the cone ^ y _ «» 5. Find the envelope of the surface {ax + Py + yz) {ax + 'by + cm) = m, o, j8, y being parameters, satisfying the equation 6. If an enveloping cone of an ellipsoid be a cone of revolution, the plane of contact will touch a hyperbolic cylinder. 7. If a cone be described with any point of a central oonicoid as vertex, and the conjugate central section as base, this cone will envelope a similar, concentric, and similarly situated oonicoid, 8. Two generating lines of a hyperboloid, of the same system, are fixed, and a third is equally inclined to the two former, and at a constant distance from the middle point of their shortest distance, prove that all the hyperboloids will touch a hyperboloid of revolution of one sheet. Prove that, in this case, the characteristic is a straight line, and the edge is two straight lines. 9. Find the envelope of a series of spheres described on parallel chords of a hyperbolic paraboloid as diameters. PROBLEMS. 377 10. Find the envelope of the locus of a point, the rectangle of whose distances from two planes is constant, these planes being at right angles respectively to two fixed planes. -11. Find the envelopes of the surfaces (1) a, b, c a',b'c' ".Ay x,y,z (2) {ax + py + yi^y. a, b, e x,y,z -■m, (3) {ax + py+yzf + ^ a, b, e a, j8, y in each case satisfying the condition 12. The envelope of the plane, whose equation is aco&{0 + (l>) + ^cos(6-4>) + yBia{e + 4>) + 8sin(0-<^) = O, 6, <^ being parameters, is the surface CHAPTER XXI. VOLUMES, AEEAS OF SURFACES, &C. 489. To find the differential coefficients of the solid contained between a surface, given in rectangular co-ordinates, the co-ordi- nate planes, and planes parallel to them, drawn through any point of the swrface. Let X, y, z and x + Ao;, y + Ay, » + As be the co-ordinates of two points P and Q upon the surface. Draw planes through P and Q parallel to the planes of yz, zx, and let Fbe the volume CRP80M cut off by these planes from the given solid. If A^F be the increment of V, when X is changed to a; + Aa;, while y remains constant, and a VOLUMES, AKEAS^OP SUEFACES, &C. 379 similar interpretation be given to the operation A„, the volume PrM=A^F; also the volume PQNM, which is the increment of A^F when y changes to y-\-L.y = Ay(Aj,F), which is easily seen to be the same as A^, (A„F). Let z^, Sj be the greatest and least values of z within the portion of the surface PQ, therefore PQNM lies between z^^xAy and gjjAajAy ; .". ■ or ■ '^ lies between s, and z,. Ay Ax . If we proceed to the limit, in which 21=22= z, we obtain or -T — T- = z. dy dx dx dy Since the volume PrM is ultimately equal to the area MM X Ax, the partial differential coefficient ' -p represents the area BM, and similarly -5- the area 8M. 490. The differential coefficient of the volume of a wedge of the solid contained between the planes of zx, xy, a plane through the axis of z, and a plane parallel to y Oz may be ob- tained as follows. If Fbe the volume included between the planes zOx, xOy, the surface, the plane whose equation is ^ = to, and a plane parallel to yOz through any point {x, y, z), AjFis the incre- ment of F when t changes to * + A*, x remaining constant ; A^(A,F) is the increment of A, F when t changes to x + Ax; and this is the volume of the prism standing on a base between xAt Ax, and {x + Ax) At Ax. Aj,(A be the polar co-ordinates of a point P in the sur- face, being measured from Oz, and + A.(j}, and therefore =A^(AgV), and similarly = Aa(A^F). Let Sj, Sjj be the greatest and least values of r'sin^ for the portion PSQToi the surface, then A^Afl Flies between - s^ A^ A^, and - s^ A^ A9, o o and proceeding to the limit, when s^ = s^ = T-'sin we obtain , = -»•' sm 6. dtj)d0 3 494. To find the differential coefficient of a surface referred to polar co-ordinates. Let r, 0, (f) be the polar co-ordinates of P, and let 8 be the surface OPB, 382 VOLUMES, AREAS OP SURFACES, &C. Agfi^ is the increment Pr when 6 changes to + A5, A^ (Afl S) is the increment PQ when ^ changes to <^ + A^. Let ■^j, •>^j be the greatest and least inclinations of the tangent planes at points taken within PQ to the corresponding tangent planes of the sphere whose radius is r and center 0. Therefore PQ is intermediate between r^ sin 6^ A6A4> sec^j, and r^ sin 6^ Ad Acj) sec ■^j , r/ sin 0^ and r/ sin 0^ being the greatest and least values of/ sin ^. Hence ^ is intermediate between r ' sin 5, sec '^, A^ and r," sin 0^ sec ilr^ , which are ultimately equal to r' sin ^ sec ■^ ; j ' 495. To find the differential coefficient of a wedge of a volwme or surface contained between a cylindrical surface, whose gene- rating lines are parallel to the edge of the wedge, and a given surface. Let Oz be the edge of the wedge, z Ox one of the faces, and let the base of the cylinder on the plane of xy be referred to polar co-ordinates p, , z). A^ (ApF) is the prism whose base is pA(f>Ap, which lies be- tween a^pjA^A/a and z^p^A<^Ap, therefore r lies between a,p„ and z^p^; hence, proceed- ing to the limit, d, ^, si), proceeding in the same way, we obtain, as before, that 1 ^ , " lies between p^ A^ Ap sec y^ and p^ A^ A/o sec y^, -^^H^)M%)'- 497. To find, the volume of a solid contained between a cylin- drical surface, and two surfaces or two portions of the same surface, whose equations are given. Let the cylindrical surface have its generating lines parallel to the axis of z, and a guiding curve traced on the plane of xy. Suppose the volume to be divided by a series of planes into slices parallel to the plane of ys, one of which (P) is contained between the planes whose distances from yz are x, x + Ax ; the length of this slice measured on the trace of the first plane will ^^y^-^i in ^liich y^=f{x), y^=f^{x), the forms oi f and/, being obtained from the equation of the trace of the cylindrical surface on xy. Let the slice (P) be subdivided, by planes parallel to the plane oizx, into slips, of which ( Q) is between the planes whose distances from zx are y, y+ Ay, the length of this slip measured along an edge will be z^-z^, where z^ = 0^ {x, y), si^ = (fy^ (», y), the forms of ^j and ^jj being obtained from the equations of the two surfaces, or of the portions of the same surface. Let the slip (Q) be subdivided by planes parallel to the plane of xy into elementary portions (B), one of which is between planes at distances z and z + As from the plane of xy, {Q)=t{B)=AxAy'Z {Az)+eAxAy, eAxAy being the two portions of (^) contained between the curved surfaces and the complete parallelepiped, therefore e vanishes when Ax, Ay are indefinitely diminished ; .-. {Q)=AxAy{z^-z^ + e). Again, if e^ , e^ be the greatest and least values of e through the slice P, (P) = 'Z{Q) lies between 384 VOLUMES, AREAS OF SUEFACES, &C. Aa3 S {Ay (», -z^ + ej} and Ax t {Ay (a, - s^ + Cj)}, or between Aa; jl^'j J't^scJTy+e, (y,-y,)| and Ax \yjj^dzdy+e^ {l/i-!/i)\ i .: (P) = Ax [j^'ll'dz dy + v), where 17 vanishes with Ate. Again, the whole volume, = 2 (P), lies between SJAajN "1 ^dzdiz + TjAi and S JAo; N j ' dz di/ + r)j[ , where nj^, rj^ are the greatest and least values of r} and ultimately vanish ; f^s CSfa C^2 .: 't{P)= i I dzdy dx, which is the volume rec[uired, the integrations being performed with respect to z first, con- sidering X and y constant, next with respect to y, considering x constant, and lastly with respect to x. The following examples will be useful to explain the method of determining the limits of integration. 498. To find the volume, contained between the ellipsoid whose g» y^ z^ equationis -^ + ^+-^=1, and the cylinder whose equation is a Ci a? + y'= 2rx, 2r being less than a. Here, using the above notation, y^=+ '\/2rx — x^, a;, = 0, x^ = 2r, and the volume is r2r/-V2;irp /- ^ y VOLUMES, AMAS OP SURFACES, &C. 385 499. To find the volume contained between the surface whose equation is [x + yY— iaz, the tangent plane at a point (a, fi, 7) , and the planes ofzx and yz. The equatioa of the tangent plane at (a, /S, 7) is (« + y8) {x-a) + (a + /3) (y - 0) - 2a (« -7) =0, or a' + y = \/^(a + 7)* In this case ....,+^(.+„, ^.(^. and for a given value of x the tangent plane meets the surface when {x+yY = 4 '^ay (x + y) — iar/, or y = 2 '/ay — x ; •■•2'i=0, y^ = 2'/ay-x; .•. «, = 0, x^ = 2^ar/; f^B fl/s f'l therefore the volume = 1 I / dzdy dx J «?, J yi J Zi /ic^rys 1 , ^] -^(a;-2Va7)'(& 4.12a 3 ^ This result may be verified thus. Let A OB be the"' surface, AOB the tangent plane along the line AB, ABB parallel to xOy, adh a section parallel to xOy of the surface of thickness dz, area adh : area ABB :: ad^ : -4i)^ :: Ot? : OB; .-. volume A OBB = I 2a7 .-dz = arf; •/ v J[CZ)i?=^2a7.27 = |ay; o .0 .'. volume required = -^ . cc 386 VOLUMES, AREAS OF SUEPACES, &C. if 2 500. To find the volume of the elliptic paraboloid j--\ — = 2a;, cut ofi hy the plane Ix + my + nz =p. Perfonn the. integration in the order a;, y, z, _ y'^ a" _p — my — nz For a given value of z, the values of y at the curve of intersection are given by the equation x^ = x^, or y^+-j-y+-B^-j{p-nz)=0 (1)> of which y^ , y^ are the roots, and z must he taken between the limits which correspond to y^—y^, or »,, z^ on the roots of the equation h z''--f{p-nz)=-j^ I l' (2). ThevoW = 0;(£^f:|-f-^,-£).,.. ==2blJ (2/ - 2^i) (y^ - !f) Ay dz by (1) , VOLUMES, A^AS OP SURFACES, &C. 387 1 ["a [Vi ^¥b\l'My^-y^y^'' {y, -yxT= (y. + y,) - ^y,y, = ^ (« - ^d («, - z) hj (2), therefore the volume = Wb\iy (!)'i("» - "') (" - «') - (« - ^^)'}' ^' "" 3 ? j ('^~'^y ^^' "^^^^® 27 = a, - »j , 4 5* , /■ 2" = 3 71 7 I cos* ^ c?^, putting m = 7 sin ^, 4 ■ c5 2* ' .-. volume = ^V5-cM±^f±£!^\ 4 ■ f* The student may verify this result by the summation of ele- mentary slices hounded by planes parallel to the given plane. 501. To find the volume contained between surfaces given by polar co-ordinates. The volume of an elementary parallelopiped is r sin Q dr dd d(j>. If we integrate this expression from r = r^ to r = r^, r,, r, being the radii of the bounding surfaces, corresponding to 8, (f), we obtain a frustum of a pyramid the angular breadths of whose faces are dd, d^, intercepted between the two sm-faces or the two sheets of the same surface, = g sin ^cZ^c^^ (j-j' — t-j'), the radii being given in terms of 6 and +d^ constant, 0=0^to0=0^, 0^,0^ being given in terms of ^ by the bound- aries of the volume considered, we obtain the portion included between the planes inclined to a Ox at angles and j'j I ir,'-r;) sin 0. The whole volume is found by integrating from ^ = ^i to ^ = ^2, the extreme planes between which the volume is in- cluded. The volume is therefore -r {r' - r^) sin dd d. 502. To find the volume of a sphere cut off ly three planes through the center. Let a = radius of the sphere, ABO the spherical triangle cut off, OC the axis of z, OOA the plane of sa;. The ec[uation of the plane OAB is cos ^ — /3 = tan a. cot 0. The limits of integration are »• = to r= a, = to 0= cot"' (cot a cos^-yS)) ^=0 to ^= C, volume = II -^ sin 0d0d^ ^C^a^ # =/, 3 Vsin'' a + cos" a cos' (<^ - /8) g-Pa° d^ -P 3 Vl — cos'asin"^ 503. To find the volume of a wedge of a sphere cut off hy a right circular cylinder, a diameter of whose base is a radius of the sphere. Let the equation of the sphere be p' + z^ = a", and that of the cylinder p = a cos ^. VOLUMES, AREAS OF SUEPACES, &C. 389 The volume is \\2p\/c^—p'dpd<^, from p = to p = acoa, ^ = to = a, = 1 - (a° - a' sin'<^) d^ Jo ^ = -a'ja— jl (3 sin ^ - sin 3 ^) (?0|• ^ I *' f I ^^ ~ "^^^ "^ "^ II ^^ ~ "^"^ ^"'H " It « = S" > the volume = — ^ . The surface =/Vp^+ (|)Vp^(|y<^.#. between the same limits, = \\^f^d<^ JJ'^a'-p' ^ = a^ j (1 — sin ^) d^ Jo = a' (a — 1 + cos a). If a= - , the surface is a" (— — 1 ] . 504. To _ywJ the volume of a solid whose houndmg surfaces are given hy Tetrahedral Co-ordinates, If X, y, z be co-ordinates referred to rectangular axes of a point whose tetrahedral co-ordinates are a, /S, 7, h. Since a, j8, 7 are linear functions of x, y, z and a-|-/8 + 7 + 8=l, llldx dydz=G {{{da dp dy, 390 VOLUMES, AREAS OP SURFACES, &C. and if V be the volume of the tetrahedron of reference, \\\dxdydz = F. If the limits of x, y, z correspond to the boundaries of the tetrahedron, and we evaluate ||IJa *^^ summation being taken over the whole sur- face. Find A(S' in terms of a, y3 if a; = a cos a, y = 5 sin a cos yS, and s = c sin a sin yS. 10. Prove that the volume cut ofi" by the plane y^k from the surface aV + & V = 2 (ax + bz) v' is =-;^-^ . ^ ' -lOflso 11. Two cones have a common vertex in the center of an ellipsoid and bases curves in which the surface is intersected by planes parallel to the same principal plane, prove that the volume of the ellipsoid contained between the cones varies as the distance between the planes. 12. Prove that the volume contained between the surface xs^ + (x — c)r'=0, and the plane z = {c—x) tan a is ire 96 ( 6 cot a cosec a - 4 cos^ a — 3^ cot ^ j . 13. Prove that the volume, intercepted between the surfaces whose equations are xyz = a", x' = by, y' = 6a! and x = c, is trisected by the planes y = c and y = x. 14. If /S' be a closed surface, dS an element about P, at a dis- tance r from a fixed point 0, (p the angle, which the normal drawn inwards makes with OF, shew that the volume contained by the rface = ^ jjrcoa ) sin e = 2ir/S'. J -IT J Adapt this formula to the case of an ellipsoid, testing its accuracy by any independent process. 16. Prove that the area of a closed surface, no plane section of which has singular points, may be expressed by the definite integral /:/: r' ain^ ddO where p is the perpendicular from the origin upon the tangent plane. ■" ' dS 17. Find // — , where dS is an element of the surface of an n- elHpsoid, f being the perpendicular from the center upon the tan- gent plane of the element, the integral being extended over the whole surface. Wh iD I ~"'^- .-'lb IRDSTAlte WOISTElkoLME'S GEOMETRY, jtuTmOant Of Gimliridge' WJHdax VSi/- CHAPTEE XXII. CUEVATUEE OF CURVES IN SPACE. 505. In this Chapter we shall exhibit some of the methods by which the degree of curvature of curves of double curvature has been estimated ; this curvature is of two kinds, one having " reference to the rapidity with which at different points the curve deflects from its tangent in the osculating plane, and the other having reference to the rapidity with which the planes contain- ing consecutive elements change their position. The first is of the same nature as the curvatm-e of plane curves, the second is called the curvature of torsion, and is peculiar to curves in space, 506. Let an equilateral polygon be inscribed in a curve," of which consecutive sides are PQ, QB, BS, 8T, and lei p, c[, r, s be the middle points of these sides. Let Aap, Bhq, Gcr be planes perpendicular to these sides forming the polygon ABGD by their intersections. If the sides P^, QB, be diminished indefinitely, their directions are ultimately those of tangents to the curve, the planes Aap, Bhq are ultimately normal planes to the curve, the planes PQB, QB8, are osculating planes, the surface generated by the plane elements Aah, Bbc, Ccd, is ulti- mately the developable surface enveloped by the normal planes of the curve, of -which. ABOD is ultimately the edge of regression. The developable enveloped by the normal planes is called by Monge the Polar Developable. 507. Osculating Circle. A circle can be described contain- ing the points P, Q, B; this circle,' therefore, lies in the oscu- lating plane, when the sides are indefinitely diminished, and its curvature may be taken as the measure of curvature of the curve 394 CUEVATUEE OF CUEVES IN SPACE. in the osculating plane. Let the plane PQB meet Aa in U and pU, qUhe joined, since AU is the intersection of planes per- pendicular to PQ, QR, it is perpendicular to the plane PQB, and therefore to pU, q U; hence, since pU, qU are perpendicular to PQ, QR, U is the center of the circle. Therefore the center of the osculating circle is the point of intersection of two consecutive normal planes and the osculating plane. 508. Angle of Contingency. The angle p Uq which is equal to the angle between the two consecutive sides PQ, QR of the polygon, is ultimately equal to the angle between two consecu- tive tangents, and is called the angle of contingence, this angle is also the angle between two consecutive principal normals. 509. Spherical Curvature. If pa, qa be drawn to any point in Aa, since Pp = Qp, a is equally distant from P and Q, and similarly from Q and R, hence any point in Aa is equally distant from P, Q and R : similarly, any point in £b is equally distant from Q, R, and 8, therefore A their point of intersection is equally distant from the four points P, Q, R, 8. Hence, it follows that a sphere can be described whose center is A and which contains the four points P, Q, R, 8, this sphere is ultimately the sphere which has the closest possible contact with the curve, since no sphere can be made to pass through more than four consecutive points, and is therefore called the sphere of curvature; the locus of the center of spherical curvature is therefore the edge of regression of the polar de- velopable. The line of intersection of two consecutive normal planes, a property of which is that all points are equidistant from the osculating circle, Monge calls the Polar line. 510. Curvature of Torsion. The plane p Uq perpendicular to A Ua contains the sides PQ, QR, and the plane qVr perpen- dicular to BVb contains the sides QR, RS, and since qU, qVare perpendicular to the line of intersection QR of the two planes, the angle C^ F is their angle of inclination. This angle which is ultimately the angle between consecutive osculating planes is called the angle of torsion. CUEVATUEE OP CURVES IN SPACE. 395 Also, since the angles UqV&ni UBV are equal, it is evident that the angle of torsion of the curve FQB, is equal to the angle of contingence of the edge of regression of the polar developahle. 511. Osculating Cone. The osculating cone at any point is a circular cone which touches three consecutive osculating planes, having its vertex at the point of the curve in which these planes intersect. 512. Evolutes. If a be any point in the intersection of the planes normal to PQ, QB,, at their middle points p, q, it has heen shewn that ap = aq^ and they make equal angles with Aa. Produce g'a to meet Eb in h, a string, placed in the position hap, would remain in that position if subject to tension, since the tensions of the portions ab, ap resolved parallel to Aa would be equal, and if fixed at a might pass through q without shifting the position of a. Similarly, if rb be produced to c in Gc, and if 8c be. produced to d in Dd. If we proceed to the limit, it follows that a string may be stretched upon the polar developable in such a manner that the free end, passing through any point in the curve, would describe the curve, if the string were unwrapped from the surface so that the part in contact with the surface remained stationary. The portion in contact lies on a curve called an evolute. Also, since the position of the point a is arbitrary, the curve which is the limit of a,h, c, d, ... will change its position ac- cording to the position of a, hence the number of evolutes is infinite. All the evolutes of a curve are geodesic lines of the polar developable. 513. Locus of Centers of Circular Curvature. Since g"?/ will not, if produced, pass through F, because q U and q V include an angle in the same normal plane, the locus of the centers of cir- cular curvature is not one of the evolutes. 514. The Bectifying Developable. If through every point of a curve a plane be drawn per- 396 CURVATURE OF CURVES IN SPACE. pendicular to the corresponding principal normal, these planes will envelope a surface on which the curve will he a geodesic line, since its osculating plane contains the normal to the sur- face at every point. This surface is called the Rectifying Developable, since if it be developed into a plane, the curve will he developed into a straight line. The Hne of intersection of two consecutive planes is called the rectifying line for any point of the curve, being the line about which the surface is turned in order to rectify the element of the curve on that point. It may be observed that the rectifying line is not generally coincident with the hinormal, which is the normal perpendicular to the osculating plane. In the figure at page (393) the surface whose edge of regres- sion is ultimately ABO ... is the rectifying surface to the curve which is the limit of abc . . . Aa is the rectifying line at a, and the hinormal does not coincide with the rectifying line unless joa is perpendicular to Aa, or a be the center of circular curvature of the involute of aic . . . 515. If the polygon PQB8 ... were transformed into a plane polygon by turning the portion QR8T ... through the angle of torsion T^ZJ about QR, and the portion R8T ... about RS through the corresponding angle of torsion, the inclination of any side 8T in the new position in the plane of PQR would be inclined to PQ, at an angle equal to the sum of the inclina- tions of the sides taken in order, and estimated in the same direction". Proceeding to the limit, we see that if, as a point moves along a curve of double curvature, the curve be turned about the tan- gent line at every position which the point assumes through the angle of torsion, the curve will be replaced by a plane curve, such that the inclination of the tangents at the starting point, and any other point, will be the sum of all the angles of con- tingence; if, therefore, e be taken for the angle between the tangents in the plane curve, de will be the angle of contingence corresponding to the extremity of the arc traversed by the moving point. 516. The rate at which the osculating plane twists about CUEVATUEE OF CUEVES IN SPACE. 397 the tangent line at any point, called the rate of torsion, is mea- sured by the limit of the ratio of the angle of torsion to the arc at the extremities of which the osculating planes are taken. If, as we pass from PQ to QB, see figure, page (393), QB be turned in the plane PQB so that FQB is a straight line, and the plane QB8 be then turned through the angle Vq U, and the same process be repeated, the perimeter becomes rectified, and the inclination of the last to the first position of the plane con- taining two elements is the sum of all angles such as Vq U. Proceeding to the limit, it follows that, if osculating planes be taken along the curve, and the elements of the arc be rectified in each osculating plane in order, the angle between the first and final positions of the osculating plane when the curve is so rectified, is the sum of all -the angles of torsion. If therefore t be this angle, dr is angle of torsion, corre- sponding to the point at which the last osculating plane is drawn. 517. To find the angle hetween consecutive radii of curva- ture of a curve. Let PQ, rQB, sBS be directions of sides of a polygon which ' are ultimately tangents to a curve. /i \. In the planes, PQr, rBs, respectively draw QU, QV perpen- dicular to r QB, sB8, these are ultimately directions of consecu- tive radii of curvature. 398 CURVATURE OP CURVES IN SPACE. Draw QV in the plane QRS perpendicular to QR. Therefore / Z7'^Z7=(?t ultimately, ^ VQU' = de and i VQ U = d'^ the angle between consecutive radii of curvature. Also, if VQ = VU = VU', VIP=VU"'+ Un" ultimately; .'. dyjrj = de f + dr] , ds and, if -Tj si? be called the radius of complex curvature, p, a the radii of circular curvature and of tprsion, -i-i ^ 518. To find the vertical angle of the osculating cone of a curve. Let joOo, qPp, rQg_ be three consecutive planes which be- come ultimately the osculating planes of a curve. These planes intersect in P. Take P as the vertex of a circular cone which touches each of the planes, and let PH be the axis, op, pq, qr the sections of the planes made by a plane perpendicidar to the axis, t, u the points of contact with pq and qr. CUEVATUB» OF CUEVES IN SPACE, 399 Draw tE, uH perpendicular to the planes pP^, qQr; there- fore if i/r be the semi-vertical angle of the cone, and de = ^ ultimately, , 2qt .•. tan •\V=^=- , which gives the vertical angle required. 519. The, rectifying line is the axis of the osculating cone at any point of a curve. For, in the figure in the last article, each of the planes through the tangent lines PQ, QR perpendicular to the osculating planes pPq^, g_Qf,, ultimately contains the axis PH. 520. The rectifying surface is the locus of the centers of principal curvature of the developable of a curve. It will be shewn in the following chapter that the normal sections of least and greatest curvature in any surface are per- pendicular to one another, and the section of least curvature in a developable surface is that through a generating line, the normal section perpendicular to this line is therefore the section of greatest curvature. Now the plane uHt is ultimately the normal section per- pendicular to PQg, and S is therefore the center of principal curvature, every point of the rectifying line is also such a center, and the rectifying surface is the locus of all the centers of principal curvature of the developable of the original curve. Also the radius of principal curvature of a point in the de- velopable whose distance measured along a tangent to the curve is c, will therefore be c tan ■\!r = c T- . ^ dr 400 CURVATURE OP CURVES IN SPACE. 521. To find the angle of contingency of the locus of the centers of circular curvature. Let 5 F be the intersection of the planes perpendicular to QB, BS the sides of a polygon which are nltimately tangents to a curve ; rV, qV perpendicular to BV; qU, rW perpendicular to BU, CW, the polar lines preceding and succeeding BV. UV, TFFwill ultimately be tangents to the locus of the centers of cur- vature, and if WV be produced to w, UVw will be the angle of contingence required. Since qVB, qUB are right angles, U, V lie in a semi-circle on Bq, and qVU= qBU= suppose ; also wVr = rCW = rGV+ VCW = ^ + d^ + dT; if therefore a sphere, center V and radius unity, meet Vq, Vr, VU, and Vw in q, r, u, w and ux be parallel to qr', uv? = ua? + xvi' ultimately, uw = angle required, uq = ^, qr' = de, xw = d(f) + dT; therefore square of angle of contingence = (cos + drf, where sin ^ = ratio of the radius of circular to that of spherical curvature. 522. To find the element of the arc of the locus of centers jof circular curvature. CUEVATUEE OP CUEVES IN SPACE. 401 Employing the figure of the last article, we see that UV is ultimately the element required, and VqU=dT; hence UV'={qV-qUy+{qUdTy; .'. da- 1 = dp\ + pdT\. Al UV _ amUBV sin UgV ^^^° liF~ sin BUV ~ sin BgV' .-. UV=Bq sin UqV; .'. da = RdT, B, being the radius of the spherical curvature. 523. To find the radius of spherical curvature. Since UBV= VqU^dr, and Bq is ultimately the radius of spherical curvature, = B suppose, qU=p, qV=rV=p + dp, let qBU=2 ) B E J 7 dw wdR and dn = -^ ^ . Also R!' = u'' + v^ + w''; .'. RdR = udu + vdv + wdw) ^7l2 J 1= 5-12 dul + TvJ + dwJ 2RdRf R'dlT? .: dl\ +dm\ +dn\ = — '■ ^ ' W^ RT^ R (du^^ + ^T + ^T) - {R dRY _ {v dw — w dv) ^+{wdu—u dw) "+ {u dv—vdv) " - ^i , and udv — vdu = u{dz d'x — dx d'z) — v {dy d'z — dx d'y) = (m d'x + V d^y + w d'z) dz.; /drV _ {u d'x + vd''y + w d'z)'' _ 1 ■"• [dsj ~ {u^ + v' + wy ~ &" where 03 = 20" 3 y = bs\ and s = cs'. dx ds' = l-2cV, dy ds' = 3bsr, dz ' ds~ 2cs; d^' = - ic\ ds"' = 6bs, d'z d^~ ■.2c; ds ds' ds ds'^ ' _ dz d'x dx d^z _ ds ds" ds ds" ' _ dx d"y dy d^x _ , ds ds" ds ds" Let e, T be the angles contained between the tangents and osculating planes respectively, at the origin and at a point P, whose distance from the origin, measured along the curve, is s. Therefore sin' e = 1 — ( -^ j = 4c V ultimately ; Also sin" T = 1 — .•. 6 = 2cs ultimately. m? _ ?' + w° 365V r + m" + n"~l" + m" + n"~ ic" ' .■. T = — ultimately. Hence, if p and o- be the radii of curvature and torsion. and the equations of the curve near the origin assume the forms «' s" s^ 6/3"' ^ 6po-' 2/3 CUEVATUEE OF CURVES IN SPACE. 407 529. To find the shortest distance between two consecutive tangents. Employing the axes chosen in the last article, the distance between the tangent at P and the axis of x is the perpendicular from the origin upon the projection of the tangent at P on the plane oiyz, and the equation of this projection is ('-^)S-(f-)|-». which reduces to therefore 8, the distance between two consecutive tangents, is ultimately equal to — — . 530. The distance, from the osculating plane at any point, of a point near the former, is twice the distance of the tangents. The distance of P from the osculating plane at = « = -il=28. •^ 6/30- 531. The angle between the line of intersection of osculating planes at two consecutive 'points and the tangent at one of the points is equal io half the angle of contingence. The equation of the osculating plane being l{^-x) + m{'n-y)+n[};-z)=Q, the line of intersection with the osculating plane at has the equation I i^-oe) + n (?- z)-my = 0, and the inclination to the axis of x is ultimately -- = cs = ^e. n 532. To find the angle between the osculating plane at any point and the tangent at an adjacent point. The angle required = ^ = Y~a- ^^^^^^^^7- 408 PROBLEMS. 533. To find the angle between two consecutive principal normals and their shortest distance. The equations of a principal normal corresponding to a distance s from are ultimately s s 1 ' P^ /JO- p The direction cosines are proportional to . - and 1, therefore the angle between this normal and that at the origin T V p- a The shortest distance is equal to the perpendicular from upon the projection on the plane of xy, whose equation is p(f-s)+a-9? = 0; therefore the shortest distance is P^ Vp' + , PEOBLEMS. 1. Prove that the locus of the center of absolute curvature of a helix is another helix, coaxial with the former. 2. Prove that the equation of the polar surface to the helix is X cos <^ + 2/ sin <^ + a tau^ a = 0, where x' + y'' = tan^ a [a' tan' a+ (z — a tan atjif], and that its edge of regression is a helix of the same inclination on a cylinder whose radius is a tan^ a. 3. Prove that the area of the hemispherical spiral in which the latitude X and longitude I are connected by the equation contained between the spiral and the base of the hemisphere is to the area of the hemisphere as 2 : tt. 4. A curve is formed by the intersection of a hemisphere, and a cylinder whose base is the circle described on a radius of the base PEOBLEMS. 409 of the hemispliere as diameter, prove that the area of the hemi- sphere included between the curve, the meridian, touching the cy- linder and a quadrant of the base of the hemisphere, is equal to the square on the radius of the hemisphere. 5. 'Prove that the volume contained between the cylinder, the hemisphere, the meridian plane touching the cylinder, and the base of the hemisphere is |ths of the cube of the radius of the hemi- sphere. 6. A hemisphere is pierced by a cylinder, whose circular base touches the base of the hemisphere, the diameter of , the base of the cylinder being less than the radius of the hemisphere. Prove that the area of the cylinder included between the hemisphere and its base is equal to the rectangle contained by the diameter of the cylinder and the chord of the base of the hemisphere which touches the base of the cylinder and is parallel to the common tangent of the bases. 7. If p, cr, R be the radii of curvature, torsion and spherical curvature of a curve at a point whose distance measured from a fixed point along the curve is s, prove that it)- 8. When the polar surface of a curve is developed into a plane, prove that the curve itself degenerates into a point on the plane, and* Lf r, p be the radius vector and perpendicular on the tangent tcJthg^ developed edge of regression of the polar surface drawn from/thi^/ point, prove that 9. Prove that the angle between the shortest distance of two Bonsecutive tangents at two points, and the binomial at one is equal to half the corresponding angle of tension. 10. Prove that the angle between the chord joining two con- secutive points and the tangent at one of them is half the angle jf contingence. 1 1. Prove that the angle between the radius of the osculating sphere, and the edge of regression of the polar surface is equal to :lie angle between the radius of the osculating circle and the locus of ;he center of curvature. CHAPTER XXIII. CCEVATUEE OP SUKFACES. LINES OF CUEVATUfiE. Def, Two surfaces are said to have a complete contact of the w* order at a common point, when the sections of 'the surfaces, made by any plane passing through that point, have a contact of the w*'' order. 534. To find the conditions necessary in order that two sur- faces whose equations are given may have a complete contact of the n* order at a given point. If P be the given point, and if PQ, PQ' be equal arcs measured along the curves in which any plane \, /i, v intersects the two surfaces, these curves will have a contact of the «*'' QQ order, if the limiting value of pQn+i be finite. Hence the values of dx dy dz d^ d^ d^ ds' ds' ds' ds"' ds"' ds"' must be respectively equal in the two curves. But the values of these differential coefficients are found from the equations dz dz dx dz dy dx dy '^^ _ a ds dx ds dy ds ' ds ds ds ' fdxV , fdyV , fdz\^ , (* ) + [i) + [ds) = '' and the equations formed by differentiating these. We shall, therefore, obtain them as functions of dz_ ds d^z d^z d^ d^ '^'"'Ja;' ^' d^' Wd^' ^' ^' CONTACT OP SUEPACES. 411 In order then that these may be respectively equal in the two curves for all values ofX: fi-.v, we must have the values of dx di/" respectively equal in the two surfaces at the given point. These are, therefore, the conditions required. We may observe that if two surfaces have a complete con- tact of the n* order at a common point, there will he n+1 directions in which a plane can be drawn meeting them in curves which have a contact of the {n + 1)* order. For, taking the origin at the common point, and referring them to their common tangent plane as that of xy, the equations of the two surfaces will each be of the form e = rx''+2sxi/ + tf + ... + A^x"^' + AX^+ ...+A^^y'''-^ + ..., the coeflScients being the same in the two surfaces for all terms of not more than n dimensions in x and y. If then we take a section of the surface by one of the n+1 planes, determined by the equation AX^' + A^y + ... + A^y^' = ^/a;"^ + ..., the difference of the ordinates of the two curves in such a plane will in the neighbourhood of the origin, be of the order of small- ness (w + 2), and the curves will have a contact of the {n + l)* order. Hence if two 'surfaces have a common tangent plane, two normal sections can be drawn which have a contact of the second order. If two surfaces have a complete contact of the second order, three normal sections can be drawn which have a contact of the third order. If the surfaces be conicoids, it may easily be found that two of these three sections are the real or impossible generating lines through the point, and that the third is always real, having the same tangent lines as their remaining curve of inter- section, which is then a plane curve .passing through the point of contact. 535. To determine the number of constants which must he involved in the general equation of a surface, in order that it may he made to have a complete contact of the w*** order with a given surface, at a given point. 412 CONTACT OF SUEFACES. Substituting the values of x and y at the given point, we must have the corresponding values of dz dz d"z ^' dx' ~dty' 1^ the same in the two surfaces, which will supply relations among the constants which must be satisfied in order that the proposed contact may take place. Thus, when w = 2, we require six disposable constants, whence we cannot, in general, find a sphere which has a complete contact of the second order with a surface at any point. Since, however, there are four constants in the general equa- tion of a sphere, our six conditions for contact of the second order will lead to two relations among the co-ordinates of the point, which together with the equation of the surface will determine in general a definite number of points at which a sphere can be drawn as required. Such points are called umbilici. The general equation of a conicoid, referred to axes parallel to its principal axes, involves six constants, and we can there- fore always determine a conicoid, with axes in any proposed directions, which shall have a contact of the second order with a given surface at any point. 536. Also the general equation of a conicoid, having a given point at the extremity of one of its principal axes, involves six constants ; and we can, therefore, always determine a coni- coid which shall, at the extremity of one of its axes, have a complete contact of the second order with a given surface at a given point. When w = 3, the number of conditions required is ten, and we cannot therefore, in general, determine a conicoid which has a contact of the third order at a given point of a surface. Two conicoids cannot have a contact of the third order at any point without complete coincidence. We can, however, find, at any point of a conicoid, any number of conicoids such that all normal sections through the point have contact of the third order. The .CUEVATUEE OF SUEFACES. 413 general form of the equation of sucli conicoids, referred to the tangent plane and normal at the given point, will be z = ax' + If + dyz + Vzx + cxy + Gs", the constants a, i, a', h', c' being the same for all, and C arbi- trary. 537. To investigate the relations hetioeen the curvatures of the different normal sections of a given surface at a given point. Let the surface be referred to the tangent plane and normal at the given point; its equation may then be written in the form z = ax' + ly^ + cz^ + 2dyz + ib'zx + 'icixy + terms of higher dimensions. Also, let p be the radius of curvature of any normal section, inclined at an angle Q to the plane of zx. Then, if (a;, y, z) be a point near the origin, we shall have l = 2lt.-,^, = 2lt.^^^±M±^, or — = a cos^ 6 + 2c' sin d cos ^ + a sin^ 9, 2p whence it appears that the radius of curvature is proportional to the square of the diameter of the conic ax^+2G'xy+hy^ = l, which is parallel to the tangent line through which the normal section is drawn. It follows from this, that -there will generally be a maximum and a minimum value of the radius of curvature of a normal section at a given point, corresponding respectively to the transverse and conjugate axis of this conic. If p^, p^ be these principal radii, and p, p the radii of curvature of any other normal sections at right angles to each other, we shall have 1111 - + -=- + -, P P Pi Pi or, the sum of the curvatures of any two normal sections at right 414 CURTATUBE OF SURFACES. angles to each other, at any point of a given surface, is constant; which is "Euler's Theorem." It appears also from the above, that the section made by a plane parallel and near to the tangent plane at any point which is not a singular point, is ultimately a conic ; this conic, from the properties here proved, is called the " Indicatrix" of the surface at the given point. The indicatrix will be an hyperbola, ellipse, or two parallel straight lines, according as the two tangents to the intersection of the surface by its tangent plane are real, impossible, or coin- cident ; and the point is called a hyperbolic, elliptic, or para- bolic point accordingly. The criterion distinguishing these different cases is U^ (u'^-vw) + ... + 2VW{uu'-vw') + ... >< = 0, writing Z/for -5-, m for -j-j, m' for , , , &c. See Art. (289). If the equation of the surface be given in the form z = (p{x,y), the corresponding criterion is «"-»•<>< = 0. At a hyperbolic point, the radius of curvature, being pro- portional to the square of the corresponding diameter of the indicatrix, will change sign on passing through the section whose tangent lines are parallel to the asymptotes. This indi- cates that the corresponding radii of curvature will be drawn on opposite sides of the tangent plane. The radius of curvature will be infinite for the limiting sections, as might have been inferred from the fact that in those directions, three consecutive points of the surface lie in a straight line. Art. (286). At an elliptic point, the whole of the surface in the neighbourhood of the point lies on the same side of the tangent plane, and only touches it in one point. At a parabolic point, -the surface in the neighbourhood of the point also lies wholly on one side of the tangent plane, but it has contact with it at points near the given one. In this case, the radius of curvature is of constant sign, and becomes infinite in only one direction. It has a mini- mum value for the normal section at right angles to this direc- CURVATUEE OP SURFACES. 415 tion, and if p^ be this minimum value, p the radius of curvature of a section inclined at an angle 6 to this, we shall have p = Pj sec" 6. The reader will observe that every point of a developable surface is a parabolic point, and every point of a skew surface a hyperbolic point. Arts. (295, 298). 538. To determine the radius of curvature of a given oblique section at any ^point of a surface. Let 0. be the given point, Ox the tangent line to the given oblique section, Oz the normal at 0, cf) the angle between the plane of section and the plane z Ox ; through N any point on the normal draw NQ parallel to Ox to meet the surface in Q, ' and through QN draw a -plane parallel to the given section cutting the surface in the curve QB. Then in the limit BQ will coincide with the given section, and the required radius of curvature will be the limit of the radius of curvature oi RQ at R. Let r be this limit, p the radius of curvature of the normal section having the same tangent line, that is, of OQ. Then 2r = It. ^ = lt. NR 2p = It. ON ON ^,^p ■.r„ - = COS ONB — COS ^ + vfj? + wv^ + .2u'fxv + 2v'vX + 2w'\iJi, ' We shall have the conditions UX + Vfj, + Wv = 0, X^ + iJ? + v' = 1, and the problem of finding the directions of the principal sec- tions, and the magnitudes of the principal radii of curvature, is the same as that of finding the magnitude and direction of the principal axes of the section of the conicoid ux' + ... + 2u'i/z+... = l, made hy the plane Ux+Vy + Wz — 0. 540. To determine, the sections of principal curvature, and the radii of principal curvature, at any point of a given surface, in terms of the co-ordinates of the point. Writing the equations of the last article, we have to make u\^ + vfj? + wi)^ + 2u'iw + 2v'vK + 2w' V = - a maximum or minimum, hy variation of \, fn, v, subject to the conditions V + iU,' + i/= = l, U\+ViJi.+Wv = 0. Difierentiating, and using undetermined multipliers A, B, we have m\ + w> + v'v + AX+ BU= 0, w'X + VIJL + u'v + Aii. + BV=0, v'X + u'jM +wv + Av + BW= 0, whence, multiplying by X, /j,, v, and adding, we have _ 1 r + A = 0; also, we obtain U[{v+A) . {w+A) -w"} + V{u'v'-w'{w +A)} + W{w'u'-v'{v+A)} fl _ V EE 418 CUEVATUEE OP SUEFACES. which leads to the quadratic equation in r, ^'{hl){-'^-']+ + 2rwL'w'-u'(u-H + =0; and smce - = , this equation gives the values of the principal radii of curvature, and the values o{X:/jl:v, corresponding to each root, are then given by the preceding system of equations. It may readily be shewn from these equations that the prin- cipal sections are at right angles, for if A^ A^ be the two values of A, and (Xj, /ij, vj, (X^, n^, v^ the corresponding direction- cosines, then multiplying the system (m + A^ \ + M)>, + v\ + BU=0, &c. by Xj, /ij, j/j respectively, and adding, we obtain X, {u\ + w>j + v'v^ + ...+A^ (XjXjj + fi^fi^ + v^v^ = 0. Treating the other system similarly, we obtain X, {u\ + w>i + v'v^ + ... + A, (X,X, + /i,/i, + v,) = 0, whence, subtracting, (^1 - A^ {XX + /^i/*, + v^v^) = 0, ■which shews that, except when the principal radii of curvature are equal, the sections are at right angles. When the principal radii are equal, all the radii of curvature of normal sections are equal, and a sphere can be described having a complete contact of the second order, or the point is an umbilicus. The principal sections are of course then indeterminate. 541. To determine the conditions for an vmhilicfus. At an umbilicus, r retains a constant value for all values of X, fi, V subject to the given conditions ; or {u - i^ X' + ^1) - i^ /i" + (w - -) i^ + 2m>v + 2v'i'X + 2w'X^ = 0, UMBILICI. 419 for all values of X, /*, v, satisfying the equation UX + Vii + Wv = 0. The left-hand member of the latter equation must therefore be a factor of the left-hand member of the former, and the other factor will therefore be Multiplying the two, and equating coefficients, which, on eliminating r, lead to the two conditions Wv + ¥"10-2 VWu' _ U^w + W^u - 2 WW V'+W " W' + U' V^u + U'v-2UVw' U' + V' These two equations, together with the equation of the surface, will, in general, determine a definite number of points, among which are included all the umbilici. It may happen that a common factor exists, so that the three equations are satisfied by the co-ordinates of any point lying on a certain curve. Such a curve is called a line of spherical curvature. It should also be observed that U, V, W have been assumed to he finite in the foregoing investigation. Should one of them, as U, vanish, we must have, in the same manner, V/i+Wv a, factor, and must therefore have (u--j'K^ + ...+ 2u'fj.v + ... ee2 420 UMBILICI. This identity gives or Vv = Ww', 2m' ={v — u)^ + {w — u) -r^, whicli with U= 0, and the equation of the surfa,ce, give four relations between the co-ordinates, and these will not, in general, be simultaneously true of any point on the surface. Solutions of the equations for an umbilicus which make U, V, or TF vanish, must therefore be excluded. 542. To determine the number of umbilici on a surface of the w**" degree. If the equations for an umbilicus be written in the form F~ Q'~ B" the degree ofP, Q', E is 2(n-l), and of P, Q, B is 3n-4. The degree of the surfaces QB' - Q'B = 0, BF - B'P = 0, is therefore 5« — 6, and the degree of their curve of intersection is (5m — ^y. But the curve B = 0, B' = is part of their inter- section, and does not lie on the surface PQ' — P'Q = 0. The degree of th& curve P_Q^_P P'~ Q'~ B' is therefore (5»i - 6)^ - 2 (w - 1) (3w - 4) = 19^" - 46w + 28. But this curve includes three curves similar to jj_^ _ W'v+V'w-2VWu' which do not meet the surface in umbilici. The degree of this curve is (m-1) (3w-4), and the degree of the curve, which meets the surface in umbilici only-, is therefore 19?i'- 46« + 28 - 3 (w - 1) {3n - 4) = lOn' - 25n + 16. Llisrp OF CUEYATUEE. 421 The whole number of real and impossible umbilici is therefore «(10w'-25« + 16). Thus, in a conicoid, this number is 12, four in each of the principal planes ; but not more than one system is real, and if the surface be a ruled surface, none are real. There can never be real umbilici on a ruled surface of any degree whatever, as every point of. a ruled surface is either parabolic or hyperbolic. Lines of Curvature. 543. Def. If on any surface a curve he traced whose tangent line at any point lies in one of the planes of principal curva- ture at that point, such a curve is called a line of curvature. Since there are two principal planes of curvature at each point, there will be two lines of curvature passing through every point, and these will cross each other at right angles. 544. To find the differential equations of the lines of curvature on any surface. Referring to the equations of Art. (540), we see that \, /*, v are the direction-cosines of the tangents to the planes of principal curvature at any point, or are the direction-cosines of the tan- gents to the lines of curvature through that point. Hence, if {x, y, z) be the point, {x + dx, y + dy, s + dz) a consecutive point on a line of curvature, we shall have dx _dy dz ~ which, since X, fi, v are determined in terms of x, y, z, are the differential equations of the lines of curvature. Since X, fi, v were determined so as to satisfy the condition we shall have Udsx, + Vdy + Wdz = 0, and one integral of these equations will be the original equation of the surface. Any other integral we may find, will involve 422 CURVATURE OP SURFACES. one arbitrary constant, which may be determined, so as to make the line of curvature pass through any proposed point on the surface. It appears from the above, that all the surfaces represented by the equation (», y, «) = c, for diflferent values of c, wiU have the same differential equations to their lines of curvature. 545. The normals to any surface, at two consecutive points of any line of curvature, intersect each other. Let (ar, y, z), {x + dx, y + dy, z + dz) be two consecutive points on the surface, the normals at which intersect. The equations of these normals will be X —x_y —y z —z = h, (1), x'-x-dx _ ^ , U+udx + w'dy+v'dz -"' ^^'' whence, we have the three equations {1c - k) U+ dx + y {udx + w'dy + v'dz) = 0, which are precisely the same equations as those of Art. (540), 1 h A being replaced by p , and jB by 1 — =-,; whence, we shall have dx dy dz A. /* 1/ ' \, fi, V having the values there determined, or the consecutive point must be taken along a line of curvature. The condition for the intersection of the two normals (1) and (2) gives at once dx{V {v'dx +iidy + wdz) - W{w'dx + vdy +u'dz)] + ... =0, or {Vv' - Ww') dx'+ ... +{U{v-w) - Vw'+ Wv] dydz + ... = 0, which, combined with the equation Udx + Vdy + Wds = 0, CUKVATUEE OF 8UKFACES. 423 ■will give the direction of the tangents to the lines of curvature at any point. These equations are then also the differential equations of the lines of curvature on the surface. A line of curvature of any surface may therefore he defined to be a curve such that the normals to the surface, at any two consecutive points of the curve, intersect ea.ch other. 546. The property of lines of curvature proved in the last article, may be easily shewn geometrically by means of the iu- dicatrix at the point. For, let be the point, xOy the tangent plane at the point, and draw a plane parallel and indefinitely near to asy, meeting the surface in the curve PQ, and the normal at Y in C. This curve will then, ultimately, be, in general, a conic whose center is G; Art. (537). Let P be a point on this conic at which the normal to the surface intersects Oz, the normal at 0. The tangent to the conic at P is then perpendicular to Oz and to the normal at P, and since these intersect, it is also perpendicular to CJP, which Kes in the same plane with them. The point P is, therefore, at the extremity of one of the axes of the conic : or POz is one of the sections of principal curvature at 0. Art. (537). Hence OP coincides with one of the lines of curvature at P, and there are therefore two directions, at right angles to each other, in which we may proceed from any given point to a con- secutive point, at which the normal to the surface intersects the normal at the given point. 424 CQEYATUEE OF SURFACES. At an umbilicus, the indicatrix becomes a circle; and these directions will be indeterminate. In this case the shortest distance between consecutive normals vanishes, to the second order of smallness, in whatever direction the consecutive points be taken. By taking account of the terms of a higher order, we can determine three directions, two of which may be impossible, in which the shortest distance between consecutive normals vanishes, to the third order of smallness. The equations giving these directions will be Udx + Vdy + Wdz = 0, {VW'-V'W)dx+ =0, U', V, W being the values of U, V, W at a consecutive point, including the terms of two dimensions in dx, dy, dz. In the second of these equations, the terms of one dimension vanish identically, those of' two dimensions vanish at an umbilicus, and there remain only the terms of three dimensions, which, combined with the first equation, give us a cubic for the determination of the ratios dx ; dy : dz. 547. The foregoing equations, for determining the principal curvatures, undergo a considerable simplification, if the equation of the surface is of the form '^i(«') + <^.(y)+<^3(«)=o. We shall then have u', v, w' all zero ; the equation giving the length of the radius of curvature of any normal section, whose tangent line is (\, /t, v), will be r p r- ) the quadratic equation for the principal radii of curvature will be U V w r r r the difierential equations of the lines of curvature will be Udx + Vdy + Wdz = 0, V [v - w) dy dz + V{w-u)dzdx+ W{u -v)dxdy = ; CUEVATUEE OP SUEPACES. 425 and the conditions for an umlailicus are F' w+ Wv _ W\+ U'w _ IPv+V'u 548. To obtain equations for the determination of the curva- tures of a surface at any 'point, when the equation of the surface gives one of the co-ordinates explicitly in terms of the other two. These equations could of course be deduced from those already obtained, but the results being considerably shorter, though not generally so easy to deal with in practice, we give a separate investigation. Let the equation of the surface be z=^{x,y), and let the first and second differential coefficients of z be de- noted by ^, q, r, s, t. Then if {x, y, z), {x + dx, y + dy, z+ dz) be consecutive points, X, yii, v the direction-cosines of the corre- sponding tangent line, the distances of the second point from' the normal, and tangent plane, at [x, y, z), are respectively dz —pdx -I- qdy _ _ Vl+/ + ^ Also dz =pdx + qdy ■\-\{rdoi?+'is dx dy + tdy^) 4- .... Hence, if p be the radius of curvature of the corresponding, normal section, p-^a. p ^^- rd!x?+isdxdy + tdf ''^■^P+i' or. 1 _ \ll-\-p^+q^ r + 2sm + im^ B "= } 1 +p^ + 2pqm+{l + q') m' ' m being the value of -^ for the particular tangent line. The equation for determining the principal radii of curvature 426 CUBVATUKE OF SURFACES. will be found hy making this equation in m have equal rootSj and will therefore be (1 + 2= - Bt) (1 +/ -Br) = {pq- Bsy, or, (rt-s") E"- {(1 +p')t- 2^qs+ {1+^) r}B+l +f+^= 0. Also, when jB is a maximum or minimum, m must satisfy the equation {\+f+2pqm+{l+c^)m^] {s+tm) = {r+2sm+trrf) {pq+{l+q^)Tn], or, m^pqt-s{l+q^)}+m{t{l+p') -r (l+g")} +(H-p'')s-i32»-=0. Replacing m\iy -j- , this is an equation which must be satisfied at all points on a line of curvature, and since it only involves x and y, it is the differential equation of the projections of the lines of curvature on the plane of xy. The conditions for an unabilicus may be determined from either of the properties, that B is independent of m, or that the equation for m is indeterminate, at such a point. They will be r s t ' They may also be found from the condition that the equation in B may have equal roots, which may easily be expressed as the sum of two squares, and for real values of the differential coefficients, these must separately vanish, and we obtain the same equations as before. 549. The lines of curvature on a comcoid are its curves of intersection with confocal surfaces. Let the equation of the surface be a? y^ z" , a c ^ " the differential equations of the lines of curvature will then be (Art. 547), — dx + ^ dy + - dz — d, a b " c ' x{b — c) dy ds+y {c — a) dzdx+s [a — h) dx dy = Q. CURVATUEE OF SURFACES. 427 But for the curve of intersection of (1) with a confocal surface, 2 2 2 , xdx _ ydy _ zdz ^® a (6 - c) (a + ;fc) ~ 5 (c - a) (6 + A) ~ c (a - 6) (c + A;) ' c^a; (j[^ c^s and subtracting (2) from (1)', we have a (a + A;) J (& H- i) c (c + 4) at all points of th& curve of intersection. Hence, we must have, at all points on such a curve, dx dy da ' or, x{b — c)dyds+y[c — a) ds dx + z {a — h) dx dy = 0, and the differential equations of the curve of intersection of^e given conicoid with any confocal surface coincide with the difefer-'' ential equations of the lines of curvature ; and the equationXef such curves, involving an arbitrary constant le, are therefor^ the complete integral of the differential equations of the lines of curvature. Having given any one point (as', y\ z'), we shall have the quadratic equation -'«"' v'^ z'" for determining h, and to each value we shall have a correspond- ing line of curvature passing through the point (a?', y', z'). We may also solve the equations by obtaining the differential equation of the projection on the plane of xy. This equation is {x (5— o) dy +y {o — a) dx] i- dx + j^dy\ = i^a-h) {l -'^-y^dxdy, 428 CURVATURE OP SURFACES, or, of wliich tlie integral is (Boole's Differ. Equations, page 135) ia-h) ^ ^ a{b-c) ' b{a — c) that is, a conic 2 2 as « ' ~ 7.' — -^J a a', V being connected by the equation — (a — c) — -r (6 — c) = a — 5, a ^ ' b ^ Now the equation of the projection of the curve of intersec- tion of the confocal conicoids (1) and (2) is x'{a-c) . f{b-c) _ a[a+k)^ b{b-^k) a conic whose semi-axes are connected by the same relation. This conic in fact coincides with the former, if we give k such a value that ~,a{b — c) _b + k b {a— c) ~ a + k' The relation between the axes of this conic shews that it touches four real or impossible straight lines, whose equation, the singular solution of the differential equation, is I b{a-c)^ a-c ) b{a-c) ^ ' If a, b, c be positive, and in descending order, these lines will only be possible in the plane of zx, and it will be seen that they are the tangent lines to the principal section in that plane at the umbilici. CURVATURE OF SURFACES. 429 If two of the three a, h, c, be negative, there will in like manner be only one system of possible straight lines; and if one only be negative, all the systems are impossible. The general result is, that the lines of curvature passing through any point on a given conicoid are the intersections of the given surface with the two confocal conicoids passing through the proposed point. This has only been shewn for central conicoids, but the paraboloids are of course included as a limit- ing case. 550. The preceding results may also be proved in the fol- lowing manner. We will only take the case of the ellipsoid. Let P be any point on the ellipsoid, PGP' the diameter through P, GL the radius 'parallel to the tangent line at P to a normal section whose radius of curvature is required, PQL the central section having the same tangent line. Through Q, a point near P at a distance ct from the tangent plane at P, draw: a plane parallel to the tangent plane, meeting CP in V. Let j? be the perpendiculaj: distance from G to the tangent plane at P. The required radius of curvature is then ilt.^^ilt,«^^ and QV PV. VP' GU GP' ■ST' > PV CP' whence QV GLWP' ., ,, QV ^OU = — TTrT- 1 and it. = 2 ra- p . GP ' «r J? 430 CURVATUEE OF SURFACES. GI? or the required radius of curvature is - — . The tangents to the sections of principal curvature at P are therefore parallel to the axes of the central section conjugate to GP, and if a, ^ be the semi-axes of the section, the values of the principal radii of curvature are — , — respectively. Now, let the equation of the ellipsoid referred to its axes be and let x', y', z be the co-ordinates of P. The equation of the conjugate central section is axe' yy' zz' _ ,,2 "■" 3,2 "r "TT — ^> a C and the equation, giving the values of its semi-axes, is ^ t. il ^r7+jTr7+^« = 0' Art. (250); and the equation giving the values of the principal radii of , curvature is therefore <^'' «/" a"" d {a^ - pp) ^ h^ {b' -pp) ^ c" {c' -pp) and, since ^2 + J2 + ^2 i. we shall also have a —pp —pp c —pp 551. ^ three confocal conicoids {A), {B), (0) intersect in a point P, the centers of principal curvattire of (A) at P are the poles with respect to {B) and (G) of the tamgent plane to {A) at P. Let the equation of {A) be ^2 + J2+^-l, CURVATUEE OF SUEPACES. 431 and let (a;', y', z') be the co-ordinates of P. The equation of a confocal surface through P will be "' +_^ + _i!- =1 a^ + k^b^ + k^c' + k k being given by the equation , T^ 7,2 I 7.^ ^ I 7. '■> a'+k^V + k^c^ + k' and, comparing these with the equations of the last article, we see that the two values of k are the two values oi pp with the sign reversed. Now, if X, Y, Z be the co-ordinates of one of the centers of principal curvature, corresponding to a radius of curvature p, we shall have X-^ Y-y' Z-^ x' ~ y' ~ z' V(X- x'Y +_( Y- y'Y + {Z- zf _ p _ ,. . / n» ft to 1 ' ^/ X' y^ z^ 1 a* "•" 6* + c* P whence, X= a,' (l -h |) , ' r=y (l + J) , Z= z' (l + ^) , But these are the co-ordinates of the pole of the plane a» + 6>i + c"--^' with respect to the conicoid a' + k^y + k^c'^ + k The result stated is therefore true for central conicoids, and therefore generally. This proposition is due to Mr Salmon. 552. At any point in a line of curvature of a central conicoid, the rectangle contained hy the central radius parallel to the tangent line at the point, and the perpendicular from the center on the tangent plane at the point, is constant. 432 CURVATURE OF SURFACES. Let the equations of the line of curvature be then the radius conjugate to the central section of (1) through the tangent line at {x, y, z) to the line of curvature will be parallel to the normal to (2) at that point. For, it is one axis of the section parallel to the tangent plane at (a;, y, z), and, therefore, perpendicular to the other axis which is parallel to the tangent line. It is also perpendicular to the normal at {x, y, z), and therefore to the plane containing the tangent line and the normal, or to the tangent plane to (2). Hence its direc- tion-cosines are proportional to X y z a^ + k' h' + h' c^ + Zc' and its length will therefore be ^/■ I y I {a' + kY ^ Qf + kf ' [e + kf 2 2 2 or y ■ ^ a'' (a"" + ky ^ V [V + kf ^ & (c" + ky which, by virtue of equations (1) and (2), is equal to 2V-i. . But, if a, /3 be the two semi-axes of the central section parallel to the tangent plane at "(a-, y, z), and j? the perpendicular upon the tangent plane, a^p = dbc. Now, of the two a, /S, one is parallel to the tangent line at {x, y, z), and the other has been proved to be constant. Hence, if d be the radius parallel to the tangent line, we have , 2a5c which is a constant quantity throughout the same line of cur- vature. 553. To apply the general equations to find the umbilici of a central conicoid. CUEVJ^JUEE OF SUEFACES. 433 It is obvious, from the preceding articles, that the umbilici are the points at which the tangent planes are parallel to cir- cular sections of the conicoid, which sufficiently determines them. But, as an example, we may use the equations of Art. (541), to find the umbilici of the conicoid _ ^j-_ — 1 In this case 67=-^, y=-f, ^=^' " = ^. «=0, &c. The equations therefore become f , s' z^ c^ x^ / Z^4„2 '^ 47,2 4 2 ~ 4 2 412 T^ 2,4 2 < c CO Ga ac ab ba 1 or, 6* + c* c*'^a* a*"*"*-* c* "^ a* a* "^ b* 1(1 l\- 1(1. -l.\ 6*U' kV~ c'W W' a* V6'' AV ~ b* W ^V ' which are satisfied by the three systems, a; = 0, h^ = a'; y = 0, k^ = b^; or z = 0, l^ = c^; only one of which however leads to real values of all the co-ordinates ; the second, if a, h, c be in order of magnitude, which gives c' g' 1 y-0, J2_g2-„2_J2-^2_g2, results which are the same as those obtained from the circular sections. 654. As an example of finding general expressions for the radii of curvature of families of surfaces, we may take the fol- lowing, FF 434 CURVATURE OF SURFACES; A surface is generated hy the motion of a straight line which always intersects the axis ofx, to prove that the radii of curvature at any point on the axis ofx are dx cos + 1 d^ sin 8 ' X being the distance of the point from the origin, 6 the angle which the corresponding generator makes with the axis ofx, and (p that which its projection upon yz makes with the axis ofy. Let P, P' be contiguous points on the axis of x, at distances X, x + dx, from the origin, 6, ; d+dO, ^ + d^; the angles for the corresponding generating lines : Q a point on the gene- rator through P' at a distance \ from P'. The tangent plane at P contains the axis of x, and the gene- rator through P, and its equation is therefore y&va.

, whence, p being the radius of curvature of the normal section through PQ, we have * (dx's^ , „dx a , i ip — It. """"^^"""^"^""^^^^^""^ J p fj, sin0 denoting the limit of -5^- by /*. The equation giving the principal radii of curvature is, therefore, fdxV /dx mY _/ \d) - V. jy cos ^ — p sin CUEV^TUEE OP SUEFACES. 435 or, ^ (cos ^ + 1) = p sin 6, giving the values for p as stated. If the surface be a right conoid, whose axis is the axis of X, these values become + ^-r , which agrees with the fact that at any point on the axis of such a surface, the asymptotes of the indicatrix, being parallel to the generators, are at right angles to each other, and the principal radii of curvature must therefore be equal and of opposite signs. 555. To find the osculating plane of a Une of curvature at any point of a surface. Let PQ, PR be small arcs of lines of curvature drawn through P, a point in the surface, BS, Q8 lines of curvature through R, Q respectively; and let PEG, QH' G, RE, 8H' be normals to the surface at P, Q, R, 8, so that PE, QE' are ulti- mately the radii of curvature of the principal normal sections PR, Q8, and PG that oi PQ; let these be R', E + dE, and R, dR' being the increment of R' due to a change ds along the principal section PQ. The tangent to PR at P is perpendicular to the plane PE', and therefore to EE', and the tangent at R is for a similar reason perpendicular to EE', which is therefore parallel to the F F 2 436 CUEVATURE OF SUEPACES. binormal at P to the line PB, and determines the osculating plane FOB. If <^ be the inclination of the principal normal section to the TT osculating plane of PB, OHP= k~ ^' Draw H'N perpendicular to PG, , ,. EN y HN PQ tan <\> = km. ffrj^ = ^^^- pQ • jfjf _dR' B ds ' B-B'' CoE. In the case of a surface of revolution, since B' is the same for all points in the circular line of curvature supposed to dlf correspond to B, —5- = 0, and the osculating plane coincides with the normal plane. Gauss' Theorems on Curvature. 556. If a portion of a surface be cut off by a closed curve, the total curvature of this portion is defined by Gauss to be the area of the surface of a sphere whose radius is unity, cut off by a cone whose vertex is the center, and whose generating lines are parallel to the normals to the surface at every point of the bounding closed curve. The measure of curvature of a surface at any point is the ratio of the total curvatm-e of any elementary portion of the surface including the point to the area of that elementary portion. 557. To shew that Gauss' measure of curvature is ajproper measure, and to calculate its value. Let any elementary area be described about the point P of the surface, and let a series of lines of curvature divide this area into sub-elementary portions such as pqrs, and let p, p be CUEVATUKE OF SUEFACES. 437 the principal radii of curvature at p in the directions pq, ps, the portion of the area on the unit sphere corresponding to pr will be on sides equal to — , -^ , and area^r=p2'.^s ultimately. Gauss' measure of curvature gives the value lim. \^ , , . X{pr) But •— ; lies between -ppj (1 + ej and -p^r (1 + e^, RE where e^, e^ vanishes in the limit ; .'. the measure of curvature is BE 1 EE> which is independent of the form of the elementary portion. COE. The measure of curvature at any point of a surface 2 =/(»', y) is therefore ^~^ , , . Art. (548). 558. To shew that the measure of curvature is not altered hy any deformation of the surface supposed inextensible. Let a surface be referred to axes such that Oz is a normal at 0, and the planes of sx, zy are principal normal sections ; and let R, E be the radii of these sections. Let a curve be described whose geodesic distances from are constant and each equal to s ; OP, OQ two of the distances whose- planes are inclined to zx at angles 6,6-\-d6; 438 CURVATURE OF SURFACES. OM, ON the traces on xy being r, r + dr, the distances of P, Q from xy being z, z + dz, and x=.lr, y = mr ; ~ 2 U -B7 ' let p be the radius of curvature of OP; , 1 P m' and - = •d' + r = psini=.-i^,, dz = sM — — ^j Imdd, and riWd0 = ^Z- 1 TT 3 . 1 TT 1 TT 2 2 4.2 2 8 2' ■CUETATUEE OP SUEFACES. 439 Hence, the perimeter the area = £.^. = ..»(l-j^). Hence, since in any deformation the distances of all points measured along the surface remain unaltered, in order that this may be the case about every point, the perimeters of geodesic circles must he ultimately unaltered : therefore the measure of curvature at any point remains unchanged. The student, who is interested in the subject of deformation of surfaces, instantaneous and permanent lines of bending, will find a most ingenious article by Maxwell in the Cambridge Phi- losophical Transactions, Vol. ix. Part 4, No. 19, 559. Certain properties of the principal radii of curvature may be conveniently investigated by considering the angle be- tween the two tangents to the curve in which a surface is inter- sected by the tangent plane at any point. In these directions three consecutive points lie in a straight line, and the radius of curvature of a normal section through one of these tangents is therefore infinite. Hence, if 6 be the angle which one of th§s^, tangents makes with the tangents to a section of principal curva- ture, we shall have cos' d sin" 0= +—r-, P P p, p being, the algebraic magnitudes of the radii of principal curvature. Thus, for points at which the radii of principal curva- ture are equal in magnitude and opposite in sign, we shall have tan°^=l, and the tangents to the curve of intersection will therefore also be at right angles. As an example of this method we shall take the following. To prove that, in the surface x{d' + f + s'') = 2a{x'+y'), 440 CURVATURE OF SURFACES. at all points lying on the plane x = a, the radii of cuxvature will be eqiial in magnitude, and of opposite signs. This, by what has been said, will be true, if we can prore that the two straight lines, drawn through any such point, to meet the surface in three consecutive points, are at right angles to each other. X i x'—ay' — y z'—z Let — 1— -- — -= = »• I m n be a straight line which meets the surface in three consecutive points. The equation (a + Ir) {z + nrf ={a- Ir) {{a + Irf + («/ + mrY] must then have all its roots equal to zero. This gives us the equations s^ = a' + f, h^ + Inaz =■ la {al + my)-l (a' + y*) 2nlz + n'a=a {!' + m") - 2l {la + my), of which the two latter become, by reason of the first, ?y — may + naz = 0, Pa — vf^a + w'a + 2m?a + Hmy = 0. The equation giving the two values of w : Z, is ay' {af + atf + Iznl) - (Z/ + ms)" + llf {ly' + naz) = 0, whence, if {\, m^, mj, (Zj, m^, n^ be the directions of the two lines, we have «A "«'(/-«')" »* ' and similarly, eliminating n, we obtain jOT^OT, _ gV + y - 2y V y V - (y" - g")" _ y V - a* whence, finally, or the two lines are at right angles. Hence at all points of the surface lying in the plane x = a, the radii of curvature are equal and of opposite sign. CUEVATUEE OP SUEFACES. 441 • 560. Dupin's Theoeem. If therele three series of surfaces such that any two surfaces of different series intersect everywhere at right angles, the curves of intersection are lines of curvature on both surfaces. Let the origia Ibe a point of intersection of three surfaces, one of each series, and the tangents to their lines of intersection the axes. The equations of the three surfaces may then he written a! + a/+25ys + ca'+... = 0, (1). y + a'ii^ + 2b'zx + c'x^+ ... = 0, (2). a + a' V + 2i"xy + c'Y + . . . = 0, (3) . At a consecutive point on the curve of intersection of (2) and (3), we have y = 0, s = 0, x = x, and tlie equations of the tangent planes are, ultimately, X . 2c'x' + y + z. 2h'x' = 0, X . 2a"x' + y . 2h"x' +z = 0, and since these also are at right angles, 4:a"c'x'' + 2h"x' + b'x' = 0, or, taking the limit, b' + b" = 6. Similarly, b" +h = 0, b + b' = 0, which lead us to 5 = 0, b' = 0, b" = 0, or the axes are tangents to the lines of curvature on each surface. Hence, the tangent lines, at any point of intersection of three surfaces, to their curves of intersection, are tangents to the lines of curvature of the surfaces through that point, and conse- quently, their curves of intersection must coincide with the lines of curvature. 561. As an example of three series of surfaces, satisfying the conditions of Dupin's theorem, we will take the locus of the points of contact, of tangent planes drawn to a series of confocal conicoids, from fixed points on the axes. For each fixed point, 442 CUEVATUEE OP SUEFACES. « we shall obtain a surface locus, and for each axis a series of such surfaces. We will now prove that these surfaces intersect every- where at right angles. Let (a, 0, 0), (0, j8, 0), (0, 0, 7) be the co-ordinates of three points from which tangent planes are drawn to the series of coni- coids, confocal with the ellipsoid a? y" z' , The equations of the surfaces may readily be found to be and -r-T^ 2 + — TTii 2 + -=l- (3). yz + a — 7» + — c 7 The points of real intersection of (1) and (2) may, by sub- tracting their equations, be found to lie in the plane a.x — c^ = ^y — V, which, combined with (1), gives the equation ^(aa!-a'+c»)^-|(J83/-5= + c') + a^ = da:-a» + c^ the equation of a sphere. The curves of intersection are then circles. Let the equation of any one of the confocal conicoids be ■K* «" ^ a^ + liV-Vh ' c' + k' and let the tangent plane at {x', y\ «') pass through the straight line aa; .CUEVATUKE OP SURFACES. 443 the point (a;', y\ z') will then lie on the intersection of (1) and (2). This gives us ' = a^+h, Pif = V + h, a-=(c^ + A=)|l-^-^| (4); and if (»', y', z') be the point of intersection of (1), (2), (3), we shall have aa;' - a" = /3y - 6" = 7»' - c' = ;fc, h being determined by the equation — ^ — •■ ai H a~ = !• Let (X, fi, v) be proportional to the direction-cosines of the tangent Hne at this point to the intersection of (1) and (2). Then, by (4), 2- ^^ + A V' + h ^ ^c+A;;(^^,+^,j 7^ a'' /3 Similarly, if (V, /i', v') be the direction of the tangent at this point to the intersection of (1) and (3), a\ =-j 5^=7", and these will be at right angles, if which is obviously true. The tangent lines to the three circles of intersection are then mutually at right angles, and therefore the planes, each of which contains two of these tangent lines, are mutually at right angles, and these are the tangent planes to the surfaces at the point. Hence, at any point which is the intersection pf three such sur- faces, the tangent planes are mutually at right angles. But, by varying 7, a and j8 remaining constant, we can make this point coincide with any proposed point on the intersection of (1) and (2), and the surfaces will therefore cut each other orthogonally at all points on their curve of intersection. 444 CURVATURE OP SURFACES. The angle between the curve of intersection of (1) and (2), and the tangent plane to (1) at the point in question will be found to be tan"' - , a value which, not depending on 7, will be a constant throughout the curves, as is always the case when a line of curvature is a plane curve. The lines of curvature on any surface (1) will then be two systems of circles, whose planes are parallel to the axes of a and y respectively, and pass each through one of two fixed points on the axis of x. XXIV. 1. The principal radii of curvature, at the points of the surface, aV = s? (3? + «/"), where x = y —z, are given by the equation 2. Prove that, in the surface a? (y — ») + ayz = 0, the radii of principal curvature, at points where it is met by the cone (a!''+6ya)2/« = («/-«)*, are equal, and opposite. 3. Deduce the conditions for an umbilicus from the equation giving the radii of curvature, by making the roots of the equation equal. The only surface of revolution, such that the radii of curvature are at every point equal and opposite, is that produced by the revolu- tion of a catenary about its directrix. 4. Prove that the only real points on the surface X {x^ + y" + z') = 2a {x' + if), at which the radii of curvature are equal and of opposite signs, are those lying on the hyperbola x = a, z^ -y^- a'. 5. A surface is generated by the revolution of a parabola about its directrix; shew that the principal radii of curvature at any point are to each other in a constant ratio. ^ PEOBLEMS. 445 6. If p, p' be the principal radii of curvature at a point of a sur- face, where the normal's direction-cosines are I, m, n, prove that 1 1 dl dm dn p p' dx dy dz ' 7. Shew how to find the lines on a surface at which one of the principal curvatures vanishes. How does the form of the surface alter in passing across such a line? Is the line "in question a line of curvature % 8. The locus of the centers of curvature of all plane sections of a given surface at a given point is the surface whose equation, referred to the normal and the principal tangents, as axes, is ■a? 9. Shew that the projection, on the plane of xy, of the indica- trix at any point of the surface s = (e" + e"") cos a; is a rectangular hyperbola. 10. If at any point of a surface p,, p^ be radii of curvature of normal sections at right angles to each other, and B^, B^ be principal radii of curvature, the sections corresponding to p^ , p^ being inclined at an angle a, prove that cos° a sin° a cos 2a and Pi Ps -^1 sin° a cos* a _ cos 2a Pi P2 ~ -S. ' 11. Find the umbilici on the surface whose equation is x^ ^ 1^ , and shew that the radius of curvature there is ^ (a' + 6' + c°)*. 12. A surface is generated by the motion of a variable circle which always intersects the axis of x, and is parallel to the plane of yz. If r be the radius of the circle at a point on the axis of X, and the inclination of the diameter through that point to the axis of z, prove that the principal radii of curvature at the point are given by the equation pV +p' (p- r) = 0, where p is the value of -73 at the point. au 446 PROBLEMS. 13. A surface is generated by a straight line ■whicli always in- tersects a given circle, and the straight line through the center of the circle normal to its plane, prove that the principal radii of curvature of the surface, at any point on the circle, are given by the equation p' {-y-A — ap cos 6 — a^ = 0, a being the radius of the circle, 6 the angle which the generator at the point makes with the fixed line, and ^ the angle which the radius of the circle through the point makes with a fixed radius. 14. Find the differential equation of surfaces possessing the pro- perty, that the projections on a fixed plane of their lines of curvature cross each other everywhere at right angles. Prove that it is satisfied by surfaces of revolution whose axes are perpendicular to the fixed plane ; and obtain the general solution. 15. Two surfaces touch each other at the point P ; if the princi- pal curvatures of the first surface at P be denoted by a ± 6, those of the second by a' ± 6' ; and if ■ct be the angle between the principal planes to which a + h, a' + h' refer, 8 the angle between the two branches at P of the curve of intersection of the surfaces, shew that . - a? — 2aa' + a'" cos' S = : 'b'-2bb'oos2-s7+b"' 16. The measure of curvature at any point of an ellipsoid is pro- portional to p*, p being the perpendicular from the center on the tangent plane. 17. The measure of curvature at any point of the paraboloid y z — +—, = x varies as ( — ) , p being the perpendicular from the origin on the tan- gent plane. 18. The planes drawn through the center of an ellipsoid, parallel to the tangent planes at points along a line of curvature, envelope a cone which intersects the ellipsoid in a spherical conic. 19. If i? he the radius of absolute curvature at any point of a line of curvature on an ellipsoid, r, r' the radii of curvature of the normal sections of the ellipsoid, and the confocal hyperboloid which contains the line of curvature, through the tangent to the line of curvature, prove that 1-i i_ ^ PROBLEMS. 447, 20. If P be any point on a surface, a, b the principal radii of curvature at P, PQ = s an indefinitely small arc taken along a normal section making an angle with the principal section to which a refers, prove that if D be the minimum distance between the normals at P and Q, c the distance from P of their point of nearest approach, sm^ Ocos^e {a -hYs' _ ah (a sin' 9 + b cos' 6) a' sin' e + b' cos" ' "" a' sin' 6 + b' cos' 6 ' Prove that the maximum value of the ratio — is | = | , the s \a^bj upper or lower sign being taken, according as the point is elliptic or hyperbolic. Prove also, that, in the case of the ellipsoid, this maxi- mum value is greatest at the extremities of the mean axis. 21. Prove that the surface generated by normals drawn to a surface, at points whose distance from the normal at a fixed point P is constant and small, will intersect a normal plane making an angle 6 with one of the principal planes at P, in two parabolas, whose radii of curvature are Pjp^-Pi) P,(Pi-P,) p^sin'e ' p^cos'd ' p , p being the principal radii of curvature. 22. If a line of curvature be a plane curve, its plane cuts the tangent planes to the surface, at points lying along it, at a constant angle. 23. If one series of lines of curvature on a surface be plane curves, lying in parallel planes, the other aeries wUl also be plane curves. 24. If i? be the radius of absolute curvature at any point of the curve, which is the intersection of two surfaces A, B, and r, / be the radii of curvature of the sections of the surfaces A, B made by the tangent planes to B, A respectively, and 6 be the angle between the tangent planes, prove that 1 _ 1 2 cos ^ J^ 25. If a plane curve be given by the equations - = cos 6 + logjtan ^ , - — sin 6; the surface, produced by the revolution of this curve about the axis of X, will have its measure of curvature constant. 448 PROBLEMS. a 26. In a surface, generated as in (13), if <^ = log tan ^ , the mea- sure of curvature -will be the same at corresponding points on the axis of z and on the circle. 27. On an umbilical conicoid, the projections of the lines of curvature on the planes of circular section by lines parallel to an axis, form a series of confocal conies, the foci of which are the pro- jections of the umbilici. 28. Prove that the three surfaces yz = ax, Ja? + f + Jx' + s^ = h, sjm^ + y' - Jx'+'i^=c, intersect each other always at right angles ; and hence prove that, on a hyperbolic paraboloid, whose principal sections are equal parabolas, the sum or the difference of the distances of any point on a line of curvature, from the two generators through the vertex, is constant. a; 29. In the helicoid, whose equation is 2/ = a; tan-, the lines of curvature are the intersections of the helicoid with the surfaces re- presented by the equation 2 Jx' + y' ? 1 _5 a c for different values of c. Also, prove that the principal radii of curvature are, at every point, constant, equal in magnitude, but of opposite signs. CHAPTEE XXIV. GEODESIC LINES. 562. Dep. a geodesic line of a given surface, between two given points on it, is a line of maximum or minimum length. Any infinitesimal arc of such a line will manifestly he the mini- mum line between its extremities, hut if the two given points be at a finite distance, a geodesic passing through them may be either a maximum or minimum, and there may be an infinite number of such maxima and minima. 563. The osculating plane at any point of a geodesic on any surface contains the normal io the surface. The distance between two indefinitely near points will be the least possible, when the curvature of the line joining them is the least, or when the radius of curvature is the greatest,-' Now the curvature of any curve on a surface will be, at any, point, the same as that of the section of the surface made by the osculating plane at .that point, since the two curves will have three coincident points. Also, of all sections, having a common tangent line, the normal section is that of least curvature, by Meunier's Theorem (Art. 538). Hence, the osculating plane of a geodesic, at any point, must be a normal section. This also appears from the consideration of a stretched weightless string, joining any two points on a surface. This will manifestly assume the form of the shortest line joining the points, and since the resultant of the tensions of two consecutive elements of the string is balanced by the normal reaction of the surface, the normal must lie in the plane of these elements, that is, in the osculating plane of the curve. '" ' 450 GEODESIC LINES. The equations of any geodesic Jine on the surface i^ (a;, y, a) = 0, may therefore be written in the form d^x d'y d'z ds' ds' ds' If" "IF'- '-If- dx dy dz One integral of these equations is, of course, F{^, y, z) = 0, and, if another can be found, it will involve two constants, to be determined so as to make the line pass through any two proposed points. The form of the equations, connecting the constants of inte- gration with the co-ordinates of the two proposed points, may be such that an infinite number of values can be given to the constants, to each of which will correspond a geodesic through the two given points. As an example of such a family of geodesies joining two given points, we will take the very simple case of a right circular cylinder. 564. To investigate the equations of a geodesic, joining two given points on a right circular cylinder. Let the equation of "the cylinder be and let the given points be (a, 0) ; (a cos a, a sin a, c), re- spectively. The general equations of a geodesic give d^x d^ d^ d^ _ ds^ _ ds'' X y ' Hence -r- is constant s qds ^S, whence (-^1 ■\-\-j-) = sin^'jS. Eliminating x and s, the equation connecting y and z is . r>dz a Bin p J- = therefore « sin /S = a sin"' (-] + C. GEODESIC LINES. 451 .To determine C we shall have the equation * C+asin"'0 = 0, and the equation of the geodesic will he, whatever possible value we assign G, y . fz sin B\ We shall then have, to determine /8, the equation (c sin /3\ sin a = sm a which admits of any solution of the form c sin/3 / ,v, , . , , = »-7r 4- (— 1) a, r being any whole number ; which give, for the geodesic, the equation I = sin ^{r7r + (^ !)••«}, and there will be a different curve for every different value of r. Thus there will be an infinite number of geodesies joining any two proposed points on the cylinder. This is obvious geometrically, for we can wind a string round the cylinder in either direction as many times as we please so as to start from one point, and pass through the other, and retain; its form under tension. If a be a positive angle less than two right angles, the value r = 1m will give a geodesic crossing every generating line m times, and a certain portion of them m + 1 times ; and r = 2»» + 1 will give a geodesic crossing every generating line m times, and a certain portion, supplementary to the former portion, m-\-\ times. ^Two points describing geodesies of different systems will start in opposite directions from one of the points to proceed to the other. 565. To investigate the equations of a geodesic, joining two given points on a cone of revolution. Let the equation of the cone be y^+»' = «" tan^ a, G G 2 452 GEODESIC LINES. and let the two given points be (a, a tan a, 0) ; (J, J tan. a cos /3, h tan a sin ^8) ; then, for the geodesic, we have — a; tan a. y z d^ii d^z „ dy dz , If we take y =r cos tji, z = r sin ^, we have r = x tan a, and the abovB equation becpmes r-^ + h^O. ds Also, we have the equation ©■-■(f)'-il"-.".4.(l)--@)'='- whence -r-^ (-^ + ^= = ^ the solution of which is r = h sec (^ sin a + C), h, G being constants, to be determined so as to make the geo- desic pass through the two given points. Hence a tan a = A sec G, h tan a = A sec {fi sin a + C), and, eliminating h, G, we obtain, for the geodesic, the equation - sin {{0 — (j>) sin «} + t sin (^ sin a) = sin (/8 sin a) tan a. C[> Now, the two points on the cone remain unchanged, if for /9 we substitute 2»i7r + /S, m being any positive or negative integer, and to each such value will correspond a geodesic line, joining the two points. The number of these lines is however limited by the condition that the angle {2imr + ^) sin a must lie between-the limits IT and — tt. For, if we take i^nm + /S) sin a = nw + S, n [To face paje 453.] ^-GEODESIC LINES. 453 being any whole number, then, if n be odd, we shall hare, when ^ sin a = S, r = — b; and if n be even, when 0=77+8, r = — b; or, in both cases, we shall have negative values of r, for a value of (j> less than the one corresponding to our second point. Hence, r must have passed through the values or oo , But, it is mani- fest, that r = does not satisfy the equation ; hence r must have passed through the value oo before arriving at the second point, and there will be no corresponding geodesic of finite length. The number of finite geodesies joining the two points will therefore ex- ceed by unity the sum of the integral parts of the two quantities " -/3 ^ + /3 sm a sin a 27r ' 27r ' and cannot difier by more- than unity from the integral part of -; — . It has been assumed that 8 lies between a;nd 27r. sin a The form of the geodesic lines which can be drawn from any point A to any other 5 on a cone, are exhibited in the figure. With the radius VJB and center V let a semicircle be described, whose base contains the point A, and let CVG^ be the sector into which the portion of the cone terminated by the circle containing JS would be developed, VA being less than V.B. Let equal sectors be placed in order whose bases are C^C^, C^G^, C^G^, which in the figure are all the complete sectors which can lie within the limits of the semicircle. Take G,B^ = GJB, = G,B^ = G,B,= Gfi = GA^ GJ>,= GB, and join A by straight lines to all the points B and S, intersect- ing the radii VO^, &c. in P,p, Q, q, B, r and 8. Take VQl=VQ„ VQ;' = VQ; = VQ„ and join Q;q;, Q^'B. One of the lines AB^ crosses intothree of the sectors, and if the second and third be placed upon the first the portions of the line corresponding will fall on Q^ Q^ and Q^'B, and if the cone be formed of the sector GVG^, these portions will form a con- tinuous line, being' a geodesic line from A to B, Similarly for AB^, AB„ and AB,. 454 GEODESIC LINES. In the same way, if the semicircle on the opposite side of CD were divided into sectors and a similar construction made, we should obtain other geodesic lines corresponding to the dotted lines Ah^ A\, &c. the figure being supposed turned about CD to the opposite side. It is evident that the number of geodesic lines on each side will be limited by the consideration that each line AB must -cross the sectors consecutively, in order that a continuous line may be obtained on the cone, corresponding to each. Hence, in the par- ticular case of the figure, there will be five setting out on the one side of VG and four on the other. Generally, if 7 be the developed angle between the generating lines through A and B, and. S the complete developed angle of the cone, the number r + 1 on one side is given by the in- equality 7 + rS < •n-, or the number is the integer next less than — jT-^ + 1, the number on the other is that next less than o s ^ + 1, or— ^. 566. To jarove that, throughout a geodesic on any surface of revolution, the distance of any point from the axis varies as the cosecant of the angle between the geodesic and the meridian. Let the axis of the surface of revolution be taken as the axis of a, and let the equation of the surface be s = fips' + y^). Then, for any geodesic, we have d^ d^ d^ df_^d£_^d£_ , dz dz —\' dx dy But from the form of the equation, dz ^^ _c\ dy ^dx~' , d^y d^x „ dv dx , whence a,^_y_ = o, xf^-y^ = h. ^EODESIC LINES. 455 This may be transformed by the formulae x = p cos , y = p sin «^, when it takes the form p^ -j- =h. '■ Now, p -? may be readily proved to be the sine of the angle between the cm-ve in question, and the meridian of the surface of revolution. If this angle be denoted by /, we see that through- out any geodesicon a surface of revolution I 00 sin J ' 567. To prove that, throughout a geodesic on a central coni- coid, pd is constant, p heing the perpendicular on the tangent plane from the center, and d the central radius parallel to the geodesic. t the equation of the conicoid be x' f z" , a be, and let (?, m, w) be the direction of the tangent line at any point of a geodesic. We shall then. have the equations (1), (2), Z' + m' + «' = 1. (3). dl dm dn ds ds Ts = k, ^— ■ = —— = - ^"^ X y z ^ ¥ "? Ix a' + J. + nz = 0, Dliferentiating (2) we obtain X dl y dm ^^ <^« , i^ , ^ , ^^ _ a'd's'^P'd^'^c'ds^a'^ii'^c' "' hence, by (1), Z'' g!+|I + ^) +5 + F + ? =^- ^56 GEODESIC LINES fhx- ley hz^ I si or tl dl m dm n dn\ fa? ,y', «'"\ which gives the first integral Also ^ -^\y\^' 1_^\'«\^' and therefore, finally, -j^^ = Q, or ^(Z is constant. The equations of a geodesic on a conicoid have not been in- tegrated farther, but the property here proved to belong to them leads to many important results. The following geometrical proof of this property, and the corresponding one for lines of curvatm-e, was published by Mr Joyce, in the Quarterly Journal, Vol, 5, page 265. Consider any curve traced on a surface as the limit of an equilateral inscribed polygon, as in Chap. xxii. The tangent planes to the surface at the different points of the curve will be the limiting positions of planes passing through the sides of this polygon, and intersecting the surface in conies, of which these sides are diameters, and the normals to the surface will be the limiting positions of lines drawn through the middle points of the chords perpendicular to these planes. Let the line of intersection of two consecutive tangent planes be called the conjugate line. Then, if two consecutive normals to ' the surface make equal angles with the osculating plane containing the corresponding sides of the polygon, these sides must also make equal angles with the conjugate line, and the angles which they make with the same part of the conjugate line will be equal, or supplemen- j^EODESlC LINES. 457 tary, according as the normals lie on the same side, or on oppo- site sides, of the osculating plane. If the normals lie on the same side they will intersect, and the limit of the polygon will then he a line of curva- ture. If the normals lie on opposite sides, then, in the limit, when they coincide, the normal to the surface, which is their common limit, must lie in the osculating plane, or the curve hecomes a geodesic. On any surface, therefore, consecutive sides of those polygons which become, in the limit, lines of curvature will make equal angles with the corresponding conjugate line; and, for those whose limits are geodesies, these angles are supplementary. Now, let LM, MN\)Q consecutive elements of such a polygon, on a conicoid whose centre is 0, MP the corresponding conju- gate line, Om the central radius vector parallel to the conjugate line, ml, mn chords of the ellipsoid parallel to ML, MN, Then ml, mn will ultimately be tangents to the central section made by & plane parallel to the tangent plane to the surface, at a point on the curve. Also, since the angles 0ml, Omn, are either equal or supplementary, the perpendiculars from on these tangents will be equal. That is, if a plane be drawn through the center parallel to the tangent plane at any point on 458 GEODESIC LINES. the curve, and a tangent line be drawn to this section parallel to the tangent line to the curve, the perpendicular from the center on this tangent line will be of constant length. Let -p' be this length. Then ffd will be the volume of a parallelo- piped enveloping the conicoid, and having its faces parallel to conjugate planes, and will therefore be constant. Hence, since />' has been shewn to be constant, pd is con- stant. 568. The constant pd has the same value for all geodesies passing through an umbilicus. For, at the umbilicus, p is common to all the geodesies, also d being parallel to a tangent line at an umbilicus, is a diameter of a central circular section, and is therefore equal (XC to ab, whence ^£?= -T- 6= «c. 569. The locus of a point on an ellipsoid, the sum, or dif- ference of whose geodesic distances from two adjacent umbilici is constant, is a line of curvature. For, let Z7, F be two adjacent umbilici, P any point on the locus, then at P, pd will be the same for the geodesies PU, PV, and p being the same, d will also be the same : that is, the central radii parallel to the tangents are equal. These radii must there- fore make equal angles with the axes of the section in which they lie, which is parallel to the tangent plane at P. But these axes are parallel to the tangents to the lines of curvature at P. The geodesies PU, PV will therefore make equal angles with the lines of curvature at P. But if the sum of the geodesic distances PU, PV be constant, the locus of P will be some curve which always bisects the external angle between PU, PV, and if the difference be constant, the locus will always bisect the internal angle, by the same proof as is used for the cor- responding property of plane conies. But the lines of curvature have been shewn to possess this property. The locus of P will therefore be one of these lines of curvature, according as the sum or the difference is constant. 570. All geodesies Joining two opposite umbilici are of equal length. GEODESIC LINES. 459 Let P be any point on a line of curvature, U, V two um- bilici for which PU+ PF is constant, U' the umbilicus opposite U. Then PU, PU', making equal vertically opposite angles, "with the tangent to the line of curvature at P, will be parts of the same geodesic. Also the line of curvature will bisect the internal angle between PU', PV, and the difference of PU', PV will therefore be constant. But the sum oi PU, PVis constant, and therefore the sum of PU, PU' is constant, or the geodesic UPU' is of constant length. 571. The constant pd has the same value for all geodesies which touch the same line of curvature. For j)d is constant throughout a line of curvature, and, at the point of contact with any geodesic, both p and d are th^ same for both curves. 572. Two geodesic tangents, drawn to a line of curvature, will make equal angles with the lines of curvature at their intersection. For, pd will be the same for both, and therefore at the point of intersection, d will be the same for both, which, as in Art. (569), leads to the property enunciated. It follows from this, by the same proof as for plane confocal conies, that if two tangents be drawn to a line of curvature, from a point lying on another line of curvature of the same system, that the sum of the tangents will exceed the intercepted arc by a constant quantity. 573. The locus of the intersection of two geodesic tangents to two lines of curvature, at right angles to each other, lies on a sphere. Let the lines of curvature be given by the equations a? a^-k^ ' V'-k^ ' c'-k,' V.2 a'-k.'^b'-kyc'-k^' ' 460 GEODESIC LINES. then, along these lines, pd=-j--, or, -j—, Art. (552), and at the intersection of the two geodesies, pd has these values for both, and p is the same for both. Let a', V he the semi-axes of the central section conjugate to their point of intersection, d, d' the central radii parallel to the tangents to the geodesies. These being at right angles, we have p'd'^fd'' " a'b'Y ~ aVc^ r being the central distance of the point of intersection. Hence, a^6V ~ dW& ,or»--a+&+c k^ Ic^, whence r has a constant value, or the point of intersection lies on a sphere. A particular case of this is, that the foot of the geodesic perpendicular from an umbilicus, on any tangent to a line of .curvature, lies on a sphere, any geodesic through an umbilicus touching the limit of the lines of curvature, obtained by taking h = h. 574, Another form of the fundamental equation for geo- desies may be found, in terms of the parameters of the lines of curvature at any point of the geodesic, and the angles which the geodesic makes with the lines of curvature. Let 6,-^ — be these angles, and \ , k^ the parameters of the lines of curvature, then \, \ are the semi-axes of the central section conjugate to the point considered, and 1 _ cos'''g siu°(9 d-'~ k^ ^ k,' • GEODESIC LINES. 461 Also, if {x, y, z) be the point, the equation whose roots are k^yk^ IS whence k^h,' = a»6V f ^ + fj + ^) = ^' . * \a b c J p From these ec[uations we have ' 212 2 —2^ = /c/ cos" 6 + k^ sin'' 6, pa ^ ' or Jb/ cos' 6 + k^. sin" 6 — const. If there be two geodesies at right angles to each other, for each of which ^cf is a given quantity, we shall have k,^co&^e + k^&We=X\ 7c,' sin" e + k,^ CDS' e = X", whence k^' + k^' ^X^' + X", or a''+b^ + c'-'d'-f-z^ = \'' + \'\ and the point of intersection lies on the same sphere as that found in the last Article. From each point of this locus, we can then draw two pairs of tangents at right angles to each other, one of each pair touching the corresponding line of curvature, and since only two geodesies can be drawn through each point (unless an um- bilicus) having a given pd, we see that every geodesic, for which pd has a given value, will touch the corresponding line of curvature. 575. ^ a point of an ellipsoid, and a line of curvature, he projected on a plane of circular section, hy lines parallel to the greatest or least axis ; the angle between the geodesic tangents drawn from, the point to the line of curvature will be equal to the angle between the tangents from the projection of the point to the projection of the line of curvature. 462 GEODESIC LINES. Let (a5, y, z) be any point on the ellipsoid, {X, Y) the co- ordinates of its projection by -a line parallel to the axis of x on a plane of circular section, the axis of Y coinciding with that of y, and the axis of X lying in the plane of circular section. We shall then have y= Y, z = Xaina, where 2 + X^^^^\^-b''~d'-W' or the projection of a line of curvature is a conic having given foci, which are readily seen to be the projections of the um- bilici. Now, let {x', y', z') be a point from which geodesic tangents are drawn to the line of curvature X, and let \, \ be the parameters of the lines of curvature through {x',y',z'). Then, if 20 be the angle between the tangents, 6 is given by the equation Let (X', Y') be the projection of {x', y', z), then k^, ^j, are the roots of the equation X'^ ys j2 i + l^-d'^li'-h^ d'-b" GEODESIC LINES. 463 and if 2^ be the angle between tlie tangents from {X', Y') to the conic. i + : we shall have , ,,, ' g^ (^°-^') (^'-''°) {x£7+x5-^F , tan»2(^ = ; ^ — ! p— ^^ i , But, from the equation in k, and V + A/ = 6' + c^ + ^^ {X" + Y"). Hence tan' 2(^ = - 4 V ^, ^ {a'-Fy {k,' + k,'~2X'r sec ^

y, k,' + k; - 2V = ± {k,' - /<;/) cos' (j,, which gives the equations k^'sm'4> + k^'cos'(f>^X', or ^i'cos'0+A;/sin''^=\', which proves that 20 is either equal to 20, or to tt - 2d, and in either case the acute angle between the geodesic tangents is equal to the acute angle between the tangents to the projection. 576. ^ a geodesic he drawn through an umbilicus, inclined at a constant angle to a geodesic tangent to a fixed line of curvature, the point of intersection will lie on one of two spheres. 464 GEODESIC LINES. .For> the locus of the projection of the point will be the locus of the point of intersection of a straight line drawn through a focus with the tangent to a fixed conic, and will therefore be one of two gircles. But, since we see that if the projection lie on a circle in the plane of cir- cular section, the corresponding point of the ellipsoid must lie on a sphere. A particular case of this will be that, if two geodesies be drawn through two adjacent umbilici, inclined at a constant angle, their point of intersection will lie on a sphere. 577. The tangent lines to a geodesic on a conicoid will all touch a fixed confocal conicoid. Let I, m, n be the direction-cosines of a tangent line to the conicoid, 0^ v" «^ - a c at a point [x, y, z). This gives the condition Ix my nz _ a c and the condition that this line may touch the confocal conicoid, x^ y' z^ _ is f Ix Ix my nz which is, on reduction, a quadratic in F, one of whose roots will of course be zero, - Making the substitutions „2/ZV mY wVN 2mnyz=Fcr. .flict aiife^ 'ap^ .^-iiie 'projection of ON on the tiipgeDit to' s^tP fit -:^,-:i3ieb "' ' ' • " \ '• } ".•"'' -\: t-''- ^.' ' " dp _d, ^f 'H ■i'v t.jW • 1'^ fll iftkfii*** f-s ^ A.*" »f»' /t' ..^ t,Uf i^ Kfi^^^'i ' (. "V-v( -('WS 3k^^^^^^ ^ (tifl l.l'l-'l.F' iK^fS'^"'' I ■^ i > 'f- • mM f l' ,n gi- . • -■ " ^ ^^u. % •I Mis