arY 997 ,m'„ IL, liSsliSSMiffliiLiSlStcffiSi 1 PHiS: SOME TTITRl JL -JUL JLi-J ITT ».««<• ffi! ! 'ri W f fflu— i» t i |LW i w imi AND 11 ■11 F- M- WHJLSQN JL lii Pi 1 lilll i lllllf alls) IlllPlillil . . .■ w .•••;';;;: .:;•::;!.' ■ lilll : -:. i: ;r.'rf«j|:i*: : ; : s ■ isfflffisr iSBHSSHH M»IHi . fyxnttt Wnfomitg |f itafg BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Hcnrg M. Sage 1891 „4>JA1S7.3 W/./3L//.fA.^ Cornell University uw-» olin.anx 997 DESCRIPTIVE GEOMETRY AND MECHANICAL DRAWING SERIES By Frederick Newton Willson, C.E., A.M. The specialized treatises constituting this series are uniform in size (nine inches by twelve inches, with text-page six and three-quarters by nine), printed on plate paper, and elaborately illustrated with photo-engravings, wood-cuts, cero- graphic process blocks, half-tone plates, etc. They are adapted both to class-room use and self-instruction. 1. fiote^Taking, Dimensioning and Iiettering. A text-book on Free-hand and Mechanical Lettering in general, and on the lettering and dimensioning of u working drawings ; " also con- taining full instructions as to the sketching of bridge and machine details, for inspection or design. Various conventional methods of represent- ing materials are also given. Roman and Gothic letters, vertical and inclined, together with the Soennecken Round Writing, Reinhardt Gothic, and other alphabets much employed by engineers and architects, receive ample illustration ; while the total of sixty-five complete alphabets affords an unusual range of choice among serviceable forms. Full instructions are given as to the proportioning of titles, spacing, mechanical "short-cuts," etc.; also a large number of designs for fancy corners and borders. $1.25 net. 2. The Third Angle JVTethod of making Working Dracaings. A practical treatise on the American draughting-office system of applying the principles of projection in the making of "shop " drawings. It contains a large number of geneial problems on projection, intersections, and the development of surfaces, together with such special problems as an upper-chord post-connection of a bridge ; standard screws, bolts and nuts, with tables of proportions ; helical springs ; rail and valve sections ; spur gear. Illustrated with eighty-five cerographic-process blocks, and eight wood engravings. $1.25 net. 3. Some JVTathematieal Curves and their Graphical Construction. This work presents in compact form the more interesting and important properties, methods of construction, and practical applications of the curves with which it is essential that the architect and engineer should be familiar. It is also adapted to class-room use in mathematical courses. Among its special features are sections on homologous plane and space figures, given in connection with the conic sections, and laying a sound foundation for work in projective geometry ; link-motion curves and centroids, as an introduction to kinematic geometry ; historic notes and problems. Among other topics treated are the Helix ; Common Cycloid and its Companion ; Curtate and Prolate Ortho-cycloids ; Hypo, Epi, and Peri-trochoids ; Special Trochoids, as the Ellipse, Straight Line, Linton, Cardioid, Trisectrix, Involute, Spiral of Archimedes ; Parallel Curves ; Conchoid ; Quadratrix ; Cissoid ; Tractrix ; Witch of Agnesi ; Cartesian Ovals ; Cassian Ovals ; Catenary ; Logarithmic Spiral; Hyperbolic Spira'i. Lituus, and the Ionic Volute. The work concludes with a chapter on the nomenclature and double generation of cycloidal curves. Illustrated with fifty-four ceiographic blocks and one half-tone. $1-50 net. z±. Practical Engineering Dracaing and Third Angle Projection. A practical course for students in scientific, technical and manual training schools, and for engineering or architectural draughtsmen. 1 1 includes not only the contents of the first three volumes of this series, but also full instructions as to the choice and use of drawing instruments and materials ; line tinting and shading ; conventional methods of representation; plane problems of frequent recurrence; blue-printing and other methods of graphic reproduction and illustration; isometric drawing; and cavalier perspective (oblique projection). One hundred and seventy-eight pages, two hundred and seventy illustrations, and sixty-five alphabets. 2.80 net. 5. Shades, Shadocas and Ltinear Perspective. A short course for students of engineering or architecture, and for professional draughtsmen. For its reading a knowledge of elementary projection drawing is assumed. The methods employed in the best American practice receive especial emphasis. Illustrated by twenty-one cerographic blocks and three half-tones. 1.00 net. 6. Descriptive Geometry — Pure and Applied, raith a chapter on Jiigher Plane Curves and the Jielix. This work contains in logical sequence not only the matter constituting volumes 2, 3 and 5 of this series, but also a chapter on the pure descriptive geometry of Monge, with elaborate illustration of the mathematical surfaces of most importance to the graphicist, and with applica- tions to Trihedrals, Spherical Projections, Axonometric (including Isometric) Projection, One-plane Descriptive Geometry, Oblique Projection, etc. , the whole constituting a broad, educational course. One hundred and ninety pages, illustrated with two hundred and eighty-four cerographic blocks, six half-tones, and twenty-one wood engravings. $3.00 net. 7. Theoretical and Practical Graphics. This work embodies the entire contents of the six preceding treatises, in a volume of three hundred pages, with four hundred and seventy eight illustrations, sixty-five alphabets, and thirty-eight border designs. It constitutes a progressive course in graphical science. $4.00 net Published by the macmillan company, 66 Fifth Ave., New York. London : macmillan & CO., Limited. MATHEMATICAL CURVES AND THEIR GRAPHICAL CONSTRUCTION. a 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/details/cu31924032184271 SOnE nATHEMATlGAL GtiRVES AND THEIR GRAPHICAL CONSTRUCTION A BRIEF TREATISE ON THE Properties, Methods of Construction, and Practical Applications, of Conic Sections, Trochoids, Link-Motion Curves, Centroids, Spirals, the Helix, and Other Important Curves. For Students in Mathematical, Engineering or Architectural Courses, Draughtsmen, Etc. Frederick flecuton CXlillson, C.E., K.fil., Professor of Descriptive Geometry, Stereotomy and Technical Drawing in the John C. Green School of Science, Princeton University ; Mem. Am. Soc. Mechanical Engineers; Associate Am. Soc. Civil Engineers; Mem. Am. Mathematical Society; Fellow American Association for the Advancement of Science. fletxt York THE MACMILLAN COMPANY LONDON : MACMILLAN & CO., Ltd. 1898 r;i'-.HT- fiE=.EF ;f:i ^ tX O "5 •- f\ ,iu^' COPYRIGHT 1896 BY FREDERICK N. WILLSON ^uhJ PREFACE r3 book has been prepared to occupy the middle ground between the extensive and abstruse treatises which appeal solely to mathematicians, and those at the other extreme, frequently styled "practical," and which give scarcely a hint of the beauty, history and relations of the more important curves. Although appearing originally as a chapter of the author's larger and more general work entitled Theoretical and Practical Graphics, the idea of its ultimate separate issue in this form was in mind at the time of writing, and with the exception of Article 58 (reproduced in substance below), and two or three unimportant allusions, it was made independent of the other matter with which it was paged. F. N. W. NOTE ON THE USE OF FRENCH CURVES FOR DRAWING NON-CIRCULAR ARCS For the accurate delineation of non - circular curves the student will need a number of the irregular forms called French Curves or Sweeps, which can be obtained of any dealer in draughtsmen's supplies, and in various materials, as pear -wood, hard rubber and celluloid. The last is particularly recommended, both on account of its transparency and its cleanliness. After obtaining a sufficient number of the points of the curve connect them by free - hand arcs, pencilled with the hand on the concave side of the curve, this tending to the avoidance of too sudden changes of curvature. The free-hand curve thus drawn should be as true as possible, since, in inking, the aim should be to fit a rigid curve to a proper form already obtained, rather than to correct errors. "When ready to ink, a portion of one of the French curves is sought, which will coincide for some distance with an arc of the desired curve. Less of this must be actually used than would seem at first sight justifiable, as some allowance must be made for the change in curvature of the next portion, and anything like angles in a curve is abso- lutely unallowable. When placed in a new position a portion of the irregular curve must coincide with a part of that last inked. It is frequently necessary to ink in curves that are too sharp for instrumental work ; for example, that at 7', Fig. 81. The fine writing pen, used free-hand, must then be brought into requisition ; but the student must handle it so that there shall be no indication of the point where its use began, imitating the quality and steadiness of the ruled line. EXERCISES FOR THE IRREGULAR CURVE. 30 CHAPTER V. THE HELIX. — CONIC SECTIONS.— HOMOLOGICAL PLANE CURVES AND SPACE -FIGURES. — LINK- MOTION CURVES.— CENTROIDS. — THE CYCLOID. — COMPANION TO THE CYCLOID. — THE CUR- TATE TROCHOID. — THE PROLATE TROCHOID. —HYPO-, EPI-, AND PERI-TROCHOIDS. — SPECIAL TROCHOIDS — ELLIPSE, STRAIGHT LINE, LIMACON, CARDIOID, TRISECTRIX, INVOLUTE, SPIRAL OP ARCHIMEDES.— PARALLEL CURVES. — CONCHOID. — QUADRATRIX. — CISSOID. — TRACTRIX. — WITCH OF AGNESL — CARTESIAN OVALS. — CASSIAN OVALS. -CATENARY. — LOGARITHMIC SPIRAL.— HYPERBOLIC SPIRAL.— THE LITUUS.— THE IONIC VOLUTE. 119. There are many curves which the draughtsman has frequent occasion to make, whose con- struction involves the use of the irregular curve. The more important of these are the Helix; Conic Sections — Ellipse, Parabola and Hyperbola; Link-motion curves or point -paths; Centroids; Trochoids; the Involute and the Spiral of Archimedes. Of less practical importance, though equally interesting geometrically, are the other curves mentioned in the heading. The student should become thoroughly acquainted with the more important geometrical properties of these curves, both to facilitate their construction under the varying conditions that may arise and also as a matter of education. Considerable space is therefore allotted to them here. At this point Art. 58 should be reviewed, and in addition to its suggestions the student is fur- ther advised to work, at first, on as large a scale as possible, not undertaking small curves of sharp curvature until after acquiring some facility with the curved ruler. THE HELIX. 120. The ordinary helix is a curve which cuts all the elements of a right cylinder at the same angle. Or we may define it as the curve which would be generated by a point having a uniform motion around a straight line, combined with a uniform motion parallel to the line. Fig- SI. The student can readily make a model of the cylinder and helix by drawing on thick paper or Bristol - board a rectangle A" B" C" D" (Fig. 81) and its diagonal, D" B" ; also equidistant elements, as m"b", n"c", etc. Allow at the right and bottom about a quarter of an inch extra for over- lapping, as shown by the lines xy and sz. Cut out the rectangle zx; also cut a series of vertical slits between D" C" and zs; put mucilage between B" C" and xy; then roll the paper up into cylindrical form, bringing A" D" t" ' h" in front of and upon the gummed portion, so that A" D" 40 THEORETICAL AND PRACTICAL GRAPHICS. will coincide with B" C". The diagonal D" B" will then be a helix on the outside of the cylinder, but half of which is visible in front view, as D'T, (see right-hand figure); the other half, T A', being indicated as unseen. To give the cylinder permanent form it can then be pasted to a cardboard base by mucilage on the under side of the marginal flaps below D" C", turning them outward, not in toward the axis. The rectangle A" B" C" D" is called the development of the cylinder; and any surface like a cylinder or cone, which can be rolled out on a plane surface and its equivalent area obtained by bringing consecutive elements into the same plane, is called a developable surface. The elements m"b", n"c", etc., of the development stand vertically at b, c, d . . . . g of the half plan, and are seen in the elevation at m' b', n' c', o' d', etc. The point 3', where any element, as c', cuts the helix, is evidently as high as 3", where the same point apjjears on the development. We may therefore get the curve D'T A' by erecting verticals from b,c,d....g, to meet horizontals from the points where the diago- nal D" B" crosses those elements on the development. D" C" obviously equals 2 -n-r, where r=OD. The shortest method of drawing a helix is to divide its plan (a circle) and its pitch (D' A', the rise in one turn) into the same number of equal parts; then verticals bm', en', etc., from the points of division on the plan, will meet the horizontals dividing the pitch, in points 2', 3', etc., of the desired curve. The construction of the helix is involved in the designing of screws and screw-propellers, and in the building of winding stairs and skew-arches. Mathematically, both the curve and its orthographic projection are well worth study, the latter being always a sinusoid, and becoming the companion to the cycloid for a 45°-helix. (Arts. 170 and 171). For the conical helix, seen in projection and development as a Spiral of Archimedes, see Art. 191. THE CONIC SECTIONS. 121. The ellipse, parabola and hyperbola are called conic sections or conies because they may be obtained by cutting a cone by a plane. We will, however, first obtain them by other methods. According to the definition given by Boscovich, the ellipse, parabola and hyperbola are curves in which there is a constant ratio between the distances of points on the curve from a certain fixed point (the focus) and their distances from a fixed straight line (the directrix). Referring to the parabola, Fig. 82, if S and B are points of the curve, F the focus and A'}" the directrix, then, if S F : S T : : B F : B X, we conclude that B and S are points of a conic section. 122. The actual value of such ratio (or eccentricity) may be 1, or either greater or less than unity. When 8F equals ST the ratio equals 1, and the relation is that of equality, or parity, which suggests the parabola. 123. If it is farther from a point of the conic to the focus than to the directrix the ratio is greater than 1, and the hyperbola is indicated. 124. The ellipse, of course, comes in for the third possibility as to ratio, viz., less than 1. Its construction by this principle is not shown in Fig. 82 but later, (Art. 142), the method of generation here given illustrating the practical way in which, in landscape gardening, an elliptical plat would be laid out; it is therefore called the construction as the "gardener's ellipse." Taking AC and DE as representing the extreme length and width, the points F and F (foci) would be found by cutting AC by an arc of radius equal to one -half AC, centre D. Pesjs or pins at F and F, , and a string, of length A C, with ends fastened at the foci, complete the preliminaries. The curve is then traced on the ground by sliding a pointed stake against the string, as at P so that at all times the parts F, P, F P, are kept straight. CONIC SECTIONS. 41 125. According to the foregoing construction the ellipse may be denned as a curve in which the sum of the distances from any point of the curve to tivo fixed points is constant. That constant is evidently the longer or transverse (major) axis, A C. The shorter or conjugate (or minor) axis, D E, is perpen- dicular to the other. With the compasses we can determine P and other points of the ellipse, by using F and F x as centres, and for radii any two segments of A C. Q, for example, gives A Q and C Q as segments. Then arcs from F and F, , with radius equal to Q C, would intersect arcs from the same centres, radius Q A, in four points of the ellipse, one of which is P. 126. By the Boscovich definition we are also enabled to construct the parabola and hyperbola by continuous motion along a string. For the parabola place a triangle as in Fig. 82, with its altitude G X toward the focus. If a string of length G X be fastened at G, stretched tight from G to any point B, by putting a pencil at B, then the remainder BX swung around and the end fastened at F, it is then, evidently, as far from B to F as it is from B to the directrix; and that relation will remain constant as the tri- angle is slid along the directrix, if the pencil point remains against the edge of the triangle so that the portion of the string from 6 to the pencil is kept straight. X-ig-. 82. \ THE HYPERBOLA \ s,y „; cor- respondingly B 1 C" and B, C 1 meet at a . But A 1 B 1 and A 2 B. 2 , being corresponding lines, lie in the plane with S 2 , though belonging to figures in two other planes; they must, there- fore, meet also at the same point, c ; and similarly for the other lines in the figures used with S 2 . 146. Were A l B 1 C\ a circle, and all its points joined with S t , the figure A 1 B l C 1 would obviously be an ellipse; equally so were A 2 B 2 C 2 a circle used in connection with S 2 . We may, therefore, 48 THEORETICAL AND PRACTICAL GRAPHICS. substitute a circle for .1 2 J3 2 C 2 , and using on the same plane with it get an ellipse in place of the triangle A l B 1 C 1 . Before illustrating this, it is necessary to show the relation of the axis to the other elements of the problem, and supply a test as to the nature of the conic. 147. First as to the axis, and employing again for a time a space figure (Fig. 93), it is evident that raising or lowering the horizontal plane cXY parallel to itself, and with it, necessarily, the axis, would not alter the Mud of curve that it would cut from the cone S. H A B, were the elements of the latter prolonged. But raising or lowering the centre S, while the base circle AHBt remained, as before, in the same place, would decidedly affect the curve. Where it is, there are two elements of the cone, SA and SB, which would never meet the plane c X Y. The shaded plane containing those elements meets the vertical plane in " vanishing line (a),'' parallel to the axis. This contains the projections, A and B, of the points at infinity where the lower plane may be considered as cutting the elements SA and SB. Were S and the shaded plane raised to the level of H, making "vanishing line (a)" tangent to the base, there would be one element, S H, of the cone, parallel to the lower plane, and the section of the cone by the latter would be the parabola; as it is, the hyperbola is indicated. The former would have but one point at infinity; the latter, two. 148. Raise the centre S so that the vanishing line does not cut the base, and evidently no line from S to the base would be parallel to the lower plane; but the latter would cut all the elements on one side of the vertex, giving the ellipse. 149. Bearing in mind that the projection of the circle A H B t is on the lower plane produced, if we wish to bring both these figures and the centre S into one plane without destroying the relation between them, we may imagine the end plane Q L X removed, the rotation of the remaining system occurring about cr x in a manner exactly similar to that which would occur were iojc a system of four pivoted links, and the point o pressed toward c. The motion of S would be parallel and equal CONIC SECTIONS AS HOMOLOGOUS FIGURES. — RELIEF-PERSPECTIVE. 49 to that of o, and, like the latter, S would evidently maintain its distance from the vanishing line and describe a circular arc about it. The vanishing line would remain parallel to the axis. 150. From the foregoing we see that to obtain the hyperbola, by projection of a circle from a point in the plane of the latter, we would require simply a secant vanishing line, M N (Fig. 94), and an axis of homology parallel to it. Take any point P on the vanishing line and join it with any point K of the circle. PK meets the axis at y; hence whatever line corresponds to PK must also meet the axis at y. OP is analogous to S A of Fig. 93, in that it meets its corresponding line at infinity, i.e., is parallel to it. Therefore yk, parallel to OP, corresponds to Py, and meets the ray OK at I; corresponding to A'. Then K joined with any other point R gives Kz. Join z with h and prolong R to intersect kz, obtaining r, another point of the hyperbola. 151. In Fig. 93, were a tangent drawn to arc A H B at B, it would meet the axis in a point which, like all points on the axis, " corresponds to itself." From that point the projection of that tangent on the lower plane would be parallel to SB, since they are to meet at infinity. Or, if SJ is parallel to the tangent at B, then / will be the projection of J 1 at infinity, where SJ meets the tangent; ■/ will be therefore one point of the projection of said tangent on the lower plane; while another point would be, as previously stated, that in which the tangent at B meets the axis. 152. Analogously in Fig. 94, the tangents at M and N meet the axis, as at F and E; but the projectors OM and N go to points of tangency at infinity; M and N are on a "vanishing line"; hence OM is parallel to the tangent at infinity, that is, to the asymptote (see Art. 134) through F; while the other asymptote is a parallel through E to N. 153. As in Fig. 93 the projectors from S to all points of the arc above the level of S could cut the lower plane only by being produced to the right, giving the right-hand branch of the hyper- bola; so, in Fig. 94, the arc MHN, above the vanishing line, gives the lower branch of the hyperbola. To get a point of the latter, as h, and having already obtained any point x of the other branch, join H with X (the original of a>) and get its intersection, g, with the axis. Then xgh corresponds to g X H, and the ray OH meets it at h, the projection of H. The cases should be worked out in which the vanishing line is tangent to the circle or exterior to it. 154. The homological figures with which we have been dealing were plane figures. But it is possible to have space figures homological with each other. In homological space figures corresponding lines meet at the same point in a plane, instead of the same point on a line. A vanishing plane takes the place of a van- ishing line. The figure that is in homology with the original figure is called the relief -perspective of the latter. (See Art. 11.) Remarkably beautiful effects can be obtained by the construction of homological space figures, as a glance at Fig. 95 will show. The figure represents a triple row of groined arches, and is from a photograph of a model designed by Prof. L. Burmester. Although not always requiring the use of the irregular curve, and therefore not strictly the material for a topic in this chapter, its close analogy to the foregoing matter may justify a few words at this point on the construction of a relief-perspective. 155. In Fig. 96 the plane P Q is called the plane of homology or picture -plane, and— adopting Cremona's notation— we will denote it by ■*. The vanishing plane M N, or <$>', is parallel to it. O is ng-- ©5. .-»() THEORETICAL AND PRACTICAL GRAPHICS. the centre of homology or perspective- centre. All points in the plane ir are their own perspectives, or, in other words, correspond to themselves. Therefore B" is one point of the projection or perspective of the line A B, being the intersection of A B with jr. The line v, parallel to A B, would meet the X'igT. 96. latter at infinity; therefore v, in the vanishing plane ', would be the projection upon it of the point at infinity. Joining v with B", and cutting v B" by rays OA and OB, gives A' B' as the relief- perspective of AB. The plane through and 4 5 cuts tt in 5"n, which is an axis of homology for AB and ^'5', exactly as ma in Fig. 92 is for ^1,5, and J,£. 2 . As DC in Fig. 96 is parallel to AB, a parallel to it through is again the line Ov. LINK-MOTION CURVES. 51 The trace of D C on -n- is C ". Joining v with C" and cutting v C" by rays D, C, obtains D' C" in the same manner as A' B' was derived. The originals of A' B' and CD' are parallel lines; but we see that their relief- perspectives meet at v. The vanishing plane is therefore the locus* of the vanishing points of lines that are parallel on the original object, while the plane of homology is the locus of the axes of homology of corresponding lines; or, differently stated, any line and its relief -perspective will, if produced, meet on the plane of homology. 156. Fig. 97 is inserted here for the sake of completeness, although its stud}' may be reserved, if necessary, until the chapter on projections has been read. In it a solid object is represented at the left, in the usual views, plan and elevation ; G L being the ground line or axis of intersection of the planes on which the views are made. The planes -n and ' are interchanged, as compared with their positions in Fig. 96, and they are seen as lines, being assumed as perpendicular to the paper. The relief- perspective appears between them, in plan and elevation. The lettering of A B and D C, and the lines employed in getting their relief- perspectives, being identical with the same constructions in Fig. 96, ought to make the matter clear at a glance to all who have mastered what has preceded. Burmester's Grundzuge der Belief- Perspective and Wiener's Darstellende Geometrie are valuable reference works on this topic for those wishing to pursue its study further; but for special work in the line of homological 'plane figures the student is recommended to read Cremona's Projective Geometry and Graham's Geometry of Position, the latter of which is especially valuable to the engineer or architect, since it illustrates more fully the practical application of central projection to Graphical Statics. LINK - MOTION CURVES 157. Kinematics is the science which treats of pure motion, regardless of the cause or the results of the motion. It is a purely kinematic problem if we lay out on the drawing-board the path of a point on the connecting-rod of a locomotive, or of a point on the piston of an oscillating cylinder, or of any point on one of the moving pieces of a mechanism. Such problems often arise in machine design, especially in the invention or modification of valve -motions. Some of the motion -curves or point -paths that are discovered by a study of relative motion are without special name. Others, whose mathematical properties had already been investigated and the curves dignified with names, it was later found could be mechanically traced. Among these the most familiar examples are the Ellipse and the Lemniscate, the latter of which is employed here to illustrate the general problem. The moving pieces in a mechanism are rigid and inextensible, and are always under certain conditions of restraint. "Conditions of restraint" may be illustrated by the familiar case of the con- necting-rod of the locomotive, one end of which is always attached to the driving-wheel at the crank -pin and is therefore constrained to describe a circle about the axle of that wheel, while the other end of the rod must move in a straight line, being fastened by the "wrist-pin" to the "cross- head," which slides between straight "guides." The first step in tracing a point-path of any mechanism is therefore the determination of the fixed points, and a general analysis of the motion. * Locus is the Latin for place; and in rather untechnical language, although in the exact sense in which it is nsed mathe- matically, we may say that the locus of points or lines is the place where you may expect to mid them under their conditions of restriction. For example, the surface of a sphere is the locus of all points equidistant from a fixed point (its centre). The locus of a point moving in a plane so as to remain at a constant distance from a given fixed point, is a circle having the latter point as its centre. 52 THEORETICAL AND PRACTICAL GRAPHICS. 158. We have given, in Fig. 98, two links or bars, MN and S P, fastened at N and P by pivots to a third link, N P, while their other extremities are pivoted on stationary axes at M and S. The only movement possible to the point N is therefore in a circle about M; while P is equally limited to circular motion about S. The points on the link NP, with the exception of its 2 MN _2 MS 3 THE LEMNISCATE AS A LINK-MOTION CURVE -^T extremities, have a compound motion, in curves whose form it is not easy to predict and which differ most curiously from each other. The figure-of-eight curve shown, otherwise the "Lemniscate of Bernoulli," is the point-path of Z, the link NP being supposed prolonged by an amount, P Z, equal to NP. Since NP is constant in length, if N were moved along to F, the point P would have to be at a distance NP from F, and also on the circle to which it is confined; therefore its new position /, is at the intersection of the circle Psr by an arc of radius PA 7 , centre F. Then Ff, prolonged by an amount equal to itself, gives /, , another point of the Lemniscate, and to which Z has then moved. All other positions arc similarly found. If the motion of N is toward D it will soon reach a limit, A, to its further movement in that direction, arriving there at the instant that P reaches a, when NP and PS will be in one straight line, SA. In this position any movement of P either side of a will drag A T back over its former path; and unless P moves to the left, past a, it would also retrace its path. P reaches a similar "dead point" at v. To obtain a Lemniscate the links A^P and PiS' had to be equal, as also the distance MS to MN. Bv varying the proportions of the links, the point-paths would be correspondingly affected. INS TA N TA NE US CEN TR ES.— CEN TR01D S. 53 By tracing the path of a point on PN produced, and as far from N as Z is from P, the student will obtain an interesting contrast to the Lemniscate. If M and S were joined by a link, and the latter held rigidly in position, it would have been called the fixed link; and although its use would not have altered the motions illustrated, and it is not essential that it should be drawn, yet in considering a mechanism as a whole, the line joining the fixed centres always exists, in the imagination, as a link of the complete system. INSTANTANEOUS CENTRES. CENTROIDS. 159. Let us imagine a boy about to hurl a stone from a sling. Just before he releases it he runs forward a few steps, as if to add a little extra impetus to the stone. While taking those few steps a peculiar shadow is cast on the road by the end of the sling, if the day is bright. The ZFigT- 99. boy moves with respect to the earth; his hand moves in relation to himself, and the end of the sling describes a circle about his hand. The last is the only definite element of the three, yet it is sufficient to simplify otherwise difficult constructions relating to the complex curve which is described relatively to the earth. 54 THEORETICAL AND PRACTICAL GRAPHICS. A tangent and a normal to a circle are easily obtained, the former being, as need hardly be stated at this point, perpendicular to the radius at the point of tangency, while the normal simply coincides in direction with such radius. If the stone were released at any instant it would fly off in a straight line, tangent to the circle it was describing about the hand as a centre; but such line would, at the instant of release, be tangent also to the compound curve. If, then, we wish a tangent at a given point of any curve generated by a point in motion, we have but to reduce that motion to circular motion about some moving centre; then, joining the point of desired tangency with the — at that instant — position of the moving centre, we have the normal, a perpendicular to which gives the tangent desired. A centre which is thus used for an instant only is called an instantaneous centre. 160. In Fig. 99 a series of instantaneous centres are shown and an important as well as inter- esting fact illustrated, viz., that every moving piece in a mechanism might be rigidly attached to a certain curve, and by the rolling of the latter upon another curve the link might be brought into all the positions which its visible modes of restraint compel it to take. 161. In the "Fundamental" part of Fig. 99 A B is assumed to be one position of a link. "We next find it, let us suppose, at A'B', A having moved over A A', and B over BB'. Bisecting A A' and B B' by perpendiculars intersecting at 0, and drawing A, A', OB and OB', we have A A' = t = B OB', and evidently a point about which, as a centre, the turning of A B through the angle d l would have brought it to A'B'. Similarly, if the next position in which we find A B is A" B", we may find a point s as the centre about which it might have turned to bring it there; the angle being 6 2 , probably different from 6 1 . N and m are analogous to and s. If s be drawn equal to s and making with the latter an angle 6 1 , equal to the angle A A' , and if Os were rigidly attached to A B, the latter would be brought over to A'B' by bringing s' into coincidence with s. In the same manner, if we bring s' n' upon s a through an angle 6 2 about s, then the next position, A" B", would be reached by A B. 0' s' n' m' is then part of a polygon whose rolling upon Osnm would bring A B into all the positions shown, provided the polygon and the line were so attached as to move as one piece. Polygons whose vertices are thus obtained are called central polygons. If consecutive centres were joined we would have curves, called centroids*, instead of polvgons - the one corresponding to s n m being called the fixed, the other the rolling centroid. The perpen- dicular from upon A A' is a normal to that path. But were A to move in a circle, the normal to its path at any instant would be simply the radius to the position of A at that instant. If, then, both A and B were moving in circular paths, we would find the instantaneous centre at the intersection of the normals (radii) at the points A and B. 162. In Fig. 98 the instantaneous centre about which the whole link NP is turning is at the intersection of radii MN and SP (produced); and calling it A' we would have XZ for the normal at Z to the Lemniscate. 163. The shaded portions of Fig. 99 illustrate some of the forms of centroids. The mechanism is of four links, opposite links equal. Unlike the usual quadrilateral fulfilling this condition, the long sides cross, hence the name "anti- parallelogram." The "fixed link (a)" corresponds to MS of Fig. 98, and its extremities are the centres of rotation of the short links, whose ends, / and /,, describe the dotted circles. For the given position T is evidently the instantaneous centre. Were a bar pivoted at T and *Keuleaux' nomenclature; also called centrodes by a number of writers on Kinematics. TROCHOIDS. 55 fastened at right angles to "moving link (a)," an infinitesimal turning about T would move "link (a)" exactly as under the old conditions. By taking "link (a)" in all possible positions, and, for each, prolonging the radii through its extremities, the points of the fixed centroid are determined. Inverting the combination so that "moving link (a)" and its opposite are interchanged, and proceeding as before, gives the points of "rolling centroid (a)." These centroids are branches of hyperbolas having the extremities of the long links as foci. By holding a short link stationary, as "fixed link (b)," an elliptical fixed centroid results; " rolling centroid (b) " being obtained, as before, by inversion. The foci are again the extremities of the fixed and moving links. Obviously, the curved pieces represented as screwed to the links would not be employed in a practical construction, and they are only introduced to give a more realistic effect to the figure and possibly thereby conduce to a clearer understanding of the subject. 164. It is interesting to notice that the Lemniscate occurs here under new conditions, being traced by the middle point of "moving link (a)." The study of kinematics is both fascinating and profitable, and it is hoped that this brief glance at the subject may create a desire on the part of the student to pursue it further in such works as Reauleaux' Kinematics of Machinery and Burmester's Lehrbuch der Kinematik. 165. Before leaving this topic the important fact should be stated, which now needs no argument to establish, that the instantaneous centre, for any position of a moving piece, is the point of contact of the rolling and fixed centroids. We shall have occasion to use this principle in drawing tangents and normals to the TROCHOIDS which are the principal Roulettes, or roll- traced curves, and which may be defined as follows: — If, in the same plane, one of two circles roll upon the other without sliding, the path of any point on a radius of the rolling circle or on the radius produced is a trochoid. 166. The Cycloid. Since a straight line may be considered a circle of infinite radius, the above definition would include the curve traced by a point on the circumference of a locomotive wheel as it rolls along the rail, or of a carriage wheel on the road. This curve is known as a cycloid* and is shown in T n a b c, Fig. 100. It is the proper outline for a portion of each tooth in a certain case of gearing, viz., where one wheel has an infinite radius, that is, becomes a "rack." Were T 6 a ceiling -corner of a room, and T V1 the diagonally opposite floor - corner, a weight would slide from T 6 to T u more quickly on guides curved in cycloidal shape than if shaped to any other curve, or if straight. If started at s, or any other point of the curve, it would reach T u as soon as if started at T 6 . 167. In beginning the construction of the cycloid we notice, first, that as T VI) rolls on the straight line A B, the arrow D RT will be reversed in position (as at D b !F 6 ) as soon as the semi- circumference T 8 D has had rolling contact with A B. The tracing point will then be at T 6 , its maximum distance from A B. When the wheel has rolled itself out once upon the rail, the point T will again come in contact with the rail, as at T V1 . ♦ "Although the invention of the cycloid is attributed to Galileo, it is certain that the family of curves to which it belongs had been known and some of the properties of such curves investigated, nearly two thousand years before Galileo's time if not earlier. For ancient astronomers explained the motion of the planets by supposing that each planet travels uniformly round a circle whose centre travels uniformly around another circle."— Proctor, Geometry of Cycloids. 56 THEORETICAL AND PRACTICAL GRAPHICS. The distance TT U evidently equals 2wr, when r—TR. We also have TD i —D i T n =vr If the semi-circumference T3Z> (equal to irr) be divided into any number of equal parts, and also the path of centres RR e (again=irr) into the same number of equal parts, then as the points 1, 2, etc., come in contact with the rail, the centre R will take the positions R U R 2 , etc., directly above the corresponding points of contact. A sufficient rolling of the wheel to bring point 2 upon A B would evidently raise T from its original position to the former level of 2. But as T must always be at a radius' distance from R, and the latter would by that time be at R 2 , we would find T located at the intersection («) of the dotted line of level through 2 by an arc of radius R T, centre R. L . Similarly for other points. The construction, summarized, involves the drawing of lines of level through equidistant points of division on a semi-circumference of the rolling circle, and their intersection by arcs of constant radius (that of the rolling circle) from centres which are the successive positions taken hj the centre of the rolling circle. It is worth while calling attention to a point occasionally overlooked by the novice, although almost self-evident, that, in the position illustrated in the figure, the point T drags behind the centre is! until the latter reaches R 6 , when it passes and goes ahead of it. From R 1 the line of level through 5 could be cut not alone at c by an arc of radius cR 1 but also in a second point; evidently but one of these points beltings to the cycloid, and the choice depends upon the direction of turning, and upon the relative position of the rolling centre and the moving point. This matter requires more thought in drawing trochoidal curves in which both circles have finite radii, as will appear later. Fig-. lOO. 168. Were points T e and T 12 given, and the semi -cycloid T s T VI desired, we can readily ascertain the "base," A B, and generating circle, as follows: Join T e with T,, ; at any point of such line, as x, erect a perpendicular, xy; from the similar triangles xyT t2 and T e D b T vl , having angle <£ common and angles equal, we see that x y : x T u : : T t , D t : D 5 T n : : 2 r : -n- r : : 2 : tt : : 1 : £; or, very nearly, as 14 : 22. If, then, we lay off xT u equal to twenty -two equal parts on any scale, and a perpendicular, xy, fourteen parts of the same scale, the line y T n will be the base of the desired curve; while the diameter of the generating circle will be the perpendicular from T 6 to y T u prolonged. 169. To draw the tangent to a cycloid at any point is a simple matter, if we see the analogy between the point of contact of the wheel and rail at any instant, and the hand used in the former illustration (Art. 159). At any one moment each point on the entire wheel may be considered as describing an infinitesimal arc of a circle whose radius is the line joining the point with the point of contact on the rail. The tangent at N, for example, (Fig. 100), would be t N, perpendicular to the normal, No, joining N with o; the latter point being found by using N as a centre and THE CYCLOID.— COMPANION TO THE CYCLOID. 57 cutting A B by an arc of radius equal to m I, in which m is a point at the level of N on any position of the rolling circle, while I is the corresponding point of contact. The point o might also have been located by the following method: Cut the line of centres by an arc, centre N, radius T R; o would obviously be vertically below the position of the rolling centre thus determined. 170. The Companion to the Cycloid. The kinematic method of drawing tangents, just applied, was devised by Roberval, as also the curve named by him the "Companion to the Cycloid," to which allusion has already been made (Art. 120) and which was invented by him in 1634 for the purpose of solving a problem upon which he had spent six years without success, and which had foiled Galileo, viz., the calculating of the area between a cycloid and its base. Galileo was reduced to the expedient of comrjaring the area of the cycloid with that of the rolling circle by weighing paper models of the two figures. He concluded that the area in question was nearly but not exactly three times that of the rolling circle. That the latter would have been the correct solution may be readily shown by means of the "Companion," as will be found demonstrated in Art. 172. 171. Suppose two points coincident at T (Fig. 101) and starting simultaneously to generate curves, the first of these points to trace the cycloid during the rolling of circle T VD, while the second is to move independently of the circle and so as to be always at the level of the point tracing the cycloid, yet at the same time vertically above the point of contact of the circle and base. This makes the second point always as far from the initial vertical diameter, or axis, of the cycloid, as the length of the arc from T to whatever level the tracing point of the latter has then reached; that is, MA equals arc THs; RO equals quadrant Tsy. Adopting the method of Analytical Geometry, and using as the origin, we may reach any point, A, on the curve, by co-ordinates, as Ox, % A, of which the horizontal is called an abscissa, the vertical an ordinate. By the preceding construction Ox equals arc sfy, while x A equals siv — the sine of the same arc. The "Companion" is therefore a curve of sines or sinusoid, since, starting from 0, the abscissas are equal to or proportional to the arc of a circle, while the ordinates are the sines of those arcs. It is also the orthographic projection of a 45° -helix. This curve is particularly interesting as "expressing the law of the vibration of perfectly elastic solids; of the vibratory movement of a particle acted upon by a force which varies directly as the distance from the origin; approximately, the vibratory movement of a pendulum; and exactly the law of vibration of the so-called mathematical pendulum."* (See also Art. 356). 172. From the symmetry of the sinusoid with respect to RR e and to 0, we have area T A R = E C R fi ; adding area D E L R to both mem- bers we have the area between the sinusoid and TD and DE equal to the rectangle RE, or one-half the rect- angle D E K T; or to ^ w r x 2 r = Tvr'', the area of the rolling circle. As T A CE is but half of the entire sinusoid, it is evident that the total area below the curve is twice that of the generating circle. The area between the cycloid and its "companion" remains to be determined, but is readily ascertained by noting that as any point of the latter, as A, is on the vertical diameter of the circle ^!gr- loi. ' Wood, Elements of Co-ordinate Geometry, p. 209. 58 THEORETICAL AND PRACTICAL GRAPHICS. passing through the then position of the tracing point, as a, the distance, A a, between the two curves at any level, is merely the semi-chord of the rolling circle at that level. But this, evidently, equals Ms, the semi-chord at the same level on the equal circle. The equality of Ms and A a makes the elementary rectangles Mss 1 m l and AA Y a x a equal; and considering all the possible similarly -constructed rectangles of infinitesimal altitude, the sum of those on semi-chords of the rolling circle would equal the area of the semi-circle TDy, which is therefore the extent of the area between the two curves under consideration. The figure showing but half of a cycloid, the total area between it and its "companion" must be that of the rolling circle. Adding this to the area between the "companion" and the base makes the total area between cycloid and base equal to three times that of the rolling circle. 173. The paths of points carried by and in the plane of the rolling circle, though not on its circumference, are obtained in a manner closely analogous to that employed for the cycloid. In Pig. 102 the looped curve, traced by the arrow-point while the circle CHM rolls on the base A B, is called the Curtate Trochoid. To obtain the various positions of the tracing point T describe a circle through it from centre R. On this circle lay off any even number of equal arcs, and draw radii from R to the points of division; also "lines of level" through the latter. The radii drawn intercept equal arcs on the rolling circle CHM, whose straight equivalents are next laid off on the path of centres, giving R i , R 2 , etc. While the first of these arcs rolls upon A B, the point T turns through the angle T R 1 about R, and reaches the line of level through point 1. But T is always at the distance R T (called the tracing radius) from R; and, as R has reached R 1 in the rolling supposed, we will find T x — the new position of T — by an arc from JJ„ radius TR, cutting said line of level. ffigr. ios. CURTATE TROCHOID. 4 — h — 4 — ^i_^ — | — + — j._ After what has preceded, the figure may lie assumed to be self-interpreting, each position of T having been joined with the position of R which determined it. 174. Were a tangent wanted at any point, as T,, we have, as before, to determine the point of contact of rolling circle and line when T reached T 7 , and use it as an instantaneous centre. T was obtained from R.; and the point of contact must have been vertically below the latter and on A B. Joining such point to T n gives the normal, from which the tangent follows in the usual way. 175. The Prolate Trochoid. Had we taken a point inside of the circle CHM and constructed its path, the only difference between it and the curve illustrated would have been in the name and the HYPO-, EPI- AND PERI-TROCHOIDS. 59 shape of the curve. An undulating, wavy path would have resulted, called the prolate trochoid; but, as before, we would have described a circle through the tracing point; divided it into equal parts; drawn lines of level, and cut them by arcs of constant radius, using as centres the successive positions of R. A bicycle pedal describes a prolate trochoid relatively to the earth. HYPO-, EPI- AND PERI -TROCHOIDS. 176. Circles of finite radius can evidently be tangent in but two ways — either externally, or internally; if the latter, the larger may roll on the one within it, or the smaller may roll inside the larger. When a small circle rolls within a larger, the radius of the latter may be greater than the diameter of the rolling circle, or may equal it, or be smaller. On account of an interesting property of the curves traced by points in the planes of such rolling circles, viz., their capability of being generated, trochoidally, in two ways, a nomenclature was necessary which would indicate how each curve was obtained. This is included in the tabular arrangement of names below, and which was the outcome of an investigation* made by the writer in 1887 and presented before the American Association for the Advancement of Science. In accepting the new terms, advanced at that time, Prof. Francis Reuleaux suggested the names Ortho-cycloids and Cyclo - orthoids for the classes of curves of which the cycloid and involute are respectively representative; orthoids being the paths of points in a fixed position with respect to a straight line rolling upon any curve, and cyclo -orthoid therefore implying a circular director or base -curve. These appropriate terms have been incorporated in the table. For the last column a point is considered as within the rolling circle of infinite radius when on the normal to its initial position, and on the side toward the centre of the fixed circle. As will be seen by reference to the Appendix, the curves whose names are preceded by the same letter may be identical. Hence the terms curtate and prolate, while indicating whether the tracing point is beyond or within the circumference of the rolling circle, give no hint as to the actual form of the curves. In the table, R represents the radius of the rolling circle, E that of the fixed circle. NOMENCLATURE 01' TROCHOIDS. Position of Tracing or Describing Point. On circumference of rolling circle. Within Circumference. Without Circumference. Circle rolling upon Straight Line. F = CO Ortho-cycloids. Cycloid. Prolate Trochoid. Curtate Trochoid. 177. From the above we Circle rolling upon circle. Straight Line rolling upon Circle. R=co External Internal contact. Larger Circle rolling-. Smaller circle rolling. ■1 R>F. 2 R < F. 2 R = F. Cyclo-orthoids. Epitrochoids. Peritrochoids. Major Hypotrochoids. | Minor Medial Hypotrochoids. Hypotrochoids. (a) Epicycloid. (a) Pericycloid. ' (d) Major Hypocycloid. (d) Minor Straight Hypocycloid. Hypocycloid. Involute. (b) Prolate Epitrochoid. (c) Prolate Peritrochoid. (e) Major Prolate Hypotrochoid. (f) Minor prolate ! (g) F ^ ol ? te , Hypotrochoid. „ LU ptlC h a ' Hypotrochoid. Prolate Cyclo-orthoid. (i) Major (c) Curtate (b) Curtate Curtate Epitrochoid. Peri trochoid Hypotrochoid. (e) Minor ! (g) Curtate w C ? rta u e -, TT Elliptical Curtate Hypotrochoid. Hypotrochoid. Cvclo-orthoid. I ve see that the prefix epi (over or upon) denotes the curves resulting from external contact; hypo (under) those of internal contact with smaller circle rolling; while peri (about) indicates the third possibility as to rolling. * Re-printed in substance in the Appendix. 60 THEORETICAL AND PRACTICAL GRAPHICS. jFigr- loa- ns. The construction of these curves is in closest analogy to that of the cycloid. If, for example, we desire a major hypocycloid, we first draw two circles, mVP, mxL, (Fig. 103), tangent internally, of which the rolling circle has its di- ameter greater than the radius of the fixed circle. Then, as for the cycloid, if the tracing -point is P, we divide the semi -circumference mVP into equal parts, and from the fixed centre, F, describe circles through the points of division, as those through 1, 2, 3, 4 and 5. These replace the "lines of level" of the cycloid, and may he called circles of distance, as they show the varying distances of the point P from F, for definite amounts of angular rotation of the former. For if the circle PVm were simply to rotate about R, the point P would reach m during a semi -rotation, and would then be at its maximum distance from F. After turning through the equal arcs P-l, 1-2, etc., its distances from F would be Fa and Fb respectively. If, however, the turning of P about R is due to the rolling of circle PVm upon the arc mxz, then the actual position of P, for any amount of turning about R, is determined by noting the new position of R, due to such rolling, as R,, R, 2 , etc., and from it as a centre cutting the proper circle of distance by an arc of radius R P. Since the radius of the smaller circle is in this case three - fourths that of the larger, the angle m F z (135°), at the centre of the latter, intercepts an arc, mxz, equal to the 180° -arc, m V P, on the smaller circle ; for equal arcs on unequal circles are subtended by angles at the centre which are inversely proportional to the radii. As a proportion we would have F m : R m : : 180°: 135°. (In an inverse proportion between angles and radii, in two circles, the "means" must belong to one circle and the '' extremes " to the other). While arc mVP rolls upon arc mxz, the centre R will evidently move over circular arc i?---i? 6 . Divide m x z into as many equal parts as m V P and draw radii from F to the points of division ; these cut the path of centres at the successive positions of R. When arc m 5-4, for example, has rolled upon its equal muv, then R will have reached R./, P will have turned about R through angle PR2 = mRA, and will be at n, the intersection of bfg — the circle of distance through 2 — by an arc, centre R 2 , radius R P. Similarly for other points. 179. General solution for rdl trochoidal curves, illustrated by epi- and peri -trochoids. To trace the path of any point on the circumference of a circle so rolling as to give the epi- or peri- cycloid, requires a construction similar at every step to that of the last article. The same remark applies equally to the path of a point within or beyond the circumference of the rolling circle. This is shown in Fig. 104, before describing which in detail, however, we will summarize the steps for any and all trochoids. Letting P represent the tracing point, R the centre of the rolling circle and F that of the fixed circle, we draw (1) a circle through P, centre R; (2) a circle (path of centres) through R, centre F; (3) ascertain by a proportion (as described in the last article) how many degrees of arc on either circle are equal to the prescribed arc of contact on the other; (4) on the path of centres lav EPI- AND PERI TROCHOIDS. 61 off — from the initial position of R and in the direction of intended rolling — whatever number of degrees of contact has been assigned or ascertained for the fixed circle, and divide this arc by radii from F into any number of equal parts, to obtain the successive positions of R, as R 1} R. 2 , etc.; (5) on the circle through P lay off — from the initial position of P, and in the direction in which it will move when the assigned rolling occurs — the same number of degrees that have been assigned or calculated as the contact arc of the rolling circle, and divide such arc into the same number of equal parts that was adopted for the division of the path of centres; (6) through the points of division obtained in the last step draw " circles of distance " with centre F, numbering them from ? ^ L GENER£rg^ P; (7) finally, to get the suc- cessive positions of P, use R P (the " tracing radius ") as a con- stant radius, and cut each circle of distance by an arc from the like -numbered position from R, selecting, of course, the right one of the two points in which said curves will always intersect when not tangent. In Fig. 104 the path of the point P is determined (a) as car- ried by the circle called "first generator," rolling on the exterior of the "first director"; (b) as carried by the "second generator" which rolls on the exterior of the "second director" — which it also encloses. In the first case the resulting curve is a 'prolate epi- trochoid; in the second a curtate peritrochoid ; but such values were taken for the diameters of the circles, that P traced the same curve under either condition of rolling.* These (before reduction with the camera) were 3" and 2" for first generator and first director, respectively. For the epitrochoid a semi- circle is drawn through P from rolling centre R; similarly with centre p for the peritrochoid. Dividing these semi -circles into the same number of equal parts, draw next the dotted " circles of distance " through these points, all from centre F. The figure illustrates the special case where the two sets of " circles of distance " coincide. The various positions of P as P lt P-i, etc., are then located by arcs of radii RP or p P, struck from the successive positions of R or p and intersecting the proper "circle of distance." * Regarding their double generation refer to the Appendix. In illustrating both methods in one flgure it will add greatly to the appearance and also the intelligibility of the drawing if colors are used, red for one construction and blue for the other. 62 THEORETICAL AND PRACTICAL GRAPHICS. For example, the turning of P through the angle PR1 about R would hring P somewhere upon the circle of distance through point 1; but that amount of turning would he due to the rolling of the first generator over the arc m Q, which would hring n upon Q and carry R to R,; P would therefore be at I\, at a distance RP from R 1: and on the dotted arc through 1. Similarly in relation to p. When s reached k, in the rolling, we would find P at P. 2 . Each position of 7' is joined with each of the centres from which it could he obtained. SPECIAL TROCHOIDS. 180. The Ellipse and Straight Line. Two circles are called Cardanic* if tangent internally and the diameter of one is twice that of the other. If the smaller roll in the larger, all points in the plane of the generator will describe ellipses except points on the circumference, each of which will move in a straight line — a diameter of the director. Upon this latter property the mechanism known as "White's Parallel Motion" is based, in which a piston-rod is pivoted to a small gear-wheel which rolls on the interior of a toothed annular wheel whose diameter is twice that of the pinion. 181. The Limacon and Cardioid. The Limacon is a curve whose points may be obtained by drawing random secants through a point on the circumference of a circle, and on each laying off a constant distance, on each side of the second point in which the secant cuts the circle. In Fig. 105 let Or and Od he random secants of the circle Ons; then if n v, np, ea and erf are each equal to some con- stant, b, we shall have v, p, a and d as four points of a Limacon. Refer points on the same secant, as a and d, to and the diam- eter Os; we then have d=p = c + c d = 2r cos 6 + b, while Oa = 2 r cos 6 — b; hence the polar equation is p=2r cosO±b. When b = 2r the Limacon becomes a CardioidJ (See Fig. 106). 182. All Limaeons, general and special, may be generated either as epi- or peri-trochoidal curves: as epi-trocboids the generator and director must have equal diameters, any point on the circumference of the generator then tracing a Cardioid, while any point on the radius tor radius produced) describes a Limacon; as peri -trochoids the larger of a pair of Cardanic circles must roll on the smaller, the Cardioid and Limacon then resulting, as before, from the motion of points respec- tively on the circumference of the generator, or within or withoid it. 183. In Fig. 106 the Cardioid is obtained as an epicycloid, being traced by point P during one revolution of the generator PHin about an equal directing circle msO. As a Limacon we may get points of the Cardioid, as y and s, by drawing a secant through and laying off s y and s z each equal to 2 r. 184. The Limacon as a Trisectrix. Three famous problems of the ancients were the squaring of the circle, the duplication of the cube and the trisection of an angle. Among the interesting curves invented by early mathematicians for the purpose of solving one or the other of these problems, were the Quadratrix and Conchoid, whose construction is given later in this chapter; hut it has been found that certain trochoids may as readily be employed for trisection, among them the Lima- con of Fig. 106, frequently called the Epitrochoidal Trisectrix. When constructed as a Limacon we find points as G and A", on any secant R X of the circle called "path of centres," by making SX and S G each equal to the radius of that circle. •Term due to Keuleaux, and based upon the fact that Cardano (Kith century) was probably the first to investigate the paths described by points during their rolling. tt'rotn Cardis, the Latin for heart. SPECIAL TROCHOIDS. 63 185. To trisect an angle, as M R F, by means of this epitrochoid, bisect one side of the angle, as FR, at m; use in R and mF as radii for generator and director respectively of an epitrochoid hav- ing a tracing radius, RF, equal to tivice that of the generator. Make RN=RF and draw NF; this will cut the Limacori* F T X RQ (traced by point F as carried by the given generator) in a jjoint 1\ . The angle T l R F will then be one -third of NRF, which may be proved as follows: F reaches T x by the rolling of arc in a on arc mn l . These arcs are subtended by equal angles, , the circles being equal. During this rolling R reaches R lt bringing RF to R 1 T 1 . In the triangles T l R 1 F and RFR l the side FR 1 is common, angles 4> equal, and side R 1 T 1 equal to side RF; the line R T l is there- fore parallel to R t F, whence angle T X RF must also equal <£. In the triangle RFR X we denote by 6 -i on o i the angles opposite the equal sides RF and R 1 F; then 2(9+<£ = 180°, or 0= R . In triangle 2 E'ig'- 106- NRF we have the angle at F equal to 6 — <£, and 2 (6 — ) + x+ = 180°, which gives x=2, by substituting the value of 6 from the previous equation. 18G. The Involute. As the opposite extreme of a circle rolling on a straight line we may have the latter rolling on a circle. In this case the rolling circle has an infinite radius. A point on the straight line describes a curve called the involute. This would be the path of the end of a thread if the latter were in tension while being unwound from a spool. In Fig. 107 a rule is shown, tangent at u to a circle on which it is supposed to roll. Were a pencil-point inserted in the centre of the circle at j (which is on the line ux produced) it would trace the involute. When j reaches a, the rule will have had rolling contact with the base circle over an arc uts---a whose length equals line uxj. Were a the initial point, we would obtain 6, c, 64 THEORETICAL AND PRACTICAL GRAPHICS. etc., by making tangent mb=arc ma; tangent nc—arc na. Each tangent thus equals the arc from the initial point to the point of tangency. 187. The circle from which the involute is derived or evolved is called the evolute. Were a hexagon or other figure to be taken as an evolute, a corresponding involute could be derived; but the name "involute," unqualified, is understood to be that obtained from a circle. From the law of formation of the involute, the rolling line is in all its positions a normal to the curve; the point of tangency on the evolute is an instantaneous centre, and a tangent at any point, as /, is a perpendicular to the tangent, fq, from / to the base circle. Like the cycloid, the involute is a correct working outline for the teeth of gear-wheels; and gears manufactured on the involute system are to a considerable degree supplanting other forms. A surface known as the developable helicoid (see Figs. 209 and 270) is formed by moving a line E'ig-- 107. so as to be always tangent to a given helix. It is interesting in this connection to notice that any plane perpendicular to the axis of the helix would cut such a surface in a pair of involutes.* 188. The Spired of Archimedes. This curve is generated by a point having a uniform motion around a fixed point — the pole — combined with uniform motion toward or from it. In Fig. 107, with as the pole, if the angles are equal, and D, OE and Oy 3 are in arith- metical progression, then the points D, E and y , are points of an Archimedean Spiral. This spiral can be trochoidally generated, simultaneously with the involute, by inserting a pencil point at y in a piece carried by — and at right angles with — the rule^ the point y being at a distance, •The day of writing the above article the following item appeared in the New Y6rk Evening Post: "Visitors to the Royal Observatory, Greenwich, will hereafter miss the great cylindrical structure which has for a quarter century and more covered the largest telescope possessed by the Observatory. Notwithstanding its size the Astronomer lloyal has now procured through the Lords Commissioners a telescope more than twice as large as the old one. . . . The optical peculiarities embodied in the new instrument will render it one of the three most powerful telescopes at present in existence.... The peculiar architectural feature of the building which is to shelter the new telescope is that its dome, of thirty -six feet diameter, will surmount a tower having a diameter of only thirty-one feet. Technically, the form adopted is the surface generated by the revolution of an involute of a circle." SPECIAL TROCHOIDS. 65 Fig-. lOB. xy, from the contact -edge of the rule, equal to the radius Os of the base circle of the involute; for after the rolling of ux over an arc ut we shall have tx 1 as the portion of the rolling line between x and the point of tangency, and xy will have reached x 1 y l . If the rolling be continued y will evidently reach 0. We see that Oy = ux, and Oy 1 =*tx 1 ; but the lengths ux and tx x are propor- tional to the angular movement of the rolling line about 0, and as the spiral may be defined as that curve in which the length of a radius vector is directly proportional to the angle through which it has turned about the pole, the various positions of y are evidently points of such a curve. 189. A Tangent to the Spiral of Archimedes. Were the pole, 0, given, and a portion only of the spiral, we could draw a tangent at any point, y , , by determining the circle on which the spiral could be trochoidally generated, then the instantaneous centre for the given position of the tracing- point, whence the normal and tangent would be derived in the usual way. The radius 1 of the base circle would equal wy — the difference between two radii vectores Oy and Oz which include air angle of 57° 29+, (the angle which at the centre of a circle subtends an arc equal to the radius). The instantaneous centre, t, would be the extremity of that radius which was perpendicular to Oy l . The normal would be ty 1: and the tangent TT t perpendicular to it. 190. The spiral of Archimedes is the right section of an oblique helicoid. (Art. 357). It is also the proper outline for a cam to convert uni- form rotary into uniform rectilinear motion, and when combined with an equal and oppo- site spiral gives the well-known form called the heart- cam. As usually constructed the act- ing curve is not the true spiral, but a curve whose points are at a constant distance from the theoretical outline equal to the radius of the friction -roller which is on the end of the piece to be raised. Qs 2 (Fig. 107) is a small portion of such a "parallel curve." 191. If a point travel on the surface of a cone so as to combine a uniform motion around the axis with a uniform motion toward the vertex it will trace a conical helix, whose orthographic projection on the plane of the base will be a spiral of Archimedes. In Fig. 108 a top and front view are given of a cone and helix. The shaded por- tion is the development of the cone, that is, the area equal to the convex surface, and which — if rolled up — would form the cone. To obtain the development draw an arc A'G"A" of radius equal to an element. The convex surface of the cone will then be repre- sented by the sector A'O'A", whose angle may be found by the proportion 0^4:0'^':: circumference of the cone's base. 0:36O C since the arc A'G"A" must equal the entire The student can make a paper model of the cone and helix by cutting out a sector of a circle <;<; THEORETICAL AND PRACTICAL GRAPHICS. making allowance for an overlap on which to put the mucilage, as shown by the dotted lines O'y and y v z in the tigure. The development of a conical helix is the same kind of spiral as its orthographic projection. PARALLEL CURVES. 192. A parallel carve is one whose points are at a constant normal distance from some other curve. Parallel curves have not the same mathematical properties as those from which they are derived, except in the case of a circle; this can readily he seen from the cam figure under the last heading, in which a point, as s n of the true spiral, is located on a line from which is by no means in the direction of the normal to the curve at s lt upon which lies the point s 2 of the parallel curve. Fig 1 - 109. Instead of actually determining the normals to a curve and on each laying off a constant distance, we may draw many arcs of constant radius, having their centres on the original curve; the desired parallel will be tangent to all these arcs. In strictly mathematical language a parallel curve is the envelope of a circle of constant radius whose centre is on the original curve. We may also define it as the locus of consecutive inter- sections of a system of equal circles having their centres on the original curve. If on the convex side of the original the parallel will resemble it in form, but if within, the two may be totally dissimilar. This is well illustrated in Fig. 109, in which the parallel to a Lemniscate is shown. The student will obtain some interesting results by constructing the parallels to ellipses, parabolas and other plane curves. THE CONCHOID OF NICOMEDES. 193. The Conchoid, named after the Greek word for shell* may be obtained by laying off a con- stant length on each side of a given line 71/ JV (the directrix), upon radials through a fixed point or pole, (Fig. 110). If mv = mn=sx then v, n and x are points of the curve. Denote by a the distance of from M N, and use c for the constant length to be laid off; then if c the curve has an undulation or wave -form towards the pole. *A series of curves much more closely resembling those of a shell can be obtained by tracing the paths of points on the piston-rod of an oscillating cylinder. See Arts. 157 and 158 tor the principles of their construction. THE CONCHOID.— THE QUADRATRIX. 67 Ov = c + Om; On=c — Om; we may therefore express the relation to of points on the curve by the equation p = c± m — c±asec 4>. -ig-. HO. 194. Mention has already been made (Art. 184) of the fact that this was one of the curves invented in part for the purpose of solving the problem of the trisection of an angle. Were m x (or ) the angle to be trisected we would first draw pqr, the superior branch of a conchoid having the constant, c, equal to twice Om. A parallel from m to the axis will intersect the curve at q; the angle pOq will then be one-third of <£: for since b q=20m we have mq=20mcos ft; also mq:0m::sind:sin ft; hence 2 m cos ft :0m: : sin : sin ft, whence sin 8 = 2 sin ft cos ft = sin 2 ft (from known trigonometric relations). The angle 8 is therefore equal to twice ft, which makes the latter one -third of angle . 195. To draw a tangent and normal at any point v, we find the instantaneous centre o on the principle that it is at the intersection of normals to the paths of two moving points of a line, the distance between said, points remaining constant. In tracing the curve, the motion of (on v) is — at the instant considered — in the direction Ov; Oo is therefore the normal. The point m of Ov is at the same moment moving along M N, for which mo is the normal. Their intersection o is then the instantaneous centre, and o v the normal to the conchoid, with v z perpendicular to o v for the desired tangent. 196. This interesting curve may be obtained as a plane section of one of the higher mathemat- ical surfaces. If two non - intersecting lines — one vertical, the other horizontal — be taken as guiding lines or directrices of the motion of a third straight line whose inclination to a horizontal plane is to be constant, then horizontal planes will cut conchoids from the surface thus generated, while every plane parallel to the directrices will cut hyperbolas. From the nature of its plane sections this surface is called the Conchoidal Hyperboloid. (See Fig. 219). THE QUADRATRIX OF DINOSTRATUS. 197. In Fig. Ill let the radius T rotate uniformly about the centre; simultaneously with its movement let M N have a uniform motion parallel to itself, reaching A B at the same time with radius T; the locus of the intersection of M N with the radius will be the Quadratrix. Points 68 THEORETICAL AND PRACTICAL GRAPHICS. Pig. Ill- exterior to the circle may be found by prolonging the radii while moving MN away from A B. As the intersection of M N with B is at infinity, the former becomes an asymptote to the curve as often as it moves from the centre an additional amount equal to the diameter of the circle; the number of branches of the Quadratrix may therefore be infinite. It may be proved analytically that the curve crosses A at a distance from equal to 2 r -r- t. 198. To trisect an angle, as TO a, by means of the Quadratrix, draw the ordinate ap, trisect p T by s and % and draw sc and xm; radii Oc and Om will then divide the angle as desired: for by the conditions of generation of the curve the line MN takes three equi- distant parallel positions while the radius describes three equal angles. THE CISSOID OF DIOCLES. 199. This curve was devised for the purpose of obtaining two mean proportionals between two given quantities, by means of which the dujjlication of the cube might be effected. The name was suggested by the Greek word for ivy, since "the curve appears to mount along its asymptote in the same manner as that parasite plant climbs on the tall trunk of the pine."* This was one of the first curves invented after the discovery of the conic sections. Let C (Fig. 112) be the centre of a circle, ACE a right angle, NS and M T any pair of ordinates parallel to P-ig-. 113. and equidistant from, CE; then a secant from A through the extremity of either ordinate will meet the other ordinate in a point of the cissoid. AT and NS give P; AS and MT give O The tangent to the circle at 7? will be an asymptote to the curve. It is a somewhat interesting coincidence that the area between the cissoid and its asymptote is the same as that between a cycloid and its base, viz., three times that of the circle from which it is derived. 200. Sir Isaac Newton devised the following method of obtaining a cissoid by continuous motion: Make A V=A C; then move a right-angled triangle, of base = FC, so that the vertex F travels along * Leslie. Geometrical Analysis. 1821. THE CISSOID.— THE TRAGTRIX. 69 the line DE while the edge J K always passes through V; then the middle point, L, of the base FJ, will trace a cissoid. This construction enables us readily to get the instantaneous centre and a tangent and normal; for Fn is normal to FC — the path of F, while nV is normal to the motion of J toward J" V; the instantaneous centre n is therefore at the intersection of these normals. For any other point as P we apply the same principle thus: With radius AC and centre P obtain x; draw Px, then Vz parallel to it; a vertical from x will meet Vz at the instantaneous centre y, whence the normal and tangent result in the usual way. The point y does not necessarily fall on nV. Since nV and FJ are perpendicular to JV they are parallel. So also must Vz be parallel to P x, regardless of where P is taken. 201. Two quantities m and n will be mean proportionals between two other quantities a and b if m'' = na and n 2 =mb; that is, if m 3 =a 2 6 and if n 3 = ab 2 If b = 2a we will find, from the relation m :i = a 2 6, that m will be the edge of a cube whose volume equals 2 a '. To get two mean proportionals between quantities, r and 6, make the smaller, r, the radius of a circle from which derive a cissoid. Were APR the derived curve we would then make Gt equal to the second quantity, b, and draw B t, cutting the cissoid at Q. A line A Q would cut off on Ct a distance Cv equal to m, one of the desired proportionals; for m 3 will then equal r 2 b, as may be thus shown by means of similar triangles: Cv: MQ::CA:MA whence Cv' = '"''^T (1) Ct:MQ::CB:BM " C<«=^£^ (2) MQ:MA::S-N:AN:: / A N. B N: A iY, whence MQ= MA ^j ^ BN (3) From (2) we have MQ = BM( °J = h) (4) .. (3, ■ - „ .5, Replacing M Q 3 in equation (1) by the product of the second members of equations (4) and (5) gives CV (i.e., m 3 ) = r 1 b. By interchanging r and b we obtain n, the other mean proportional; or it might be obtained by constructing similar triangles having r, b and in for sides. THE TRAGTRIX. 202. The Tractrix is the involute of the curve called the Catenary (Art. 214) yet its usual con- struction is based on the fact that if a series of tangents be drawn to the curve, the portions of such tangents between the points of tangency and a given line will be of the same length; or, in other words, the intercept on the tangent, between the directrix and the curve, will be constant. A practical and very close approximation to the theoretical curve is obtained by taking a radius Q R (Fig. 113) and with a centre a, a short distance from R on Q R, obtaining b, which is then joined with a. On a b a centre c is similarly taken for another arc of the same radius, whence c d is obtained. A sufficient repetition of this process will indicate the curve by its enveloping tangents, or a curve may actually be drawn tangent to all these lines. Could we take a, b, c, etc., as mathematically consecutive points the curve would be theoretically exact. The line Q S is an asymp- tote to the curve. 70 THEORETICAL AND PRACTICAL GRAPHICS. P%. 11-4- The area between the completed branch RPS and the lines QR and Q S would be equal to a quadrant of the circle on radius Q R. M_ 9 =Si-D 203. The surface generated by revolving the trac- trix about its asymptote has been employed for the foot of a vertical spindle or shaft, and is known as Schiele's Anti- Friction Pivot. The step for such a pivot is shown in sectional view in the left half of the figure. Theoretically, the amount of work done in overcoming friction is the same on all equal areas of this surface. In the case of a bearing of the usual kind, for a cylindrical spindle, although the pressure on each square inch of surface would be constant, yet, as unit areas at different distances from the centre would pass over very different amounts of space in one revolution, the wear upon them would be necessarily unequal. The rationale of the tractrix form will become evident from the following Fig. us. consideration : If about to split a log, and having a choice of wedges, any boy would choose a thin one rather than one with a large angle, although he might not be able to prove by graphical statics the exact amount of advantage the one would have over the other. The theory is very simple, how- ever, and the student ma}' profitabl}' be introduced to it. Suppose a ball, c, (Fig. 114) struck at the same instant by two others, a and b, moving at rates of six and eight feet a second respectively. On a, c and b c prolonged take ce and ch equal, respectively, to six and eight units of some scale; complete the parallogram having these lines as sides; then it is a well-known principle in mechanics* that cd — the diagonal of this parallel- ogram — will not only represent the direction in which the ball c will move, but also the distance — in feet, to the scale chosen — it will travel in one second. Evidently, then, to balance the effect of balls a and b upon c, a fourth would be necessary, moving from d toward c and traversing d c in the same second that a and b travel, so that impact of all would occur simultaneously. These forces would be represented in direction and magnitude (to some scale) by the shaded triangle c'd'e', which illustrates the very important theorem that if the three sides of a triangle — taken like c'e', e'd', d'c', in such order as to bring one back to the initial vertex mentioned — represent in magnitude and direction three forces acting on one point, then these forces are balanced. Constructing now a triangle of forces for a broad and thin wedge, (Fig. 115) and denoting the force of the supposed equal blows by F in each triangle, we see that the pressures are greater for the thin wedge than for the other; that is, the less the inclination to the vertical the greater the pressure. A pivot so shaped that as the pressure between it and its step increased the area to be traversed diminished would therefore, theoretically, be the ideal; and the rate of change of curvature of the tractrix, as its generating point approaches the axis, makes it, obviously, the correct form. •For a demonstration the student may refer to Kankine s Applied Mechanics, Art. 51. THE TRACTRIX.— WITCH OF AGNESI.-C ARTESIAN OVALS. 11 204. Navigator's charts are usually made by Mercator's projection (so-called, not being a projection in the ordinary sense, but with the extended signification alluded to in the remark in Art. 2). Maps thus constructed have this advantageous feature, that rhumb lives or loxodromics — the curves on a sphere that cut all meridians at the same angle — are represented as straight lines, which can only be the case if the meridians are indicated by parallel lines. The law of convergence of meridians on a sphere is, that the length of a degree of longitude at any latitude equals that of a degree on the equator multiplied by the cosine (see foot-note, p. 31) of the latitude; when the meridians are made nan- convergent it is, therefore, manifestly necessary that the distance apart of originally equi- distant parallels of latitude must increase at the same rate; or, otherwise stated, as on Mercator's chart degrees of longitude are all made equal, regardless of the latitude, the constant length repre- sentative of such degree bears a varying ratio to the actual arc on the sphere, being greater with the increase in latitude; but the greater the latitude the less its cosine or the greater its secant; hence lengths representative of degrees of latitude will increase with the secant of the latitude. Tables have been constructed giving the increments of the secant for each minute of latitude; but it is an interesting fact that they may be derived from the Tractrix thus: Draw a circle with radius Q R, centre Q (Fig. 113); estimate latitude on such circle from R upward; the intercept on QS between consecutive tangents to the Tractrix will be the increment for the arc of latitude included between parallels to QS, drawn through the points of contact of said pair of tangents." On map construction the student is referred to Chapter XII, or to Craig's Treatise on Projections. THE WITCH OF AGNESI. 205. If on any line S Q, perpendicular to the diameter of a circle, a point S be so located that S Q:AB::PQ: QB then S will be a point of the curve called the Witch of Agnesi. Such point is evidently on the ordinate PQ prolonged, and vertically below the intersection T of the tangent at A by the secant through P. E-ig-- US. y\ ^!^^Je\ z \^-^ --^fKWVt-iN asymptote ^^^^B The point E, at the same level as the centre 0, is a diameter's distance from the latter. The tangent at B is an asymptote to the curve. The area between the curve and its asymptote is four times that of the circle involved in its construction. The Witch, also called the Versiera, was devised by Donna Maria Gaetana Agnesi, a brilliant Italian lady who was appointed in 1750, by Pope Benedict XIV, to the professorship of mathematics and philosophy in the University of Bologna. THE CARTESIAN OVAL. 206. This curve, also called simply a Cartesian, after its investigator, Descartes, has its points connected with two foci, F' and F", by the relation m,p'±np"=hc, in which c is the distance between the foci, while m, n and k are constant factors. ♦ Leslie. Geometrical Analysis. Edinburgh, 1821. 72 THEORETICAL AND PRACTICAL GRAPHICS. Salmon states that we owe to Chasles the proof that a third focus may be found, sustaining the Figr- 117*. same relation, and expressed by an equation of similar form. (See Art. 209). The Cartesian is symmetrical with respect to the axis — the line joining the foci. 207. To construct the curve from the first equation we may for convenience write m p' ± n p'' = k c in the form p' ± — p" = — ; or ' m rn by denoting; - : by b and - by d, it takes the yet more simple form p'±bp"=d. Then p" will have two values, according as the positive or negative sign is taken, being respectively — . — and — . — ; the former is for points on the inner of the two ovals that constitute a complete Cartesian, while the latter gives points on the outer curve. ,1—p' Fig. us. To obtain p' b take F' and F" (Fig. 118) as foci; F'S = d; SK at some random acute angle 8 with the axis, and make SH=-; that is, make F' S : S H: : b : 1. Then from F' draw an arc tfP, of radius less than d ) and cut it at P by an arc from centre F", radius S T, Tt being a parallel to F' II; then P is a point of the inner oval; for St=d — p' , and ST=p"; there- ~~ f fore p":d — p' : :-: d, whence p"=——. — - 208. If an arc x y K be drawn from F' , with X \ radius, F' x, greater than d, we may find the second value of p", viz., ■ , by drawing x Q parallel to F' H to meet H S prolonged; for QS will equal p , in which p' — F'z. Again using F" as a centre, and a radius QS=p", gives points R and M of the larger oval. The following are the values for the focal radii to the four points where the ovals cut the axes. (See Fig. 117). For A, p' — d , , d + b c ■ — j- — = c + p whence p = I A = — — o l—o » „ ,t d — p' , a, p = — — = c + p P ,i p' — d , ■£>, P = — r — = c — p (I — p p' = F' a-. d~bc 1 + 6 1 + 6 0, p - ■ - j . - - C — p p = I b \ r hav< and i>' above; and for the inner oval between those of a and 6 d — b i b '' ' ' " ~ 1 _ b The construction - arcs for the outer oval must evidently have radii between the values of p' for A CAR TESIAN VALS.-CA USTIGS. 73 rig-. 1A.&. The numerical values from which Fig. 118 was constructed were m = Z; n=2; r = l; h = 3. 209. TAe Third Focus. Fig. 118 illustrates a special case, but, in general, the method of finding a third focus F'" (not shown) would be to draw a random secant F'r through F', and note the points P and G in which it cuts the ovals — these to he taken on the same side of F', as two other points of intersection are possible; a circle through P, G and F" would cut the axis in the new focus sought. Then denoting by G the distance F' F'", wc would find the factors of the original equation appearing in a new order; thus, k p' ± n p" = mC, which — for purposes of construction — may lie written p'±6'p'"=d'. If obtained from the foci F" and F'" the relation would he m p'" — kp" = ± n 0', in which C" equals F" F'". Writing this in the form p"' — B p" = ± D we have the following interesting cases: (a) an ellipse for D positive and B = — 1; (h) an hyperbola, for D positive and B = + 1; (c) a limagon for D=C'B; (d) a cardioid for B=+l and D = C" 210. The following method of drawing a Cartesian by continuous motion was devised by Prof. Hammond: A string is wound, as shown, around two pulleys turning on a common axis; a pencil at P holds the string taut around smooth pegs placed at random at F l and F.,; if the wheels he turned with the same angular velocity, and the pencil does not slip on the string, it will trace a Cartesian having F 1 and F 2 as foci.* If the pulleys are equal the Cartesian will become an ellipse; if both threads are wound the same way around either one of the wheels the resulting curve will be an hyperbola. 211. It is a well-known fact in Optics that the incident and reflected ray make equal angles with the normal to a reflecting surface. If the latter is curved then each reflected ray cuts the one s-ig-. i2o. next to it, their consecutive intersections giving a curve called a caustic by reflection. Probably all have occasionally noticed such a curve on the surface of the milk in a glass, when the light was 2">roperly placed. If the reflecting curve is a circle the caustic is the evolute of a limagon. In passing from one medium into another, as from air into water, the deflection which a ray of light undergoes is called refraction, and for the same media the ratio of the sines of the angles of incidence and refraction (8 and <£, Fig. 120) is constant. The consecutive intersections of refracted. ra) r s give also a caustic, which, for a circle, is the evolute of a Cartesian Oval. The proof of this statement t involves the property upon which is based the most convenient method of drawing a tangent to the Cartesian, viz., that the normal at any point divides the angle between the focal radii into parts whose sines are proportional to the factors of those radii in the equation. If, then, we have obtained a point G on the outer oval from the relation mp'±np" = kc, we may obtain the tangent at G by laying off on p' and p" distances proportional to m and n, as Gr and Gh, Fig. 118, then bisecting r h at j and drawing the normal Gj, to which the desired tangent is a perpendicular. At a point on the inner oval the distance would not be laid off on a focal radius produced, as in the case illustrated. * American Journal of Mathematics, 1878. f Salmon. Higher Plane Curves. Art. 117. 74 THEORETICAL AND PRACTICAL GRAPHICS. CASSIAN OVALS. 212. In the Cassian Ovals or Orals of Cassini the points are connected with two foci by the relation p'p" = k' ! , i.e., the product of the focal radii is equal to some perfect square. These curves have already been alluded to in Art. 114 as plane sections of the annular torus, taken parallel to its axis. Fig-. 121. ng-. 122. In Art. 158 one form — the Lemniscate — receives special treatment. For it the constant k 2 must equal w'\ the square of half the distance between the foci. "When k is less than m, the curve becomes two separate ovals. 213. The general construction depends on the fact that in any semicircle the square of an ordinate equals the product of the segments into which it divides the diameter. In Fig. 122 take F 1 and F. 2 as the foci, erect a perpendicular F 1 S to the axis F 1 F 2 , and on it lay off F l R equal to the constant, k. Bisect F 1 F 2 at and draw a semicircle of radius R. This cuts the axis at A and B, the extreme points of the curve; for k 2 = F l AxF l B. Any other point T may be obtained by drawing from F, a circular arc of radius F t t greater than F i A; draw t R, then R x perpen- dicular to it; xF s will then lie the p", and F 1 t the p', for four points of the curve, which will be at the intersection of arcs struck from I\ and I\ as centres and with those radii. To get a normal at any point T draw T, then make angle F 2 T& the desired line. THE CATENARY. F.TO; Ts will be 214. If a flexible chain, cable or string, of uniform weight per unit of length, be freely sus- pended by its extremities, the curve which it takes under the action of gravity is called a Catenary, from catena, a chain. A simple and practical method of obtaining a catenary on the drawing-board, would be to insert two pins in the board, in the desired relative position of the points of suspension, and then attach to them a string of the desired length. By holding the board vertically, the string would assume the catenary, whose points could then be located with the pencil and joined in the usual manner with the irregular curve. Otherwise, if its points arc to be located by means of an equation we take axes in the plane of the curve, the ?y-axis (Fig. 123) being a vertical line through the lowest point T of the catenary, while the z-axis is a horizontal line at a distance m below T. The quan- tity m is called the 'parameter of the curve, and is equal to the length of string which represents the tension at the lowest point. THE CATENARY.— THE LOGARITHMIC SPIRAL. 75 The equation of the catenary ' is then y = — \e m + e logarithms" and has the numerical value 2.7182S18+. By taking successive values of x equal to m, 2 m, 3 m, etc., we get the following values for y: — ( e + ) which for m = unity becomes 1.54308 m) in which e is the 6a.se of Napierian IFig-- 1S3. .C m...y c = 2 m x = 3 m . . . y = !(■■+}) " ■ " 3.76217 " 10.0676 " 27.308 s = 4m...j,=|(«*+£) " « " " To construct the curve we therefore draw an arc of radius B = m, giving T on the axis of j/ as the lowest point of the curve. For OB we have y =B P = 1.54308; for 0. a n 1.03142. -r we have y The tension at any point P is equal to the weight of a piece of rope of length B P = P C + m. At the lowest point the tangent is horizontal. The length of any arc TP is proportional to the angle between T C and the tangent P V at the upper extremity of the arc. 215. If a circle RL B be drawn, of radius equal to m, it may be shown analytically that tangents P S and Q R, to catenary and circle respectively, from points at the same level, will be parallel: also that PS equals the catenary -arc Pr T; S therefore traces the involute of the catenary, and as SB always equals R and remains perpendicular to PS (angle ORQ being always 90°) we have the curve TSK fulfilling the conditions of a tractrix. (See Art. 202.) If a parabola, having a focal distance m, roll on a straight line, the focus will trace a catenary having m for its parameter. The catenary was mistaken by Galileo for a parabola. In 1669 Jungius proved it to be neither a parabola nor hyperbola, but it was not till 1691 that its exact mathematical nature was known, being then established by James Bernouilli. THE LOGARITHMIC OR EQUIANGULAR SPIRAL. 216. In Fig. 124 we have the curve called the Logarithmic Spiral. Its usual construction is based on the property that any radius vector, as p, which bisects the angle between two other radii, O M and ON, is a mean proportional between them; i.e., p' z = OS 2 = OMx ON. If M and G are points of the spiral we may find an intermediate point K by drawing the ordinate K to a semicircle of diameter OM+OG; a perpendicular through G to GK will then give D, another point of the curve, and this construction may be repeated indefinitely. Radii making equal angles with each other are evidently in geometrical progression. This spiral is often called Equiangular from the fact that the angle is always the same between 1 Rankine. Applied Mechanics. Art. 175. 2In the expression 102=100 the quantity "2" is eallefl the logarithm of 100, it being the exponent of the power to which 10 must be raised to give 100. Similarly 2 would be the logarithm of 64, were 8 the base or number to be raised to the power indicated. 76 THEORETICAL AND PRACTICAL GRAPHIC S. a radius vector and the tangent at its extremity. Upon this property is based its use as the out- line for spiral cams and for lobed wheels. The curve never reaches the pole. The name logarithmic spiral is based on the property that the angle of revolution is proportional to the logarithm of the radius vector. This is expressed by p = a 9 , in which 6 is the varying angle, and a is some arbitrary constant. To construct a tangent hy calculation, divide the hyperbolic logarithm ' of the ratio M : K (which are any two radii whose values are known) by the angle between these radii, expressed in circular measure; 2 the quotient will be the tangent of the constant angle of obliquity of the spiral. 217. Among the more interesting properties of this curve are the following: Its involute is an equal logarithmic spiral. Were a light jjlaced at the pole, the caustic —whether by reflection or refraction — would be a logarithmic spiral. The discovery of these properties of recurrence led James Bernouilli to direct that this spiral be engraved on Iris tomb, with the inscription — Eadein Mutata Resurgo, which, freely trans- lated, is — i" shall arise the same, though changed. Kepler discovered that the orbits of the planets and comets were conic sections having a focus at the centre of the sun. Newton proved that they would have described logarithmic spirals as they travelled out into space, had the attraction of gravitation been inversely as the cube instead of the square of the distance. THE HYPERBOLIC OR RECIPROCAL SPIRAL. 218. In this spiral the length of a radius vector is in inverse ratio to the angle through which it turns. Like the logarithmic spiral, it has an infinite number of convolutions about the pole, which it never reaches. The invention of this curve is attributed to James Bernouilli, who showed that Newton's conclusions as to the logarithmic spiral (see Art. 217) would also hold for the hyperbolic spiral, the initial velocity of projection determining which trajectory was described. To obtain points of the curve divide a circle m 5 8 (Fig. 125) into any number of equal parts, and on some initial radius On lay off some unit, as an inch; on the second radius 2 take On 2 IFig-- 125. ,-; on the third -tt : , etc. For one -half the angle 6 the radius vector would evidently be 2 On, giving a point s outside the circle. The equation to the curve is 1 in which r is the radius vector, a some numerical con- stant, and 6 is the angular rotation of r (in circular measure) estimated from some initial line. 'To get the hyperbolic logarithm of a number multiply its common logarithm by 2.3026. 'In circular measure 360° = 2w7-, which, for r = 1, becomes 6.28318; 180 ° = 3.14159; 90° = 1.5708- 60°=1.0472- 45°=07S>4- an ° — 0.5236; 1 ° = 0.0174533. THE HYPERBOLIC SPIRAL.— THE LITUUS. 77 The curve has an asymptote parallel to the initial line, and at a distance from it equal to — units. E"ig-- 12®. To construct the spiral from its equation, take as the pole (Fig. 26); Q as the initial line; a, for convenience, some fraction, as — ; and as our unit some quantity, say half an inch, that will make — of convenient size. Then, taking Q as the initial line, make P= — = 2", and draw PR a o 'a parallel to Q for the asymptote. For = 1, that is, for arc KH=radius OH, we have and r = — = 2", giving H for one point of the spiral. Writing the equation in the form r a e ' expressing various values of in circular measure we get the following : e = 30° = 0,5236; r=OM=3'.'8 + : 6 = 45° = 0.7854; r= ON=2'.'55; 0=9O° = 1.57O8; r = OS = l'.'2+: 6 = 180° = 3.14159 ; r = T= .6366, etc. The tangent to the curve at any point makes with the radius vector an angle <£, which is found by analysis to sustain to the angle 6 the following trigonometrical relation, tan = 6; the circular measure of may therefore be found in a table of natural tangents, and the corresponding value of 4> obtained. , THE LITUUS. — THE IONIC VOLUTE. 219. The Litims is a spiral in which the radius vector is inversely proportional to the square root of the angle through which it has revolved. This relation is shown by the equation r = _ also written a 2 0-- 1 When 8 = we find r = oo , which makes the initial line an asymptote to the curve. In Fig. 127 take Q as the initial line, as the pole, ft = 2, and as our unit 3"; then — = U". a For 6 = 90° = it (in circular measure 1.5708) we have r = 31=1" 2 +. For 6 = 1 we have the radius T making an angle of 57°. 29 + with the initial line, and in length equal to - units, 78 THEORETICAL AND PRACTICAL GRAPHICS. 1. e.. U". For = 45°=- (or 0.7854) r will 4 0R= 1". 7 +. Then OH = OR for in rotating to H the radius vector passes over four 45° angles, and the radius must therefore be one- half what it was for the first 45 ° described. Similarly, OK= ^=; 0i»/ = -7T-, etc. ; this rela- tion enabling the student to locate any number of points. To draw a tangent to the curve we employ the relation tan = 2 6, being the angle made by the tangent line with the radius vector, while 6 is the angular rotation of the latter, in circular measure. Architectural Scrolls. — The Ionic Volute. The Lituus and other spirals are occasionally employed as volutes and other architectural ornaments. In the former application it is customary for the spiral to terminate on a circle called the eye, into which it blends tangentially. Usually, in practice, circular-arc approximations to true spiral forms are employed, the simplest of which, for the scroll on the capital of an Ionic column, is ^-gr- is?' (a). probably the following: Taking A P, the total height of the volute, at sixteen of the eighteen "parts" into which the module (the unit of proportion = the semi- diameter of the column) is divided, draw the circular eye with 'radius equal to one such part, the centre dividing A P into segments of seven and nine parts respectively. Next inscribe in the eye a square with one diagonal vertical; parallel to its sides draw (see enlarged square mnop) 2—4 and 3 — 1, and divide each into six equal parts, which number up to twelve, as indicated. Then (returning to main figure) the arc A B has centre 1 and radius 1 — .1. With 2 as a centre draw arc B C; then CD from centre 3, etc. In the complete drawing of an Ionic column the centre of the eye would be at the intersection of a vertical line from the lower extremity of the cyma reversa with a hori- zontal through the lower line of the echinus. To complete the scroll a second spiral would be required, constructed according to the same law and beginning at Q, where A Q is equal to one -half part of the module. THE NOMENCLATURE AND DOUBLE GENERATION OF TROCHOIDS. THE NOMENCLATURE AND DOUBLE GENERATION OF TROCHOIDS. [The anomalies and inadequateness of the pre-existing nomenclature of trochoidal curves led to an attempt on the part of the writer to simplify the matter, and the following paper is, in substance, that presented upon the subject before the American Association for the Advancement of Science, in 1887. Two brief quotations from some of the communications to which it led w r ill indicate the result. From Prof. Francis Reuleaux, Director of the Eoyal Polytechnic Institution, Berlin: "I agree with pleasure to your discrimination of major, minor and medial hypotrochoids and will in future apply these novel designations." From Prof. Richard A. Proctor, B.A., author of Geometry of Cycloids, etc.: "Your system seems complete and satisfactory. I was conscious that my own suggestions were bat partially corrective of the manifest anomalies in former nomenclatures " The final outcome of the investigation, as far as technical terms are concerned, appears on page 59, in a tabular arrange- ment suggested by that of Kennedy, and which is both a modification and an extension of his ingenious scheme. The property of double generation of trochoids, when the tracing -point is not on the circumference of the rolling circle, is even at present writing not treated by some authors of advanced text-books who nevertheless emphasize it for the epi-, hypo- and peri -cycloid. This fact, and the importance of the property both in itself and as leading to the solution of a vexed question, are my main reasons for introducing the paper here in nearly its original length; although to the student of mathematical tastes the original demonstration presented may prove to be not the least interesting feature of the investigation. The demonstrations alone might have appeared in Chapter V — their rightful setting had this been merely a treatise on plane curves, but they w r onld there have unduly lengthened an already large division of the work, while at that point their especial significance could not, for the same reason, have been sufficiently shown.] That would be an ideal nomenclature in which, from the etymology of the terms chosen, so clear an idea could be obtained of that which is named as to largely anticipate definition, if not, indeed, actually to render it superfluous. This ideal, it need hardly be said, is seldom realized. As a rule we meet with but few self-explanatory terms, and the greater their lack of suggestiveness the greater the need of clear definition. Instances are not wanting of ill -chosen terms and even actual misnomers having become so generally adopted, in spite of an occasional protest, that we can scarcely expect to see them replaced by others more appropriate. Whether this be the case or not, we have a right to expect, especially in the exact sciences, and preeminently in Mathematics, such clearness and comprehensiveness of definition as to make ambiguity impossible. But in this we are frequently disappointed, and notably so in the class of curves we are to con- sider. Toward the close of the seventeenth century the mechanician De la Hire gave the name of Roulette — or roll -traced curve — to the path of a point in the plane of a curve rolling upon any other curve as a base. This suggestive term has been generally adopted, and we may expect its complementary, and equally self- interpreting term, Glissette, to keep it company for all time. By far the most interesting and important roulettes are those traced by points in the plane of a circle rolling upon another circle in the same plane, such curves having valuable practical applications in mechanism, while their o-eometrical properties have for centuries furnished an attractive field for investigation to mathematicians. The terms Cycloids and Trochoids have been somewhat indiscriminately used as general names for this class of curves. As far as derivation is concerned they are equally appropriate, the former being from kwcXos, circle, and elSos, form ; and the latter from rpb x os, wheel, and c'ldos. Preference has, however, been given to the term Trochoids by several recent writers on mathematics or mechanism, among them Prof. R. H. Thurston and Prof. De Yolson Wood' also Prof. A. B W. Kennedy of England, the translator of Keuleaux' Theoretische Kinematik, in which these curves figure so largely as een- troids. Adopting it for the sake of aiding in establishing uniformity in nomenclature I give the following definition: If two circles are tangent, either externally or internally, and while one of them remains f.red the other rolls upon it without sliding, the curve described by any point on a radius of the rolling circle, or on a radius produced, will be a Trochoid. Of these curves the most interesting, both historically and for its mathematical properties, is the cycloid, with which all are familiar as the path of a point on the circumference of a circle which rolls upon a straight line, i. «., the circle of infinite radius. The term "cycloid" alone, lor the locus described, is almost universally employed, although it is occasionally qualified by the adjectives right or common. Of almost equally general acceptation, although frequently inappropriate, are the adjectives curtate and prolate, to indicate trochoidal curves traced by points respectively without and within the circumference of the rolling circle (or generator as it will hereafter be termed) whether it roll upon a circle of finite or infinite radius. As distinguished from curtate and prolate forms all the other trochoids are frequently called common. Should the fixed circle (called either the base or director) have an infinite radius, or, in other -words, be a straight line, the curtate curve is called by some the curtate cycloid; by others the curtate trochoid; and similarly for the prolate forms. Since uniformity is desirable I have adopted the terms which seem to have in their favor the greater number of the authorities consulted, viz , curtate and prolate trochoid. It should also be further stated here, with reference to this word "trochoid," that it is usually the termination of the name of every curtate and prolate form of trochoidal curve, the termination cycloid indicating that the tracing point is on the circumference of the generator. With the base a straight line the curtate form consists of a series of loops, while the prolate forms are sinuous, like a wave line ; and the same is frequently true when the base is a circle of finite radius ; hence the suggestion of Prof. Clifford that the terms looped and wavy be employed instead of curtate and prolate. But we shall see, as "we proceed, that they would not be of universal applicability, and that, except with a straight line director, both curtate and prolate curves may be, in form, looped, wavy, or neither. And we would all agree with Prof. Kennedy that as substitutes for these terms "Prof. Cayley's kru-nodal and ac-nodal hardly seem adapted for popular use." It is therefore futile to attempt to secure a nomenclature which shall, throughout, suggest both the form of the locus and the mode of its construction, and we must rest content if we completely attain the latter desideratum. "We have next to consider 'the trochoids traced during the rolling of a circle upon another circle of finite radius. At this point we find inadequacy in nomenclature, and definitions involving singular anomalies. The earlier definitions have been summarized as follows by Prof. P. A. Proctor, in his valuable Geometry of Cycloids: — ( epicycloid ) ... "The 1 v is the curve traced out by a point in the circumference of a circle which rolls without sliding ( hypocycloid j ( external ) on a fixed circle in the same plane, the two circles being in \ \- contact." ( internal j As a specific example of this class of definition I quote the following from a more recent writer: — "If the gen- erating circle rolls on the circumference of a fixed circle, instead of on a fixed line, the curve generated is called an epicycloid if the rolling circle and the fixed circle are tangent externally, a hypocycloid if they are tangent internally." (Byerly, Differential Calculus, 1880.) In accordance with the foregoing definitions every epicycloid is also a hypocycloid, while only some hypocycloids are epicycloids. Salmon [Higher Plane Curves, 1879) makes the following explicit statement on this point: — "The hypo- cycloid, when the radius of the moving circle is greater than that of the fixed circle, may also be generated as an epicycloid." To avoid any anomaly Prof. Proctor has presented the following unambiguous definition : — f epicycloid ) " An \ )- is the curve traced out by a point on the circumference of a circle which rolls without sliding (_ hypocycloid ) a ( outside ") on a fixed circle in the same plane, the rolling circle touching the i I of the fixed circle." ( inside j This certainly does away with all confusion between the epi- and hypo-curves, but we shall find it inadequate to enable us, clearly, to make certain desirable distinctions. By some writers the term external epicycloid is used when the generator and director are tangent externally, and, similarly, internal epicycloid when the contact is internal and the larger circle is rolling. Instead of internal epicycloid we often find external hypocycloid used. It will be sufficient, with regard to it, to quote the following from Prof. Proctor : " It has hitherto been usual to define it (the hypocycloid) as the curve obtained when either the convexity of the rolling circle touches the concavity of the fixed circle, or the concavity of the rolling circle touches the convexity of the fixed circle. There is a manifest want of symmetry in the resulting classification, seeing that while every epicycloid is thus regarded as an external hypocycloid, no hypocycloid can be regarded as an internal epicycloid. Moreover, an external hypocycloid is in reality an anomaly, for the prefix 'hypo,' used in relation to a closed figure like the fixed circle, implies interiorness." To avoid the confusion which it is evident from the foregoing has existed, and at the same time to conform to that principle which is always a safe one and never more important than in nomenclature, viz., not to use two words where one will suffice, I prefer reserving the term " epicycloid " for the case of external tangeney, and substituting the more recently suggested name pericycloid for both "internal epicycloid" and "external hypocycloid." The curtate and prolate forms would then be called peritrochoids. By the use of these names and those to be later presented we can easily make distinctions which, without them, would involve undue verbiage in some cases, and, in others, the use of the ambiguous or inappropriate terms to which exception is taken. And the necessity for such distinctions frequently arises, especially in the study of kinematics and machine design. Take, for example, problems like many in the work of Reuleaux already mentioned, relating to the relative motion of higher kinematic pairs of elements, the centroids being circular arcs and the point- paths trochoids. In such cases we are quite as much concerned with the relative position of the rolling and fixed circles as with the form of a point-path. In solving problems in gearing the same need has been felt of simple terms for the trochoidal profiles of the teeth, which should imply the method of their generation. Although they have not, as yet, come into general use, the names pericycloid and peritrochoid appear in the more recent editions of Weisbach and Eeuleaux, and will undoubtedly eventually meet with universal acceptance. Yet strong objection has been made to the term "pericycloid" by no less an authority than the late eminent mathematician, Prof. W. K. Clifford, who nevertheless adopted the "peritrochoid." I quote the following from his Elements of Dynamic: — "Two circles may touch each other so that each is outside the other, or so that one includes the other. In the former case, if one circle rolls upon the other, the curves traced are called epicycloids and epitrochoids. In the latter case, if the inner circle roll on the outer, the curves are hypocycloids and hypotrochoids, but if the outer circle roll on the inner, the curves are epicycloids and peritrochoids. We do not want the name pericycloids, because, as will be seen, every pericycloid is also an epicycloid; but there are three distinct kinds of trochoidal curves." As it will later be shown that ever}- peri - trochoid can also be generated as an epi-trochoid we can scarcely escape the conclusion that the name peritrochoid would also have been rejected by Prof. Clifford, had he been familiar with this property of double generation as belonging to the curtate and prolate forms as well. But it is this very property, possessed also bv the hypo- trochoids, which necessitates u. more extended nomenclature than that heretofore existing, and I am not aware that there has been any attempt to provide the nine terms essential to its completeness. These it is my principal object to present, and that they have not before been suggested I attribute to the fact that the double generation of curtate and prolate trochoidal curves does not seem to have been generally known, being entirely ignored in many treatises which make quite prominent the fact that it is a property of the epi - and hypo -cycloids, while, as far as I have seen the only writer who mentions it proves it indirectly, by showing the identity of trochoids with epicyclics and establishing it for the latter. As it is upon this peculiar and interesting feature that the nomenclature, as now extended, depends, the demonstra- tions necessary to establish it are next in order. For the epi- and hypo-cycloid probably the simplest method of proof is that based upon the instantaneous centre, and which we may call a kinematic, as distinguished from a strictly geometrical, demonstration. It is as follows:— X'isr- l. E'ig'. 3- Let P (Figs. 1 and 2) be the centre of the fixed circle, and r that of a rolling circle, the tracing point, P, being on the circum- ference of the latter. The point of contact, <7, is — at the moment that the circles are in the relative position indicated — an instanta- neous centre of rotation for every point in the plane of the rolling circle; the line Vq, joining such point of contact with the tracing point, is therefore a normal to the trochoid that the point P is tracing. But if the normal P 7 be produced to intersect the fixed circle in a second point, Q, it is evident that the s^nTinfmitesimal ar of the trochoid would be described with Q serving as instantaneous centre as when g fulfilled that office. The point P will, therefore, evidently trace the same curve, whether it be considered as in the circumference of the circle r, or in that of a second and larger circle, R, tangent to the fixed circle at Q. It is worth while, in this connection, to note what erroneous ideas with regard to these same loci were held by some writers as late as the middle of this century, — ideas whose falsity it would seem as if the most elementary geometrical Fig-- 3. construction would have exposed. Eeuleaux instances the following statement made by Weissenborn in his Cyclischen Kurren (1850) : " If the circle described about ?n Q roll upon that described about M, and if the describing point, B , describe the curve BjP,?, as the inner circle rolls upon the arc B 6, then, evidently, if the smaller circle be fixed and the larger one rolled upon it in a direction opposite to that of the former rotation, the point of the great circle which at the beginning of the operation coincided with B describes the same line BqP^j." The fallacy of this statement is to us, perhaps, in the light of what has preceded, a little more evident than Weissenborn's deduction; although, as Eeuleaux says, "his 'evidently' expresses the usual notion, and the one which is suggested by a hasty pre-judgment of the case. In point of fact B describes the pericycloid BjB'B", which certainly differs sufficiently from the hypocycloid 3 1 > 1 'P 2 ." We have next to consider the curtate and prolate epi-, hypo- and peri - trochoids. As previously stated, I have seen no direct proof that they also possess the same property of double generation, but find that the kinematic method lends itself with equal readiness to its demonstration. For the hypotrochoids, let B, Big. 4, be the centre of the first rolling circle or generator, F that of the first director, and P the initial position of the tracing point. The initial point of tangency of generator and director is m. Let the generator roll over any arc of the director, as m Q. The centre B will then he found at B 2 , and the tracing point P at P 2 . The point of contact, Q, will then be the instantaneous centre of rotation for P 2 , and P 2 Q will, therefore, be a normal to the trochoid for that particular position of the tracing point. The motion of P is evidently circular about B, while that of B is in a circle about P. The curve PPjPj P 6 is that portion of the hypotrochoid which is described while P describes an arc of 180° about B, the latter meanwhile moving through an arc of 108° about F, the ratio of the radii being .3:5. Now while tracing the curve indicated the point B can be considered as rigidly connected with a second point, p, about which it also describes a circle, p meanwhile (like B) describing a circle about P. Such a point may be found as follows: — Take any position of P, as P 2 , and join it with the corresponding position of B, also join B 2 to F. Let us then suppose P 2 B and B 2 F to be adjacent links of a four-link mechanism. Let the remaining links, F/> 2 and /3 2 P 2 , be parallel and equal to P 2 B 2 and B 2 F respectively. Taking F as the fixed point of the mechanism let us suppose F 2 moved toward it over the path P. 2 P 3 ....P 6 . Both B. 2 and p 2 will evidently describe circular arcs about F; while the motion of F 2 with respect to p 2 will be in a circular arc of radius p 2 B 2 . We may, therefore, with equal correctness, consider p 2 as the centre of a generator carrying the point P 2 , and p. 2 F a new line of centres, intersected by the normal P 2 Q in a second instantaneous centre, q, which, in strictest analogy with Q, divides the line of centres on which it lies into segments, p 2 q and F g, which are the radii of the second generator and director respectively ; q being, like Q, the point of contact of the rolling and fixed circles for the instant that the tracing point is at P 2 . The second generator and director, having p 2 q and q F respectively for their radii, are represented in their initial positions, p being the centre of the former, and p. the initial point of contact. The second generator rolls in the opposite direction to the first. It is important to notice that whereas the tracing point is in the first case within the generator and therefore traces the curve as a prolate hypotrochoid, it is without the second generator and describes the same curve as a curtate hypo- trochoid. If we now let B and F denote no longer the centres, but the radii, of the rolling and fixed circles, respec- tively, we have for the first generator and director 2 E > F, and for the second 2 B < F. It occurred to me that a distinction could very easily be made between trochoids generated under these two opposite relations of radii, by using the simple and suggestive term major hypotrochoid when 2 E is greater than F, and minor hypotrochoid when the opposite relation prevails. We would then say that the preceding demonstration had estab- lished the identity of a major prolate with a minor curtate hypotrochoid. Similarly the identity of major curtate and minor prolate forms could be shown. If the tracing point were on the circumference of the generator the trochoids traced would be, by the new nomen- clature, major and minor hypo - cycloids. It is worth noticing that for both hypo - cycloids and hypo - trochoids the centre F is the same for both generations, and that the radius F is also constant for both generations of a hypo - cycloid, but variable for those of a hypo- trochoid. Fig. Ss. DOUBLE GENERATION OF H YPOTROOHOIDS. Having given the radii of generator and director for the construction of a hypo -trochoid, the method just illustrated will always give the lengths of the radii of the second rolling and fixed circles. The accuracy of the values thus obtained may be checked by simple formulae derived from the same figure, as follows : — Eadii being given for generation as a major bypotrochoid, to find corresponding values for the identical minor hypo- trochoid. Let F ; denote the radius PQ [=PmJ of the first director. " F 2 '■ " " ~g q [ = Fm ] " " second " 11 r " " " E 2 Q [=Km] " " first generator. " p " " " p,q [= pp. ] " " second " " rr ll " tracing radius of the first generation, i. e., the distance E,P, (or KP) of tracing point from centre of first generator. Let tp equal the second tracing radius = p 2 P 2 = p P. Prom the similar triangles Q P q and Q K 2 P 2 we have P 2 Fj . : tr r F, (tr) whence F 2 = , (1) ?■ also p = P 2 — t r = - — < r = fr- j — — 1 \ ........ (2) r ( '• ) and ?p = p 2 P 2 = PP., = d, the distance between the centers of first generator and director (3) If the radii be given for a minor bypotrochoid then FQ : p 2 P 2 : : Fi-trochoid and desired for the peri- trochoid) : F, (tr) F 2 = -^~ (7) " = tr {~ + ^ • (8) t p = d = distance between centres of given generator and director = Pj + r (9) If given as a ^eri-trochoid and desired as an epi - trochoid the tracing radius will again equal the distance between the given centres (in this case, however = R — F) ; the formula of the radius of desired director will be of the same form ^ -r^GENERATo,, -in as equations (1) and (7); but f F. ) radius of second generator = tr < 1 I .... .. ... (10) With the tracing point on the circumference of the generator, and letting R = radius of the same for a peritrochoid and r for an epitrochoid, we have for the epicycloid r = E — F . . „ ..(11) " " pericycloid R =: F + r ■ , . . , (12)