-8g?^^7 -^■- "" •' '-' 1 i^Lf^^i €mml\ mmtrmitg ^ilravg THE GIFT OF l^rr\*i,eVrK,<<*_ . . A>...lr..±.^...U..O L'J^../..ZL.J.c, 6896-2 Cornell University Library eometry, and of arV19519 Elements of analvjical 3 1924 031 293 420 olin.anx The original of tiiis 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/cu31924031293420 ELEMENTS AMLITICAL GEOMETEY, AND OE THE DIFFERENTIAL AND INTEGRAL ELEMENTS ANALYTICAL GlEOMETEY, AND OP THE DIFFERENTIAL AND INTEGRAL CALCULUS. BY GERAEDIS BEEKMAN DOCHARTY, LI.D., PEOFESSOH OF MATHEMATICS IN THE NEW TOBK FEEE AOADEMT, AND AUTHOE OF A PKAOTIOAL AND COMMERCIAL AMTUMEnO, THE INBTXTHTES OF ALGEB&A, AND THE ELE- MESTB OF GEOMETRY AND TBIOONOMBTBr. NEW YORK: HARPER & BROTHERS, PUBLISHERS, FRANKLIN BQUABE. 186 5. Entered, according to Act of Congress, in the year one thousand eight hundred and sixty^five, by HARPER & BRoHtEES, . In the Clerk's Office of the District Court of the Southern District of New Yorls. TO JAMES M. McLEAN, Esq., FOR THE GREAT INTEREST HE HAS MANIFESTED IN THE CAUSE OF COK- EECT EDUCATION, AND PARTICULARLY FOR HIS PER- SONAL EFFORT^^N THE PROMOTION OP LITER- ATURE, SCIENCE, AND ART IN THE NEW YORK FREE ACADEMY, tl)i3 tJolnnte is most res^iectfitilg b^bicatcb BY THE AUTHOR PREFACE. The following Treatise on the Elements of Ana- lytical Geometry and the Differential and Integral Calculus is the result of long experience in educating young men. It is, in fact, the substance of that which the Author has taught in the Free Academy for sev- eral years ; and one great object in bringing it before the scientific instructors of our country is an endeav- or, on his part, to render the student's path, particu- larly through the Calculus, as smooth, and plain, and easy as the nature of that science will admit. The Author well knows that a correct system of education requires constant mental effort on the part of the student who aspires to be a thorough scholar, and that a subject which is simplified too much in a text-book becomes unfitted for the object for which it was intended — namely, to enlarge and strengthen the powers of the mind. But he thinks that this fault can scarcely occur in any treatise on the " Theory of the Variation of Variables and their Functions," a sci- ence which, under the most favorable light that can be thrown upon it, is suf&ciently obscure for the mind, at the early age in which the young men of this country begin the study of it. It will be seen that the Author has used the old symbols exclusively, and that he has confined his sys- tem to the Method of Limits. He has done this, not on account of any partiality he feels for that system, for PREFACE. the Infinitesimal Method generally reaches the conclu- sions in a far less time and a much shorter space, but solely from his belief that the former is better adapted to the great purpose of teaching, and that it has a greater tendency to draw out the energies of the stu- dent, expand his genius, invigorate his thoughts, and impart a keener perception to his intellectual powers. In discussing the Ellipse and Hyperbola together, the Author hasijbllowed the example of several emi- nent writers on* Analytics. In this, as also in the method he has adopted in the discussion of the circle and the transformation qf co-ordinates, and elsewhere, he has been enabled to reduce the Treatise to its min- imum size without the omission of a single essential proposition. In the recitation-room, perha^ it would be better for one studerJt to take the proposition in reference to the ellipse, while 'Sfeother student takes the same for the hyperbola ; in this way the number of examples becomes doubled. With these prefatory remarks, the volume is respect- fully submitted to the learned and scientific gentlemen who are engaged in teaching those useful and import- ant branches of pure Mathematics. They will find it precisely what it purports to be — nothing more and nothing less — a Treatise on the Elements of Analytics and the Differential and Integral Calculus, prepared at the request, and for the use of the students in the New York Free Academy, but equally a:d^pted to the instruction given in our colleges and other seminaries of learning. May, 1865. -€-Q.l M-N^T S. ANALYTICAL GEOMET lOMETRl '1 On the Point and StraiglitLine. — On"fflH|||]Jfcnation of Co-or- dinates .jj,-«.-r?!it ...rTQ^j^.-.Fage 9-26 SlPTEB II. On the Circl e. — Exam ples. — (»Jii^Parabola.— lOn^e Ellipse and Hyperbolar-^^^nffiMBNii|Me..an(f"H^fcbola referred to their Centre and Conjugate Diameters. — Probl*s. — Gsajei'al Equation of the Second Degree containing two Varilbles ' 26-81 CHAPTEK ttCAI, GEOMETRY OP TH Equations Q£.a — Equations 6F -Distancfl jPoints in Space. Tlane... 82-101 DIFFEI ILUS. Definitions BHAIntroductbry Remarks!^^Br!i^ons defined. — Differ- entiation ofiy^gebraic Functions.— ^W^rential Coefficient of a Function of alfop«<*J|t— Examples ..r^..; 103-130 On successive Differential Coeffi^Sts 131-132 Taylor's Theorem -aHb- 133-136 Maclanrin's Theorem ..."„...";. 136-137 Derelopment of Functions of two or faiore Variables when each rec eives an I ncMB ment. 137-144 TranscendenSrTumHPP^fJjjerential of the Logarithm of a Q1iantitf.tTnT..T?fi'. 145 Examples 146-147 Differential of an Exponential Function 147-150 Circular Functions 150-159 Implicit Fn«etions,...-. 159-161 CONTENTS. CHAPTER III. Application of the Calculus to the Tlieoiy of Curves. ...Page 161 Asymptotes 171-175 Singular Points of Curves 175-185 The Logarithmic Curve 185 The Cycloid 186 Spirals 187 TheCissoid 189 Differentials of the Lengths and Area of Curves 190-191 Differentials of Surfaces and Volumes of Revolution 191-193 Differentials of Areas, etc., referred to Polar Co-ordinates... 193-197 Asymptotes to Polar Curves 197-199 Osculatory Circle. — ^Radius of Curvature 200-206 Radius of Curvature to Spirals 206-207 EvoluteS and Involutes 208-215 CHAPTER IV. Vanishing Fractions 215-223 Maxima and Minima of Functions of one Variable 224^231 Maxima and Minima of Functions of two Variables 232-235 To change the Independent Vniiable 235-236 On Tangent Planes and Normal Lines to Curved Surfaces.. 236-238 Radius of CuiTature of a Curve of doubl^^jSurvature 238-239 ELEMENTS OF THE INTEGRAL CALCULUS. Preliminary Remarks. — Integral of a:»(fe 240-243 Integration of Binomials 244-247 " of Circular Functions 247-252 " by Series - 252 " of Binomials 253-256 " of Rational and Irrational Fractions 256-260 by Parts 260-265 ' of Logarithmic Functions 265-269 " of Exponential Functions 269-271 " of Trignometrical Functions 271-272 " of Sines and Cosines of Multiple Arcs 273-274 " between Limits 274-275 Rectification and Quadrature of Plane Curves 275-284 Surfaces and Volumes of Revolution 284-292 Double Integrals 292-295 The Volume a Function of three Variables 295-296 The Surface a Function of three Variables 296 The Theory of Infinitesimals 296-300 Miscellaneous Exaufles 300-306 ANALYTICAL GEOMETRY. Peoblems in Geometry, which, by the method of Eu- clid, would require for their solution long and laborious reasoning, and constant reference to propositions pi-evi- ously established, are often solved by analysis in a brief and easy manner, and from the most elementary princi- ples. This is called the application of Algebra to Ge- ometry. Analytical Geometry has for its object a more gen- eral and extensive field than that of solving determinate problems by means of algebraical equations ; it is more generally understood to be the analytical investigation of the general prope]^ies of geometrical magnitudes by means oi indeterminate equations. ANALYTICAL GEOMETRY OF TWO DIMENSIONS. CHAPTER I. (1.) On tJie Point and straight Line. Assume the two straight lines AX, AY, whose position is known, situ- ated in the same plane, and forming a given angle with each other. Let P be any point in the same plane. We are required to determ- ine its position. Prom the point P<||raw the lines '^ "° PC, PB respectively parallel to AX and AT. Then it is evident that the point P will be determined if we know the length of the lines CP and PB, or, which is the same thing, the lines AB and AC, of the parallelogram AP. 2. The two fixed lines AX, AT, extending indefinitely A2 10 ANALYTICAL GEOMETEY. right and left, up and down from it, are called the axes, or the axes of co-ordinates, AX being the axis of x, and AY the axis of y. * The distances AB or CP, and AC or BP, are called the co-ordinates of the point P; the former being the abscissa, and the latter the ordinate. The variable abscissa is denoted by x, and any known abscissa by x', x", x'", etc. The variable ordinate is denoted by y, and any known ordinate by y', y", y'", etc. The point A is called the origin of co-ordinates. The axes may be drawn making any angle with each other ; when they form a right angle they are called rectangular, and when they do not form a right angle they are called oblique axes. 3. For every point situated on the axis of x we must have y=0, because that equation indicates that the distance of the point in question from that axis is nothing ; likewise, for every point situated on the axis o^y, x=0. Therefore, the system of the two equations x—0,y=0, characterizes the point A, the origin of co-ordinates, since these equations can hold good at the, same time for no other point. In general, the two equations x=:x', y=y , when considered together, characterize a point situated at the distance x' from the axis of y, and at a distance y' from the axis of x. The first of these equations, when considered separate- ly, belongs to all the points of a straight line drawn pai'- allel to the axis of y at a distance AB=£b'/ and the sec- ond, to all points of a straight line drawn parallel to the axis of a; at a distance AC=y'. Hence the system of the two ■liquations together be- longs to the point P, where these lines intersect, and to this point alone. These expressions are the analytical representations of the point, and for this reason are called the Equations of a Point. ON THE POINT AND STRAIGHT LINE. 11 4. Not only the absolute values of the distances of the point from ^e two axes must be known, but the alge- braic signs pltis and »imt which is to be laid off from the origin A below the axis of 9!, and on the axis of y. 14 ANALyTICAI,. GEOMETRY. Let y—Q. :.x—-, a which is to be laid off from the origin on the axis of x to the right, and the line joining those two points will be the Ene re- quired. 2. Let 2/=— aa;+5. Make x=zO, then y— +5. 2/=0, a;=+l a Lay off on the axis of y above the origin a distance =S, and on the axis of a; to the right a distance =-, and join those a points, as in the figure. Let y=—ax—b. Make a; =0, then 2/== — b. y=0, x±z-~. a • 3. Let y=ax. Make £b=0, then y=.0. The line passes through the ori- ^ gin. Now, to find a second point, let x=l, then y=ffl. Lay off on the axis of x to the right a distance equal to 1, and erect a perpendicular =(3!. Join the extremity of this perpendicular and the origin by a straight line, which will be the line required. 4. Let y-=b- Then the line is parallel to the axis of x. Examples. 5. Construct the line whose equation is . y=2x-f5. 6. Construct the line whose equation is 2/=— 3a;+9. 7. Construct the line whose equation is y—'ix—S. 8. Construct the line whose equation is ON THE POINT AND STRAIGHT LINB. 15 2/=— SCK— 8. 9. Construct the line whose equation is y=2x. 10. Construct the line whose equation is 2/=— 303. Peoposition III. (8.) Eoery equation of the first degree containing two variables is the equation of a straight line. Every equation of the first degree containing two va- riables may be reduced to the form Ay-Bai+C. B , ^ A ^A B C If we put -T-=«5 and -t-=5, we shall have A. A. y=.ax-\-h. * T> When the axes are rectangular, — is the tangent of the angle which the litie makes with the axis of x : and — is the distance from the origin to the point where the line cuts the axis of y, whatever angle the axes may make with each other. If we solve the equation for a;, we shall have A C "'-B^-g' where — is the tangent of the angle which the line B C makes with the axis of y, and — — is the distance from B the origin to the point where the line cuts the axis of aj. Therefore, if we determine the value of either of the variables, the vo-efficient of the other variable will be the tangent of the angle which the line makes with the axis of that variable, and the absolute term will denote tile dis- tance from the origin to the point where the line cuts the axis of the variable by iohose co-efficient we have divided the equation. 16 ANALYTICAL GEOMETEY. Illustration. If 3i/=6a;— 9, then y=2x—Z. Here 2 is the tangent of the angle which the line makes with the axis of a;, and —3 is the distance from the origin to the point where the line cuts the axis of y. Or, solving the equation for x, 1 , 3 2^ 2 Here - is the tangent of the angle which the line makes with the axis of y, and +- is the distance from the ori- gin to the point where the line cuts the axis of a;. (9.) If the co-ordinates of any point of a line he sub- stituted for the variables in the equation of that line, the equation will be satisfiea. Peoposition- W. (10.) The equation of a straight line which passes through a given point, whose co-ordinates are x' and y', is y-2/'=a(a5-«'). Let P be the given point. the equation of a straight line is of the form yz^ax-\-b. _ (1) If we substitute the co-ordinates of the point P in this equation, we shall have y'=aa;'+5. (2) Each of these equations must be satisfied at the same time ; therefore, by subtracting the second from the first, we shall have y-y'=a{x—x') for the equation of a line passing through the given point. Since the quantities x, y, and a are unknown, it fol- lows that a, the tangent of the angle which the line makes with the axis, is indeterminate. There may, there- ON THE POINT AND STRAIGHT LINE. 11 fore, be an infinite number of straight lines drawn through a given point. (ll!) If it be required that the straight line, drawn through the given point, shall be parallel to a given line, then the angle which the line makes with the axis is known, and only one line can be drawn through a given point parallel to a given line. Jllustration. If we were required to draw a line through the point P, whose abscissa AB is 8, and ordinate BP ^ =3, parallel to the line AD, whose equar tion is 1 we shall have in the equation y-3=l{x-8), 2y—Q=x—8, 1 1/^-x — 1. 2 Lay off from A a distance AC, equal to 1 below the axis of X on the axis of y, and 6n the axis of x, to the right of the origin, a distance equal to 2, and join those points by the line CP, which will be the line required. Proposition V. (12.) The equation of a straight line which passes through two giv- en points whose co-ordinates are £»", v", is I' ' _2/ y- Let C and .2/ -y' -y («-«'). 83—33 B be two given points, the co-ordinates of C being a;', y\ and the co-ordinates of B being a;", y'\ and x and y the co-ordinates of any other point whatever. 18 ANALTTICAL SEOIIETET. The general equation of the straight line is y=ax-{-h. _ (1) But, since x', y', and x", y" are points in the straight line, these co-ordinates must satisfy the equation. Hence y'=aiiS-\-h y"^ax:'+b. Subtract (2) from (3), then y"-y'=a{x"-;^). (4) From equation (4), a— -','~^, . X —X Again, subtract (2) from (1), y—y'=a{x—a^). (5) Substitute in (5) the value of a obtained from (4), we have y—y'— y~y {x—x!), X — X which is the equation of the line passing through the two given points C and B. (13.) In the figure we see that CD is parallel to AX, and the line BC cuts them ; therefore the angle BCD is equal to the angle which BC ma^ds with the axis of x. Also that y"—y' is equal to DB, and ce"— x' is equal to CD. Hence ^Lz:;^z=-__=tangent BCD, to ra- X —x CD dins unity. If y"=2/', the line would be parallel to the axis of x, and ^^!zV^O. a; — aj If x"^x\ then y ~y , = y ~^ =oo, and the line would be perpendicular to the axis of x. Examples. 1. Find the equation of a straight line which passes through the two points whose co-ordinatfes are a;' =6, 2/' =3, and a;" =6, 2/" =2, and determine the angle which it makes with the axis of a. 2. Find the equation of the straight line which passes ON THE POINT AND STKAIGHT LINE. 19 through the two points whose co-ordinates are a;' =3. y'^A;x"=5,y"-2. . Peoposition VI. (14-) The tangent, of the angle which two lines make hy their intersection with each other is a'— a 1+aa'' where a and a' are the trigonometrical tangents of the angles which the two lines respectively make with the axis of abscissas, or the axis of x. Let ED and CB be any two lines intersecting in P. Let the equation of the line ED be y—ax+b, and the equation of the line CB be y-a'x+b'. Then a will be the tangent E~ of the angle PEX, and a' the tangent of the angle f ^X. Let the angle PEX be represented by a, and the an- gle PCX be represented by a. Then, because PCX is the exterior angle of the trian- gle EPC, it is equal to the sum of the two interior op- posite,angles CPE and CEP. Put, therefore, V=EPC. ,-. \ ^ =a —a, J . XT i "' / ' \ tan. a • — tan. a and tan.V=tan. (a —a)= l-l-tan. a, tan. a Hence tan.V="'~^. 1 + aa If the lines are parallel, they will not intersect, and tan.V= — ^-=0. .-. a'— a=0, or «'=«. l-\-aa If the lines intersect at right angles, then tan.V= " ~° =roo. .*. l+a'«=0, or a'=: — 1 + aa a which is the equation of condition for two lines to be at right angles with each other. (15.) Hence it appears that if the equation to a straight line be 2/=aa3+&, the equation to a straight 20 ANALYTICAIi GEOMETEY. line perpendicular to it, and passing through a given point whose co-ordinates are as', y', will he Pkoposition VII. (16.) ITie length of a straight line drawn from a given point perpendicular to a given straight line is y'—ax'—b VT+a^ ' where x' and y' are the co-ordinates of the given point. Let EB be the given line, and P the given point. The equation of the given line is y=zax-\-h. (1) Then, since PQR is drawn through a point whose co-or- dinates are cc' and y\ and per- pendicular to a straight line whose equation is y=iax-\-h, -^-^ the equation to PR is, hy the last proposition (Art. 15), y-y'= — (a!-a;'). (2) Now, to find the length of PQ, we must find the co- ordinates of the point Q, in which PR' meets EB, and then substitute x', y', and the co-ordinates of Q in the general expression for the distance between two points, ^'^" T> = V{^-x"Y+(y[-y"Y. (3) Let us call the co-ordinates of the point Q x" and y". Then, since Q is a point in the straight line EB, the co- ordinates of that point must satisfy equation (1). .•.y"=ax!'+b. For the sake of convenience, we will put this equation under the form y"-y'=a{x:'-x')-y'+ax'+b, (4) which is done by subtracting y' from each member of the equation, and adding ax' to, and subtracting it from the second member. ON THE TKANSFOEMATION OF CO-OEDINATES. 21 But, since Q is a point in the straight line PR, its co- ordinates x", y" must satisfy equation (2). Now equations (4) and (5) must both hold good for the point Q. Therefore, subtracting (5) from (4), we tave o=l(£B"-a!') + «(cc"-£8')-2/'+«a!'+5. a „ , y'—ax'—b l+a2 Substitute this value of x"—x' in equation (5), we shall have v"—i '__2/'— «^'— 5 ^ ^~ l+a2 Substituting the values of x"—x' and y" —y' in equa- tion (3), we have TA / „{y' —ax' —by , (y'—ax'—hY P_y' —ax' —b — — / "' which is the length of the perpendicular required. On the Teansfoemation of Co-oedinates. (17.) When a line is represented by an equation in reference to any system of axes, we can always trans- form that equation into another in reference to a new system of axes, chosen at pleasure, which equation shall equally express the relations between the co-ordinates of every point of the line. This is called tJie transformation of co-ordinates. We may either change the origin without altering the relative position of the axes, or alter the relative posi- tion of the axes without changing the origin ; or we may change the position of the origin and the angle of inclination of the axes to each other at the same time. 22 ANiLLYTICAL GEOMETRY. Peoposition VIII. (18.) The formulas for passing from one system of co-ordinate axes to another system respectively parallel to tlie first are x=a+x^, y=b+y,. liet AX, AY be the original axes ; and A'X', A'T' be the new axes parallel to the former. Let 8!=:: AM, the abscissa of the point P, referred to the original axis, and y='MP, the ordinate, re- ferred to the same. Let a;,=i:A'M'=:BM, the abscissa referred to the new axis; and.y,=zM.'F=the ordinate. Let the co-ordinates of the Hew origin AB, BA' be represented.by a and b. Then we shall have AM=AB+BM, or x=a+x^, and MP=MM'-fM'P, > or MP=BA'+M'P, or y=b+y„ which are the formulas required. Peoposition IX, (19.) The formulas for passing from a system of ob- lique co-ordinates to another system, also oblique are, ^_ x, sin. (/3-q)+y, sin. (/3-a') ^ — -, — , sin. p Xj sin. a+2/, sin. a y= "liErT? '' , wTiere j3 is the angle contained between the original axes, and a and V are the inclination of the new axes to the original axis of abseis- y . sas. Let AX, AY be the primitive axes; AX', AY' the new axes ; and P any point in the plane. Let the co-ordinates of the point P referred to the A ON THE TEASrSFOEMATION OF CO-OEBINATES. 23 primitive axes be denoted by x and y ; and its co-ordi- nates referred to the new axes be x^ and y,. Through the point P draw PP' parallel to AY' and PR parallel to AY ; also, through P' draw P'B parallel to AX, and P'R' parallel to AY. Denote the angle YAX by /3, the angle X'AX by a, and the angle Y'AX by a'. Then, by Trigonometry, AR' sin. (/3-a) . AT?/_ A-P' ^™- (/3- ") sin. (/3-a) AF- sin./3 "^"-^J^ sin./3 -'^' sin./3 ' FB _ sin. (/3-a') . p sin. (/3-a ') sin. (/3-a') P'P sin./3 •• ~ sin./3 "'''' sin./3 ' P'R'_ sin.a p,r},_ Ap,sin^ sin, a AF~sin.j3" sin. /3' sin. /3' EP ^ sin. g' ^p pp, sin.a' sin, a' P'P sin./3" ~ sin.jS"'^' sin./3" But «.= AR'+R'R= "' ^'°- (^-"1+^' ^^°- ^^-" \ sm. /3 " y=.RB+BP^ ^'^'"-"+^/^°-"' , the formulas required. Cm. 1. If /3=:90°, the primitive axes are at right angles with each other, and the formulas become (20.) x=Xi cos. a+2/, COS. a', 2/=£t;^ sin. a+y^ sin. a', which are th^ formulas for passing from a system of rectangular to a system of oblique co-ordinates, the origin remaining the same. Cor. 2. If a'— a=90°, or a'=90°+a, then cos. a'= cos. (90°+a) = — sin. a ^rig., Eq. 12), and sin. a'=sin. (90° + a) = COS. a. Substituting these in the above for- mulas for COS. a and sin. a', we have (21.) »=£«;, COS. a— y, sin. a, y=-Xi sin. a.-\-y, cos. a, which are the formulas for passing from a system of rectangular co-ordinates to another system also rectan- gular, the origin remaining the same. If the position of the origin is changed also, then (20) becomes (22.) a;=a-|-a;, COS. a-f-y, cos. a', y=h-\-x, sin. a+2/, sin. a. 24 ANALYTICAL GEOilETEY. And (21) becomes (23^ a5=a+£B, cos. a—y, sin. a, y=b+x^ sin. a+y^ cos. a. POLAE Co-OEDINATES. (24.) In the preceding propositions we have determ- ined the position of a Une upon a plane by means of an equation between two variables expressing the distances of each of its points from two fixed straight lines, the distances being reckoned parallel to each of these lines respectively. There is, however, another method for de- ' termining a series of points, which, in certain cases, is more convenient. To explain this mode of representing lines analytically, let us take any point, P, and let A'D be a given straight line in the plane of the curve, and A' a given point in that line. From A' draw the straight line A'P. Put A'P= r, and the angle PA'D=0. fr Then it is evident that if we can obtain a relation between r andS " '^ which shall hold good for every point in the line, the line will be entirely determined. The variable quantities »• and are called Polar Co- ordinates. The point A' is called the pole, r the radius vector, and the relation between r and is termed the Polar Equa- tion. The Polar Equation is, sometimes determined at once from some known property ; it is, however, more usually deduced fi-om an equation between rectilinear co-ordi- nates, which may easily be effected by a transformation. Peoposition X. (25.) The formulas for passing from a system of rec- tangular to a system, of polar co-ordinates are x=a-\-r cos. 6, y—b+r sin. 0, r being the radius vector, and 6 the variable angle which it makes loith the axis of x. POLAK CO-OEDINATES. 25 D B l^t AX and AY be the rectan- y gular axes, A' the pole, and A'D, parallel to AX, he the line from which the variable angle is to be estimated. Put PA'D=e, A'P=»-. Let the co-ordinates of the point P, referred to the prim- itive axes, be x and y, and the co-ordinates of A', referred to the same, be a and b. Then a!=AB+BR, and y-WD+jyP. But BR=A'D=:A'P cos. PA'D=>' cos. 9, and DP= AT sin. PA'D =»• sin. 6. Hence x—a+r cos. 0, i/=b-\-r sin. d. 26. ScHOLixiM. If A' be placed at the origin A, then a=0, 6=0, and the equations become x=r cos. 0, y^r sin. 0. Frdblem. (2 It.) To find the co-ordinates of the point of intersec- tion of two lines. Let the equation of the line ED be y=ax-\-h, and the equation of the line BC be y=a'x+b'. To find the abscissa and ordi- nate of the point P, their inter- section. If we call the co-ordinates of the point P x, y, then, because that point is common to both lines, the co-ordi- nates of that point must satisfy both equations ; hence, eliminating y, we shall have V — b 'a— a'' aV-a'b a— a! -a', the two lines become paral- This is evident from the anal- B 0!=- Hence y-- ScHOLiDM, Kow if a= lei, and will never meet. 20. AU-ALTTICAL GEOMETRY. ysis also; for «=«' renders both x and y equal t<^ in- finity. If we have, respectively, the equations of any two lines whatever on the same plane, referred to the same axes, we can find the point or points of intersection simply by finding the values of x and y from those equations. As these values are co-ordinates to both lines, they must be the values of the co-ordinates of the point of inter- section. CHAPTER II. ON THE CIECLE. (28.) The circumference of the circle is a plane curve, every point of which is equally distant from a point within called the centre. Peopositiok I. (29.) The equation of the circle referred to oblique axes is (x—xy+{y—i/'y±2{x—x') {y—y') cos. (i=R\ where R is the radius of the circle, x and y are the co-or- dinates of any point in the circumference, x', y' the co- ordinates of the centre, andfj the inclination of the axes. Let P be any point in Y , the circumference of the circle whose centre is C Draw AY, AX, the axes of co-ordinates, making an angle equal to /3. Through C and P draw CB and PE both parallel to AY, and through C z c z l ^X; draw FCG parallel to ^ - E BE, AX. Let the co-ordinates of the centre, AB, BC, be de- noted by x', y', and AE, EP, the co-ordinates of the point P, by X, y. Then, because FG is parallel to AX, and PE is parallel to AY, the angle PDG is equal to YAX equal to j3; and therefore PDC is the supplement of PDG=180°-/3. But by Plane Trig., Th. V., ON THE CIECLB. 27 cos.rui^- 2CDxDP - 2{x-x'} {y-y') 5 .■■Cos.PDC^-coM=. ^ 2(i-J)(y-V) ; .-. (st;-a;')^+(y-2/')2±2(a;-£c') (y-y') cos.^S^R^, (1) the equation required. (30.) Scholium 1. If /3=90°, the axes are rectangular, and (Eq. 1) becomes {^x-x'y+{y-y'Y^-B?, (2) which is the most general equation of the circle referred to rectangular axes. Scholium 2. If the origin is removed from A to F, and the axes are rectangular, x' becomes equal to R, and y'=o, and the equation becomes 2/2— 2Ra;— a;2, (3) the equation of the circle when the origin of co-ordinates is on the cir&imference. (31.) Scholium 3. If the origin is transferred to the centre, then a;'=0, 2/'=0, and the equation is 2/2+a;2-R2, (4) which is the equation of the circle when the origin of co-ordinates is at the centre. (32.) If we wish to find the points in which the curve cuts the axes, first make a;=0, and we shall have, from (Eq. 4), y=±n, which shows that the curve cuts the axis of y in two points on different sides of the origin at a distance equal to the radius. If we make y=0, then a;=:±R, which shows that the curve cuts the axis of a; at a dis- tance equal R on the right and R on the left of the origin. (33.) If we wish to trace the curve through the inter- mediate points, we find the value of x or y thus : x=±-VW^^, or yz^dtzVW—x^. 28 ANALTTICAL GEOMETET. Now, since every value of one variable gives two equal values of the other, with contrary signs, it follows that the curve is symmetrical in regard to both axes. If we make x or y greater than R, the value of the other becomes imaginary, which shows that the curve does not extend on either side of the origin beyond the value of a or y=±R. In a manner entirely similar, if in (Eq. 3) we make we have 0=2Ra;— a;^ or (2R— a;)£C=0, whence a;=0, and a:=2R, which shows that the curve passes through the origin of co-ordinates, and also cuts the axis of a; at a distance from the origin equal to twice the radius. Peoposition II. (34.) ITie equation of a tangent line to the circle is yy"+icx"=W, where R is the radiics of the circle, and x", y" are the co- ordinates of the point oftan- gency, and^x, y are the gener- al co-ordinates ofthjs, tangent line. Let the co-ordinates of the point P' be denoted by a;', y\ and the co-ordinates of P" by a;", y" ; then the equa- tion of the line BC, passing through these two points, will be v"— v' But, since these two points are on the circumference, their co-ordinates will satisfy the equation of the curve. Therefore a!"2-f-y"z=R2, (2) a;'2 +y'2 =R2. (3) Subtracting (3) from (2), we have c(!"2— ai'2-fy"^— 2/'^=0. Factor these, we have (a!"+a!') (a:"_!B') + (2/"-h2/') (j/"-y')=0; ON THE CIECLE. 29 _ y"—y' __ x"+x' "x"-x'~ y"+y'' Substitute this value in equation (1), we have Now, as the secant line BC moves toward P, the points P" and P' approach each other, and ultimately coalesce at the point P ; then the secant becomes a tangent, and x' becomes equal to x" and y' equal to y", and equation (4) becomes y-y"^-'^Xx-x"), which is the equation of the tangent line. This may be reduced to a more simple form. Clearing fractions, yy" —y"^-=—xx" -^x"\ or yy" -'t-xx" —x"^+y"'^; but x"^+y"-^=B?; .■.yy"+xx"=B?, x" , R2 . y—y^+Y'- By comparing this with the equation of a. right line, we see that ;; is the tangent of the angle which the y R2 . tangent line makes with the axis of X, and -p is the dis- tance from the centre to the point where the tangent line cuts the axis of Y. (35.) Definition. A normal is a line perpendicular to the tangent at the point of contact, and limited by the axis of abscissas. Peopositiok ni. (36.) Every normal line in a circle passes through the centre. The equation of the tangent line is y-y"=-^(«'-«'"); and as the normal line is perpendicular to the tangent, we have, from (Art. 15), 30 ANALYTICAL GEOMETET. y-y"=hi^^-^")- X Reducing, yx" —x"y" =xy" —x"y" ; v" .■.y=^^.x. Since this has no absolute term, it is the equation of a line passing through the origin ; and as the origin is at the centre of the circle, the normal passes through the centre. (37.) Definition. Two lines which are drawn from the extremities of any diameter of a curve, and which inter- sect the curve at the same point, are called supplementary chords. Peoposition IV. (38.) The supplementary chords in a circle are perpen- dicular to each other. Let A be the origin of co-ordi- nates, and A, B the extremities of a diameter through which the supplementary chords AP, BP are drawn. We are re- quired to prove that these chords are perpendicular to each other. The equation of a straight line passing through a given point is of the form y-y'=a(x-x'). (1) If the line pass through A, the co-ordinates of which are a3'=0, y'=0, the equation becomes y—ax. (2) If the line pass through B, whose co-ordinates are x'=2r, y'=0, the equation becomes y=za^{x—2r). (3) Now, if these two lines intersect each other, the co-or- dinates of the point of intersection will satisfy (2) and (3). Hence, multiplying (2) and (3) together, the result- ing equation, y^=aa'{x^—2rx), (4) will express the condition that the two straight lines shall intersect on the plane of the co-ordinate axes. But ON THE CIRCLE. 31 if the point of intersection "is to be in the circumference of a circle, x and y must satisfy the equation y^^2rx—x^, — — {x^—2rx). (5) Equating these two values of y^, we have aa' {x^ — 2rx) — — (sfi— 2rx) ; .-. aa' = — l, which is the equation of condition for two lines to be perpendicular to each other. Hence the supplementary- chords in a circle are at right angles to each other. PKOPOsmoM' V. (39.) The polar equation of the circle lohen the pole is on the circumference is r=±2R cos. e, where R denotes the radius of the circle, r the radius vector, and 6 the variable angle. The equation of the circle referred to rectangular axes when the origin is at the centre, Art. 28, is y^+x^=E?. (1) And the formulas for passing from a system of rec- tangular to polar co-ordinates. Art. 22, are a;=a+?" cos. Q, y=h-\-r sin. 0. Squaring these equations, and substituting the values of a;^ and y'^ in (1), we have b^+2b.r sin. S+r^ sin.^ d+a'^+2ar cos. e+r^ cos.^ e=R2 ; or, arranging according to the powers of r, r2(sin.2 e-f-cos.2 e) + 2(5 sin. e+a cos.'e)j-+a2+S2 =R^ . (2) which is the general polar equation of the circle. If we place the pole at the right extremity of the diam- eter on the axis of x, a=-|-R, and 5=0. The equation (2) reduces to r'^-\-2Rr cos. 6=0, because sin.2e+cos.2 0=l; .-.»•=— 2R cos. 0. (3) If we place the pole at the extremity of the diameter to the left of the centre, «=:— R, and 6=0; and (2) reduces to r'^—2Rr cos. 6=0 ; .■.r=2R cos. e. (4) 32 ANALYTICAL GEOMETEY. In (4), when 6=0, cos. 6=1 ; .-. r=2 R. As 6 increases from to 90°, the radius vector de- termines all the points in the upper semicircumference ; and when 6=90°, cos. 6=0. From 90° to 270° the ra- dius will he negative, and will determine no point of the circumference. From 270° to 860° the radius vector will again he positive, and determine all the points in the lower semicircumference. The student may take equation (3) and illustrate the manner jn which the radius vector will determine all the points of the circumference. Mcamples. 1. Construct the line whose equation is 2. Construct the line whose equation is , 3 5 _ ^ 4 8 3. Construct the line whose equation is ■ _ yz=—3x+6. 4. Construct the lines whose equation is y^—5x=0. 5. Construct the lines whose equation is 2/^=7y— 12. 6. Find the equation of the straight line which passes through two points whose co-ordinates are a;'=l, y'=3, x"=4:,y"=5. _ Ans.y—x+1. 7. Descrihe the circle whose equation is y^+x^+ij/—4x=8. 8. Describe the circle whose equation is y2+x^-6y+8x=n. 9. Prove that the straight lines drawn from the angu- lar points of a trian- gle to bisect the sides, pass through the same point. Let the lines CF, BE, and AD bisect the opposite sides of the triangle ABC ; we are ON THE CIKCLE. 33 required to prove that they pass through the same point, P. Let FB, FC be the axes of co-ordinates, the origin being at F. Then the co-ordinates of the point C are a;'=0, and y' ; " " " Barea;'and2/'=0; and, consequently, y'—Q and —x' will be the co-ordinates of the point A. But, since AE is half of AC, the co-ordinates of E will be — ja;' and ^y', and the co-ordinates of D, for a like reason, will be \x and Jy'. Therefore the equation of the line AD, which is of the •-y"-yLi.- form y-y becomes a; —83 And the equation of BE becomes ^'), y=- .,{x-x'). (1) (2) ix'+x' Now, where AD cuts CF, x^O, and where BE cuts CF, ' x—0. Making these substitutions, each equation gives y=W- Hence each of the lines intersects the line CF in the same point P. 10. Prove that the perpendiculars drawn from the angular points of a triangle to the opposite sides pass through the same point. Let the perpendiculars AD, BD, and CF be drawn from the angular points A, B, and C to the sides of the triangle ; we are required to prove that they pass through the same point P. Let AX and AY be the rec- tangular axes. Now if it can be proved that AF is the abscissa of the point of inter- section of the perpendiculars AD, BE, as well as of the point C, the truth of the problem will be established. Let the co-ordinates of the point C be x', y' B2 BbetB", 0=y" 34 ANALYTICAL GEOMETEY. Because AC passes through the origin and through the given point C, its equation is y' X The equation of the line BC, passing through two points whose co-ordinates are x" and y"=0, will be The equation of the perpendicular AD is x'-x" y= r-3'- y The equation of the perpendicular BE is y=--i^-o^")- And at the point P, where these two perpendiculars in- tersect, the ordinates must be identical. :.^ix-x)^-^x; :. !«=£»". That is, the abscissa of the point of intersection of AD and BE is the same as the abscissa of the point C ; _ hence the perpendicular CF passes through the inter- section P. On the Paeabola. (40.) Definition. A parabola is a plane curve, every point of which is equally distant from a fixed point and a given straight line. The fixed point is called th^ focus of the parabola, and the given straight line is called the directrix. Peopositioit I. (41.) The equation of the parabola referred to rectan- gular axes whose origin is at the vertex of the axis is where x and y are the general co-ordinates of the curve, and 2p is the parameter of the axis. Let F be the given fixed point, and DC the given straight line. ON THE PARABOLA. 35 Draw FB perpendicular to DC, and bisect BF in A ; then, by the definition, A is a point in the par- ^ ' abola. Take any other point, P, in the curve, and join PF. Prom P dra^7 PD perpendicular to DC. From A draw AY perpendicular to AFX. A will then be the ori- gin, and AX, AY the axes of co- ordinates. From P let fall the perpendicu- lar PR on AX. Put AR=a;, RP=2/, BF^p; then AF=J/j. Then, by the definition of the curve, PF=DP=AR+AB=a;+ii', and ¥B,=x'—ip. But PR^+FR^= ^PF^ or y2 + (x-ipy={x+ipy .: y^=2px, which is the equation of the parabola. In order to find the value of the ordinate passing through the focus, make a; =ij?/ then ••■y =±P, which shows that the double ordinate passing through, the focus is equal to the parameter. Solving the equation for y, we have y=±'\/2px, which shows that the curve is symmetrical with respect to the axis of x ; and for all values of x negative, the value of y is imaginary ; hence there is no part of the curve to the left of the origin A. If 33=0, 2/=0, and the curve passes through the origin. If we give a succession of values for x, we perceive that as X increases, y increases also, and that for each value of X there will be two values of y numerically equal with opposite signs. Hence the curve extends indefin- itely to the right of A. If we put the equation into a proportion, we have x:y::y:1p. (42.) That is, the parameter of the axis is a third pro- portional to any abscissa and its corresponding ordi- nate. 36 ANALYTICAL GEOMETET. (43.) If we take any two points on the curve, and des- ignate their co-ordinates by »', y\ and a:", y'\ we shall have Dividing these equations member by member, we shall have y'^ _x' or y'^:y"^::x':x". (44.) That is, the squares of the ordinates are to each other as their corresponding abscissas. Peoposition II. (45.) The equation of a tan- gent line to the parabola is yy"=p{x+x"), where p denotes half the param- eter, and x", y" are the co-ordi- nates of the point of tangency. ■ Put a!"=AR, 2/"=:EP; then the equation of a line passing through P will be y-y"=a{^x-x"). But since the point P is on the curve, its co-ordinates must satisfy the equation of the curve. (1) V y'^=2px is the equation of the parabola where X and y denote the co-ordinates of any point of the curve. (2), .:y"^=2px". Subtracting (2) from (1), we have y^-y^'^^2p(x-x"), or . {y-\-y") (y-y")=1p{x-x'\ Substituting for y—y" its value as above, we have (y+y")a{x-x")=2p{x-x'% (3) .■.■.=^.. But, in order that the line TP shall be tangent to the curve at P, x and y, the general co-ordinates, must, in this case, become the co-ordinates of the point P. .-. X—x" and y=y". ON THE PAEABOLA. 37 _ 2p p p (4) :.y—y"=-rn[x—x") is the equation of the tan- gent line. This, however, can be reduced. By clearing the equation of fractions, we have yy"-y"'^=px—px"; but y"^—'2px". Substitute this, yy" —2px" =px—px" , yy"=p{x+x"). To find the point at which the tangent cuts the axis of abscissas, make 2/=0, and we have p{x+x!')=0, or . »=: —x", where a;=:AT, the distance from the origin to the point in which the tangent line cuts the axis of X. (46.) Definition. The suhtangent is the projection of the tangent on the axis of abscissas. The subtangent TR is therefore bisected at the vertex of the parabola. Hence we are enabled to draw a tangent at any point of the curve, thus : From a given point let fall a perpendicular on the axis, and from the origin lay ofij on the axis, to the left a distance equal to the distance from the origin to the foot of the perpendicular. Join this last point and the given point by a straight line, which will be the tangent required. (47.) Definition. A subnormal is the projection of the normal on the axis of abscissas. Proposition III. (48.) The equation of a normal line to the parabola is y-y"=-j{^-o^"). where x" and y" are the co-ordinates of the point of tangency. For the equation of a line passing through P is of the form y-y"=a'{x-x") ; (1) 38 ANALYTICAL GEOMETRY. and since the normal PN is perpendicular to the tangent TP (Art. 34), we must have 1 a But hy Prop. II., ' P' a—-. , a = — - Substituting this in (Eq. 1), y" we have y—y"^—^—{x- 49. To find the point in which the normal cuts the axis of X, make y=0 in (Eq. 2), and we have x—x"=p. Here x=AN and a;"=AR ; .-. «— a;"=R!Nr=suhnormal z=jo. Hence the subnormal is constant, and equal to half the parameter. (50.) This furnishes us with a simple method of draw- ing a tangent to any point of a parabola. Thus : From the given point let fall a' perpendicular on the axis, and from the foot of the perpendicular lay off a distance equal to half the parameter, on the axis, to the right ; join this and the given point by a straight line, and at the given point draw a perpendicular to this latter line, which will be the tangent required. Peopositioit IV. (51.) A tangent to tJie parabola at any point makes equal angles with the axis and with the line drawn from the point of tangency to the focus. Let TP be a tangent to a pa- rabola, PF the line drawn from the point of tangency to the focus F. Then will the angle TPF be equal to the angle PTF. ^ Designate the co-ordinates of T^ the point P by x" and y", and the co-ordinates of the focus by x' and y'. ON THE PARABOLA. 39 The equation of a line passing through the focus P ■will be of the form . y-y' = a!(x-vi); (1) and if it passes through the point of tangenoy P, we shall have y"-y'=.a!{^'-^). (2) But the co-ordinates of the focus give a3'=|-^, and ^'=0. Therefore Eq. 2 becomes y" y"=a'{x"-ip). .: a'= ^„_^^ . Now the tangent of the angle which the two lines TP and FP make at their intersection, by Art. 14, gives „^„ a'— a tan. TPF=— p-T-. 1 + aa v" Substitute for a' its value — -^ , and for a its value » ' x'-ip -IT, obtained in Prop. 11., and we shall have \x'-^^p y") \ -^x"-ip)' which, by reduction, putting y"^— 2j3a;", becomes tan.TPF=^=:PTF. Hence the truth of the proposition is manifest, and FT=FP. Peopositioit v. (52.) If from the focus of a parabola a line he drawn perpendicular to the tangent, the point of intersection will be on the axis of ordinates. The equation of a line passing ^^ through a point is of the form P.X'^*^ y-y"=a\x-ff:'). (1) ^^ If the line pass through the ^■-■iy/ \ focus F, whose co-ordinates are ^^ K/ ! a;"=J^, 2/"=0, the above equa- ^ STp S *ion becomes V y=a\x-\p). _ (2) \^^ And the condition that this line ^-x..,^^ shall be perpendicular to the tan- ^~~~- 40 ANALYTICAL GEOMETRY. gent requires that the equation a'a+ 1=0 shall be ful- filled. a y_ ~ p' Equation 2 becomes y But the equation of the tangent line is y -—^i?^-\p)- (3) yy"=p{x+^'). (4) Combining equations (3) and (4), substituting for y "^ its value 2pa!\ and reducing, we have x{2x"+p\=0; i;=0, and x =—^x ■whence x=Q\ &ndix'^'=—\p. Alg., Art. 105. Therefore the point B, at which the perpendicular in- tersects the tangent, is on the axis of Y. (53.) Cor. By Geom., B. IV., Theor. IX., FA:FB::FB:FT; or FA:FB::FB:FP, because FT=FP (Prop.IV.). Hence tlie perpendicular FB is a mean proportional between the distance from thefoeus to the vertex and the distance from tJie focus to th^ point oftangency. Peoposition VI. (54.) Tlie equation of the parabola referred to a tan- gent line and the diameter passing through thepoint of tangeney, the origin being the point oftangency^ is y'^—lp'x, in which 2p' is the parameter of the diameter passing through the origin. The equation of the parabola referred to rectangular axes, of which A is the origin, is y^z=2px. (1) And the formulas for pass- ing from these to a system of oblique axes, of which A' is T^ the origin, are x=:a+x, COS. 0+2// cos. a', y=b+x, sin. a+y, sin. a, where a and b are the co-ordi- nates of the point A'. ON THE PAEABOLA. 41 Substitute these in (Eq. 1), -we have b^+2b{x, sin. a+y^ sin. a')+x^^ sin.^ a+2x^y^ sin. a sin. a' +2/,2 sin.2 a'=2pa+2pa;, cos. a+2py, cos. a'. (2) But, because the point A' is on the curve, the co-ordi- nates of that point will satisfy the equation of the curve, and we shall have And since the new axis A'X' is parallel to the primitive axis AX, we shall have a=0. .*. sin. a=0, and cos. a=:l. Making these substitutions in (2) gives 2bi/, sin. a'+y^ sva? a'=2/)a!^+2^y, cos. a. (3) Now the tangent of the angle which the tangent makes P with the axis is -7, where y' is the ordinate of the point of tangenoy, and is therefore equal to b. p p , sin. a' .'. 77='T=tan. a = ' y' b ' COS. a" .". p COS. a' ==5 sin. a'. Substituting this value of^ cos. a in (3), we have 2by' sin. a-\-y"^ sva? d ^2px' -\-2by sin. a'. 2px' 2px, .-.y'^- .\ , , or y^= .' , . 2o Or, to simplify its form, put -j— ^ — 7=2i>', and drop- ping the accents on the variables, y'^—2p'x. , In this equation, for every value we assign to x we shall have two values of y, numerically equal with con- trary signs. The curve is therefore symmetrical with respect to the axis A'X'. (55.) This axis bisects all chords of the parabola which are parallel to the tangent line A'Y', or axis ofA'Y'. It may therefore be considered as a diameter, and hence all diameters of the parabola are parallel to each other, and the centre of the parabola is at an infinite distance from the vertex. 42 ANALYTICAL GEOMBTEY. Peoposition VII. (56.) The parameter of any diameter is equal to four times the distance from the vertex of that diam.eter to the directrix, or four times the distance from the vertex to the focus. From the last Proposition we have ^_sin. a' T i' COS. a P^ sin.2 a 62- 1— sin.2a' Clearing of fractions and reducing, ,2„'_ P' but b'^=.2ap, from the equation of the curve. .-. sin.2 a'--£—. 1a-\-p And from the last Proposition, sm.^ 'CL sin.2 a .-. sin.2a'=^. P Equating these two values of sin.^ a', we have P _P 2a+p p' .■.p'=2a+p, and ^ 2p'=4(a+Jp). But (a+ip) =to the distance from the vertex A' to the directrix, or from the vertex to the focus. Hence the truth of the proposition is manifest. Peoposition Vin. (57.) The polar equation of the parabola, the pole be- ing at the focus, is r= P , 1 ±cos. 6 where p represents half the parameter, and 6 is the angle which the radius vector makes with the axis. ON THE PAEABOLA. 43 1. Let DB be the directrix, P any point in the curve. Draw the radius vector PP. Now, by the D definition, PP=DP, and DP= BR; hence FP=BR. Put the radius vector, FP=r, and the angle PFR=0. Then (Trig., Art. 5), FR=FP cos. B—r cos. 6. But BR=BF+FR, or r—p+r. cos. 6. B . >•= P 1— cos. 0" p If 6=0, cos. 6=1, and r—^= the radius vector is equal to infinity. IS'ow, as the angle 6 increases from zero, the radius vector marks points in the curve. When 6=90°, cos. 6=0, and r=p. At this point the radius vector becomes the ordinate at the focus, and is equal to half the parameter. If 6=180°, cos. 6= — 1, and rz=^p, which marks the vertex A. In a manner entirely similar, the radius vector will describe the points of the curve below the axis ; for when 0= 270°, cos. 0=0, and r—p; when 0=360°, cos. 0=1, and P r=-=co. 2. If we put the angle PFA=0, then FR= — n cos. 6, and r—p—r. cos. 6. P •'•** 1 + cos. 6" In this case the values of r begin at the vertex A, and increase from the left to the right, from zero to 360. Peoposition IX. (58.) 77ie area of any portion of a parabola is equal to two thirds of the rectangle described on its abscissa and ordinate. 'Let ADB be a segment of a parabola referred to rec- tangular axes. 44 ANALYTICAL GEOMBTEY. Bisect DB in E, and draw EQ parallel to DA ; join AB. Through Q draw QN perpendicular to AD ; then QN will be an ordinate to the diameter at the point N. Now, because NDEQ is a rectangle, QN is equal to ED=^DB; .•.QN^:BD^:1:4. But (Art. 40) ANiADxQN'iBD^; .-.AN: AD:: 1:4, or AN=:iAD; hence ND or QE=|AD. Again, since EG is par- aUel to DA, and BE=iBD, .•.EG=iAD; or N. D EG:GQ::2:1. Join AE, AQ, and QB ; then, because BD is bisected in E, the triangle ABE is equal to half the triangle ABD ; and since GQ=iGE, the triangle AGQ is equal to half the triangle AGE, and the triangle BGQ is equal to half the triangle BGE ; hence the ti-iangle ABQ=i the triangle ABE, and, consequently, the triangle ABQ =i the triangle ABD. Now, suppose BE and DE were each bisected, and from the points of bisection lines were drawn parallel to DA, these lines would evidently bisect BG and GA respectively. Then, by drawing lines from the points in which those parallel lines would cut the curve to the extremities of the chords BQ and QA, the sum of the triangles formed within the spaces BQ and QA will be equal to J the triangle ABQ, or -jlg-th of the triangle ABD. By bisecting the halves of BE and ED, and drawing lines as before parallel to AD, and joining the points of their intersections with the curve to the extremities of the chords, a series of triangles would be formed in the remaining spaces, the sum of which would be equal -to ^th of the triangle immediately preceding it, or equal to ■^th of the triangle ABD. Continue this bisection indefinitely, and we shall have the triangles filling the whole parabola, in which case the area of the parabola would ultimately be equal to ON THE ELLIPSE AND HYPERBOLA. 45 the sum of the areas of all the triangles thus formed within it. Put a=;the area of the A ABD. Then the area of the parabola is —a+ia+^a-\-^^+, etc., ad in/In., which is a geometrical progression whose first term is a, and ratio =:i, the sum of which (Alg., Art. 132) is _ a a _ -ri:;=T3|-^«- Hence the area of the parabola is equal to four thirds the area of the triangle ABD. But ttie area of the tri- angle ABD=iAD X BD. .-. area par. = § AD x BD. Q. E. D. (59.) On the Ellipse and Htpeebola. D^nitions. 1. An Ellipse is a plane curve, such that, if from any point in the curve two straight lines be drawn to two fixed points, the sum of these lines will be constantly equal to a given line. 2. The two fixed points are called the foci. If F' and P be two given fixed points, and A'A the given line, then, if P'P+FP be constantly equal to A'A for every position of the point P, X^ the curve A'BAB' will be an ellipse. 3. The distance of the point P from either fo- cus is called the radius vector. A:JShQ point C at the middle of the straight line join- in/theSfoci is called the centre of the ellipse. fe. Any "straight line drawn through the centre and limited by ihe curve is a diameter. ■%. The points. jn which any diameter meets the curve arfe called the vertices of that diameter. %. The diameter which passes through the foci is the 46 ANALYTICAL GEOMETET. transverse diameter, and the diameter perpendicular to this is the conjugate diameter. These are also called the major and m,inor axes. 8. A straight line which touches the curve in any point, hut which does not cut it, is called a tangent to the curve at that point. 9. A diameter drawn parallel to the tangent at the vertex of any diameter is called the conjugate diameter to the latter, and the two diameters are called a pair of conjugate diameters. 10. If, at the extremity of the parameter of the trans- verse diameter, a tangent be drawn to meet the axis of X, and at the point of intersection a line be drawn per- pendicular to that axis, the tangent is called the focal tangent, and the perpendicular line is called the directrix. 11. A line drawn perpendicular to the tangent at the point of tangency, and limited by the transverse axis, is called a normal. 12. The subtangent is the projection of the tangent on the axis of X, and 13. The subnormal is the projection of the normal on the same axis. Peoposition I. (60.) The equation of the ellipse referred to its centre and axes is where A and B are the semiaxes and x and y the general co-ordinates. Let F and P' be the two fixed points, or the foci. Join PP', and bi- sect PP' in C. Let P be any point in the curve, and join PP,F'P. Draw PR and CY perpendic- ular to ex. Let C be the origin, and CX, CY the axes of co-ordinates. Let the line to which PP-fP'P is constantly equal be de- noted by 2A, and the distance FP' by 2c. ON THE ELLIPSE AND HYPBEBOLA. 47 Then put CR=x, RP=2/, F'P=r', FP=r; then r'^=y^+(c+xy, (1) r^—i/^+{c—xy, (2) r'+r=2A. (3) Subtracting (2) from (1), we have r'^—r^—icx, or (r' -{-r) {r' —r) —4cx. But, because r'+r=2A, 2cx and /+r=;2A. Combining these, we have /=A+;^ ; .-. r'^^A^+1cx+^, ex . c^a;2 r—A—-^; .-.r^^A^— 2ca;+--^; .•.r'H»-^=2A2+-^. Now, by adding (1) and (2) together, we have r'2 + r2 = 2 (2/2 + ajz + c2) . Equate these two values of r'^+r^, and we obtain 2/2+a;2+c2=A2+^, or Ay+(A=-e2)£B2=A2(A2-e2). (4) Since r'+r or 2A is always greater than FP' or 2c, therefore A is always greater than 6, and hence the quan- tity A^— c^ is essentially positive. then AV+B2a;2=A2B2 (5) is the equation of the ellipse refei-red to its centre and axes, B2 or y'^^-^^iA^-x^). (6) Solving the equations for y and x in succession, we have y^+jVA^^^, {1) x=±^VW^\ (8) When x=0, y= ±B ; and when y=0, x= ±A. 48 ANALYTICAL GEOMETRY. Hence it appears that the curve cuts the axis of X at the points A to the right and A' to the left of the origin, at a distance from it equal to A, and that it cuts the axis of Y at a distance above and below the origin equal to B. Hence, also, the quantities A and B are the semiaxes. From equations (7) and (8) it is evident that the curve is symmetrical in regard to both axes. For we perceive that every value assumed for x gives two values for y numerically equal with opposite signs; and for every value we may assign to y, we shall have two values for X Numerically equal with opposite signs. (61.) And as BF is equal to BF', each being equal to A, it follows that the distance from the extremity of the conjugate axis to either foaus is equal to half the trans- verse axis. (62.)InEq.6, y^=-^^{A^-x'^). Let x=c=CF; B2 then 2/2=— (A2_c2). Now A2-c2=B2, by Eq. 5 ; .-. A2«/2=BS or Ay=B\ Hence B is a mean proportional between A and y. But y represents the ordinate at the focus. Putting the above into a proportion, we have A:B::B:y, or 2A:2B::2B:2y. (63.) That is, the double ordinate through the focus is a third proportional to the transverse and conjugate axes. But the double ordinate ^through the focus is called the parameter; hence the parameter of the trans- 2B2 verse axis is =~t-. (64.) The ratio of c to A is usually denoted by the symbol e, and is called the eccentricity of the ellipse. c Thus, ^=6; :.c=Ae, or c2=AV, and 2c=2Ae. But ON THE ELLIPSE AND HTPEEBOLA. 49 c'^^A^-B^; .•.AV=A2-B2: .-.B^r^AMl-e^), or B2 . ^ ■" A^=l-^\ Substituting this in Eq. 6, we have y2=(l-e2) (A2-a!2). We have also from the preceding ex (65.) Let us substitute e for -x, and then /=A+ea;, r=zA—ex. These last equations represent the distances from any point of the ellipse to either focus. The Htpeebola. (66.) A7i hyperbola is a plane curve, such that, if from any point in the curve two straight lines be drawn to two given fixed points, the difference of these two lines will constantly be equal to a given line. The two given fixed points are called the foci. Thus, if P' and F are the two fixed points, and P any point in the hyperbola, then will FT-FP=A'A, A'A being the given line, called the transverse axis. The middle point, C, of the straight line F'F, is called the centre of the hyperbola. Any straight line drawn through the centre, and term- inated by two opposite hyperbolas, is called a diameter, and the points in which it meets the hyperbolas are caUed the vertices of that diameter. The transverse di- ameter, when produced, passes through the foci, and the points in which it meets the curve are called the prin- cipal vertices of the hyperbola. The parameter of the 50 ANALYTICAL GEOMETET. transverse axis is the double ordinate -vrhicli passes through one of the foci. Peopositioit II. (eT.) The equation of the hyperbola referred to its cen- tre and axes is where A and B are the semi-axes, and x and y the gen- eral co-ordinates. Let F' and P be the foci, and bisect P'P in C, at which point draw the axis of Y perpendicular to P'X. Let P be any point of the curve, and draw PR perpendicular to the axis of X. Put 2A equal to the difference of the distances of the point P from the foci; and PE'=r', PP=>', and CF'=CP=c. Also, let x and y denote the co-ordinates of the point P. Then F'P2=PR2+F'RS or »-'2=y2^(a;-fc)2. (1) Again, FP2=PR2+FR2, or r2=y2+(a;-^c)2. (2) Adding (1) and (2), we have »-'2+r2=:2(y2+!BHc^)- (S) Subtracting (2) from (1), we have or (»■'+»') {r' —r)=i^cx. (4) But /— ?-=2A, 2ca: .■.r'+r=-^. By addition and subtraction, the last two equations give us /=A+|, (5) r=A-J. (6) Squaring these values of r' and r, and substituting in (3), we obtain A2+£^=yH!»Hc='; or, by reduction, A2i/2+(A2-c2)a!2=A2(A2-c2). (7) ON THE ELLIPSE AND HYPERBOLA. 51 No"w 2A must always be less than 2c, and therefore A^ ^ by x', y', and those of the point P" by cc", y". The equation of a line passing through these two points will be y-y"^^^{x-x"). (1) Since the points are on the curve, we shall have for the equations of the curves A2y'2±B2a;'2=±A2B2; (2) AV"^±B2a!"2z=±A2B2. (3) Subtracting (2) from (3), we have A2(2/"2-2/'2) ±B2(a:"2-a!'2) =0, or AHy"-\-y') {y"-y')±B%x"+x') {x"-x')=(i. y"-y' -- B^(ai"+i«') •'■ £b"-£B'~"^A2(2/"+2/')" Substituting this value in (1), the equation of the se- cant line becomes y-y"^::^E^^^ix-x"). But when the secant becomes a tangent the pomts coalesce, and then x'—x" and y'—y"; this reduces the last equation to ^"^ =^AY'^''~'"^• The term — ^ is the tangent of the angle which the 56 ANALYTICAL GEOMETET. tangent line makes with the axis of x ; the upper sign (— ) belonging to the ellipse, the lower sign (+) belong- ing to the hyperbola. Clearing of fractions, we have A2w"-A22/"2= -B2a!aJ"+BV^ or A22/2/"+B2a!£B"=A22/"2+B2a;"2=A2B2 ; .-. A2y2/"+B2£«"=A2B2, the equation of the tangent to the ellipse. Using the other sign, we have A^yy" -A^y"'^=&xx" -'Wx'"^, or A22/y"-B2aH!"=A2y"2-BV2= -A^B^ ; .■.A^yy"-Wxx" = -A^W, , the equation of a tangent line to tlie hyperbola. (77.) To find the point in which the tangent intersects the axis of x, make y—0 in the equation of the tangent line ; then for the ellipse we have which is equal to the distance from the centre of the ellipse to the point required ; and to find the length of the subtangent, we must subtract the abscissa of the point of tangenoy from this. That is, ^2 A2 y."2 subtangent=--^,—x = — X X (78.) As this expression for the subtangent is inde- pendent of the conjugate axis, the subtangent is there- fore the same for all ellipses having the same transverse diameter, and consequently belongs to the circle de- scribed upon the transverse axis. From this we are en- abled to draw a tan- gent to an ellipse at any given point. Let P be the given point, and on the transverse axis describe the cir- cle A'P'A. Through P draw the ordinate PR, and produce it to meet the circle in P'. From P' draw the tangent to ON THE ELLISPE AND HTPEEBOLA. B1 the circle, cutting the axis of a; in T ; join PT, which will be the tangent required. . (79.) To find the subtangent of the hyperbola : as the expression — ^ is the distance from the centre of the curve to the point in which the tangent cuts the axis of X, we must subtract this from the abscissa of the point of tangency. That is, sicbiangent=x" - A2 ,"2_A2 (80.) Since the normal is perpendicular to the tangent at the point of tangency, the tangent of the angle which it forms with the axis of x must be the reciprocal of tliat which the tangent line makes with the same axis, and also have a contrary sign. But the tangent of the angle which the tangent line • therefore the tan- makes with the axis of x is AY gent of the angle which the normal makes with the axis of X in the ellipse is AV B^a;"- Substituting this in (1), we have for the equation of the normal line to the ellipse. (81.) To find the subnor- mal, we must first find the point in which the normal cuts the axis of x. This is done by making y=Q ; then, by a little reduction, we have CN=a;= A=-B2 a A2 Now, subtracting this from CR, the abscissa of the point of tangency, which is represented by «", we shall have 42 7^2 -RB,-x"- . . x". A2 C2 58 ANALYTICAL GEOMETEY. NR=- (82.) If we put A2- A2 -B2 -=subnormal. A2 e\ we shall have CN=:eV. And adding c to this, which is =Ae, we get F']Sr=Ae+eV =e(A+ex") for B^ cos.^ a'. The denominator (A^ sin.^a'— B^ cos.^a') is therefore positive; and as the numei-ator is negative, .•. CE^ must be negative. If we put CD^=A'2, and CE^= — B'^, we shall have from (3) ON THE ELLIPSE AND HTPEEBOLA. 65 A2B2 „ , A2B2 „ , ,,^„ -B'2" A'2 .-. A'2y,2 -B'2a;,2 = - A'^B'^ ; or, makiDg the co-ordinates general, A'V-B'2a!2= -A'2B'2, which is the equation of the hyperbola referred to its centre and conjugate diameters. (94.) We see that the equation of the ellipse, as well as the equation of the hyperbola, when referred to the centre and conjugate diameters, is of the same form as that when referred to the centre and axes ; it follows, therefore, that every value of a; will give two values of y numerically equal, but with opposite signs ; or, if B' were real, every value of y would give two equal values of x with opposite signs ; hence the curves are symmetrical with respect to the diameter which it intersects. That is, either diameter will bisect all chords drawn parallel to the other and terminated by the citrve. Peoposition VIII. (95.) The squares of the ordinates to either of the con- jugate diameters of an ellipse or hyperbola are to each other as the rectangles of the corresponding segments from the foot of the ordinates respectively to the vertices of the diameter. The equation of the curves is A'^y^dzB'^'x^- ± A'2B'2, where the upper signs belong to the ellipse, and the lower signs to the hyperbola. If we designate any two ordi- nates by y' and y", the corre- sponding abscissas by x', a;", we shall have for the ellipse y'2 _ (A'+jg') (A'-a;') 2/"2~(A+a;") (A'-a")' and for the hyperbola y'^_ {x'+A'){x'-A') y'^~{x"+A'}{x"-A'y (96.) The parameter of any diameter is a third pro- 66 ANALYTICAL GEOMETET. portional to the diameter and its conjugate. Thus, if P denote the parameter of the -o i' ■> „,-„ 2 \ r ( +C sin.2 a y '^' (.+B (cos.2 a-sm.^ a) j ( A sin.2 a ) f n /./^o 1 +«,0+Bsin.acos.a +2,. D^°:«4 ( +C COS.^a ) ^ ' +'«^{+Ers.„}+F=o- (2) We can give to a. such a value as will render the co- efficient of a:,2/; equal to zero. Thus, 2(A— C) sin. a cos. a+B (cos.^ a— sin.^ a)=0 ; Trig. (18) and (20), (A— C) sin. 2a-l-B cos. 2a— 0', .-.tan. 2a=— - — — A— C Hence, if we make a of such a value that tan.2a=-^_^, the term containing x y, will vanish, and we shall have Ay+C'a!2+D'y+E'a;+F=0. (3) We may simplify this farther by transferring. the ori- gin to a point whose co-ordinates are a and 5, by putting a;=a+a;,, and y=.'b-\-y^. Then, substituting these in (3), we shall have AV;^+C V+(2A'6+D')2/,+ (2C'a-f E>,-f A'52 ' +CV I -f D'6 > =0. +E'a (4) +F Now, if we make 2A'5+D'=0, and 2C'a+E'=0, 80 ANALYTICAL GEOMETEY. the terms containing x, and y, will vanish, and we shall have E' ^=-2(7' 2A" Using these values of a and b, and putting R=A'S2+CV+D'5+E'a+F, equation (4) will become of the form P2/2+Q£B2=-R. (5) 1°. If P, Q, and R have the same sign, the locus is im- possible. 2°. If P and Q have the same sign, and R an opposite sign, the curve is an ellipse, and the semiaxes are ^and./=^ Q V P 3°. If P=Q, the curve is a circle. ^ 4°. If P and Q have different signs, the curve is an hyperbola. All these remarks are made under the supposition that R is not equal to zero. 6°. If R=0, and P and Q have the same sign, each of the variables must be equal to zero, and the equation characterizes a point, and that point is the origin of co- ordinates. 6°. If P and Q have different signs, and R=0, it char- acterizes two straight lines, represented by the equation y=±^J=^.x. (B.) 1£ in (3) C was zero, or, which is the same thing, if the term containing C were wanting, we should have R= A'52+D'&+E'a+F ; and, making R=0, we would find A'b^+T>'b+¥ «-- g7 ; and we also have b= — — 2A'" Substituting these in equation (3), it would become of the form GENERAL EQUATION OP THE SECOND DEGBEE. 81 Ay+E'a;=0, or 2/2= -_a!, the equation of a, parabola. (C.) Because the equation of the circle when referred to the extremity of a diameter is and that of the parabola referred to the vertex is and the ellipse y^— — {2Ax—x^), and the hyperbola y^=-—(2Ax+x'^), •we may put one general equation to characterize them all. Thus y^=mx-j-nx^, in which m is the parameter of the curve, and n the square of the ratio of the semiaxes. "We might, at first, have transferred the origin to a point whose co-ordinates are a and b, and then assigned such values to a and b as would have caused the co- efficients of a;^ and y^ to vanish. In that case we would V. r Ail. * 2AE— BD have found that a= h: B2-4AC 2CD-BE B2-4AC- We shall see that the curves represented by (1) may be divided into two classes, 1°. Those which have a centre, and 2°. Those which generally do not have a centre. In the first case, the value of B^ — 4AC does not be- come equal to zero, but in the second case it does. For if B2— 4AC=0, then a—ao and b=!x>. Removing the terms which have vanished, (1) reduces to A2/2-i-B£i;y+Ca;2-f R=0, (2) where R=A52+Ba6+Ca2+D5+Ea+F. Now it is evident that if (2) is satisfied by any ^os«- tive values of x and y, it is also satisfied by the same, values taken negatively. Hence the new origin of co-ordinates must be at the centre of the curve. D2 82 ANALYTICAL GEOMETEY. CHAPTER m. ANALYTICAL GEOMETEY OF THEEE DIMENSIONS. (115.) We have seen that the position of a point on a plane is determined when its distances from two straight lines drawn in that plane are known ; in a manner very similar, we shall now proceed to show that the position of a point in space will be determined when its distances from three planes are known. (1 16.) Take the three planes Z TAX, XAZ, YAZ, which are ' ^Y> supposed to be at right angles ^,'''' with each other, and whose ■^i_ ^^ -y intersections are the three ^^^^'^ straight lines AX, AY, and ^^ \ AZ. These three straight lines y''^ Z are each perpendicular to the other two. The plane YAX, called the plane yx, is sup- posed to be horizontal, or the plane of the floor on which the student is standing; the planes XAZ and YAZ, called the plane tea and yz respectively, are both vertical, and may be represented by any two adjacent walls of the room. A is the origin, and the three lines AX, AY, and AZ are the co-ordinate aaes, called the axis ofx, the axis of y, and the axis of z respectively. The co-ordinate planes are indefinite, and hence they will divide all space into eight equal parts or triedral angles, each having its vertex at A. Four of these angles are above the horizon- tal plane YAX, and four below it. Hence it is necessa- ry, when we oljtain the equation of a point in space, to express analytically in which of these eight angles the point is situated. In order to do this, we must apply the principles laid down in the Algebra, Art. 3, which we also applied to distances from points and straight lines in Analytical Geometry of two dimensions: that '\%, if dis- tances reckoned along AX to the right of Abe consid- ered positive, distances along the same line to the left of ANAlTTICAl GEOMETRY OF a?HEEE DIMENSIONS. 83 A mtest be regarded as negative. The same principle must govern in the two other co-ordinate axes. (117.) The eight triedral angles are numbered as fol- lows : TAX is called the 1st angle. YAX' « 2d " X'AY' « 3d " Y'AX « 4th " The fifth angle is directly under the first, the sixth under the second, the seventh under the third, and the eighth under the fourth. (118.) If we call the distances of a point in space from the three co-ordinate planes x', y', z', and suppose these distances known, then the position of the point will be completely determined, provided that we have previous- ly ascertained in what triedral angle the point is situ- ated. Let us suppose that the point is in the first angle. On the three co-ordinate axes take the distances AD = x', AF=y', AB = »', and through the points D, F, B pass planes parallel to the co- ordinate planes. Then, since the plane DG has all its points situated at the distance x' from the plane YAZ, and the plane FGhas all its points at a distance y' from the plane XAZ, it follows that all the points of the straight line GE, which is the common intersection of these two planes, have exclusively the property of being at the same distances from the planes YAZ, XAZ. Therefore the point to be determined must be situated in the straight line GE. Again, the point sought must be situated somewhere in the third plane GHBC, which is parallel to XAY, since all the points in this plane have exclusively the property of being at the distance z' from the plane XAY. Hence the point to be determined must be the point G, in which the third plane- cuts the common intersection of the first two, and thus its position is entirely determined. The co-ordinates of the point G are AD, DE, EG. (119.) If the point be in the first angle, we shall then have for its equations x=x', in the 2d angle, 03=— as' « 3d x=—x' " 4th x=x'. " 5th x—x', « 6th x=—x' « nh. x=-x' « 8th x=x'. 84 ANALTTICAIi 6E0MBTBY. y=y', s=«V y=—y\ «=«V y=-y', z=z'; y=±y', e=-z'; y=y\ »=-sV y=—y\ »=-»'/ , 2/=-y', »=-»'._ _ (120.) There are also some particular positions of the point which it is proper to notice. For example, in or- der to express that a point is situated in the plane xy, we must make its distance from that plane zero ; then we shall have for the equations of a point in the plane xy, 85=83', y~y\ s=0. Similarly, a point on the axis of x, CB=aj', y=0, s=0. (121.) If from any point in space a straight line he drawn perpendicular to a given plane, the foot of the perpendicular is called the projection of the given point upon the given plane. (122.) In like manner, if from every point of any line in space, whether straight or curved, perpendiculars he drawn to any given plane, the line traced out by the feet of the perpendiculars upon the given plane is called the projection of the given line upon the given plane. (123.) If we say that a;=a3', y=y', z=z' are the equa- tions of the point G, then the co-ordinates of the point E are 83=83', y=y', z=0. (1) The co-ordinates of the point C are 83=83', y=0, g=»', (2). which give for the co-ordinates of the point H 83=0, y=y', s=s'. (3) Hence, if the projection of a point G upon two of the co-ordinate planes be given, the third projection will also necessarily be known. (124.) When the co-ordinate planes are not at right an- gles to each other, the axes AX, AY, AZ make oblique angles with each other, and are called oblique axes ; the equations of apoint, however, are a3=£B', y=^y', e=z'; but 8! , y', and z' express distances reckoned along lines re- spectively parallel to these oblique axes. ANALTTICAl GBOSIETEY OF THBBE DIMENSIONS. 85 In Other respects, the remarks made with regard to rectangular axes are applicable to oblique axes also. Proposition I. Pboblem. (125.) To find the distance between two points in space when the co-ordinates of these points are known. Q" z p' y y •n' / / Q / A E s / y / 3/ -^ Let Q, Q', Q" be the projections of one of the points on the three co-ordinate planes, and P, P', P" be the pro- jections of the other point. Then AR=a;', RQ=y', RQ'=s' are the co-ordinates of the point Q, and AS^a;", SP=y", and SP'=s" the co-ordinates of the point P. Now, in the triangle Qi^P, right-angled at jo, we have QP2=Qi9H^P'= («"-»')'+ (y"-2/')'- But QP is the base of a right-angled triangle whose perpendicular is p'Y\ and hypothenuse the line whose length is required. Putting D equal to the length of this line, we shall have Vi^^ix - -x'Y^ {y"-y'Y+ {z"-zy ; .-.D =V(x"-x'y+{y"-y'y+{z"-z'y. _ (126.) If the point Q be at the origin, its co-ordinates become each equal to zero, and the above expression be- comes D=Va;"2+2/"H3"'. (A) (127.) From which it appears that the square of the 86 ANALYTICAL GEOMETEY. diagonal of a rectangvlar paraUelopipedon is equal to the sum of the squares of the three edges. ; (128.) From this we may easily determine a relation which exists between the cosines of the angles which a straight line makes with the co-ordinate axes. Thus, let j-^the line passing through the origin ; X, Y, Z the angles which this line makes with the axes of x, y, and z respectively. Then the co-ordinates of the other extremity of this line being x", y", z", we shall have x"z=r cos. X, y"~r. cos. Y, z"=r. cos. Z, .Squaring these equations and adding, them together, we have a."2^.y3_|.g"2_y2(cos.2 X-f C0S.2 Y-Hcos,2 Z). But from (A) we have the square of the length of a line equal to x^^+y"^+z"^, and, as wo represented this length by r, therefore ^2_^2(coS.2 X+C0S.2 Y-)-cos.2 Z), or cos.^ X+C0S.2 Y+c&s.^ Z=l. Pboposition II. To find the equations of a straight line in space, (129.) The projections of a straight line on two planes are sufficient to determine its position, and hence it fol- lows that a straight line in space will be determined an- alytically if we know the equations of its projections upon two of the three co-ordinate planes. We generally consider the projections of the straight line on the planes XZ and YZ; and since these two planes have AZ for their common axis, this line is re- garded in each of the planes as the axis of abscissas. Therefore AX is the axis of ordinates in the plane XZ, and AY is the axis of ordinates in the plane YZ. Let DE be the projection of a straight line on the co- ordinate plane XZ, and KG its projection on the plane YZ. Through the origin A draw AP and AL respect- ively parallel to DE and KG. Put BP=a3, AB=s, and a=the tangent of the angle EHZ, orPAB. Then, in the triangle BAP, we have 5E=tan. BAP, ANAITTICAL GEOMETRY OE THKEE DIMENSIONS. 87 or X -=a; .: x=dz. But the point E in the line ED is farther from the axis of Z than P is by the distance PE or AD, which may- be called a ; hence the equation of the line ED becomes x—az+a. (1) In a manner entirely similar, we find for the equation of the line GK. These, being the equations of the projections of the line on the co-ordinate planes XZ and YZ, are called the equations of the line in space. Now the equation x=:az-{-a expresses not only the relation between the co-ordinates of any point in the line DE, but it also expresses the relation between the co- ordinates of any point in the plane drawn through the line in space perpendicular to the plane XZ. In like manner, the equation belongs not only to the straight line GK, but also to all the points of the plane drawn through the line in space perpendicular to the plane '^Z. This system of equa- tions, therefore, holds good for all the points of the straight line in space, which is the intersection of the two planes perpendicular to the planes XZ and TZ, and holds good for the points of this straight line alone. Therefore these equations are the equations of the straight line itself. If we eliminate z from these equations, we shall have y-(i=!i{x-a) CO (3) for the equation of the projection on the plane XY. (130.) "When the straight line passes through the or- igin, the equations which we obtained first, viz., 88 ANALYTICAL 6E0METET. x—az, y=.U, are the equations of that line. (131.) If the straight line be situated in one of the co- ordinate planes, as, for instance, in the plane xz, we shall have y=^^, and a;=:as+«. That is, in this case, 5=0, and /3=0, which is evident, for the projection of the straight line on the plane yz will coincide with AZ. (132.) When the constants a, 6, a, /3 are given, the po- sition of the straight line is completely determined. In order to obtain its different points, we must give a suc- cession of particular values to one of the variables in each of the equations £(;=«» -|- a, 2/=5s+./3, by means of which we shall obtain corresponding values for the two other variables. Pboposition JII. (133.) To find the equations of a straight line in space which passes, throvgh one given point, and then for a straight line through two given points. Let the co-ordinates of the point be x\ y\ z'. The equations to the straight line will be of the form "^ x—az-\-a, (1) 2/=5«-f/3. (2) Now, since the line passes through this point, we shall have x'=a^-\-a, (3) y'^bz'+fi. (4) And, since these equations hold good together for the straight line required, we shall have x—x'=a{z—z'), (5) ^ v~y'=H^-^'h , (6) the equations of a line m space which passes through a given point. Again, let the co-ordinates of one given point be x', y\ z', and the co-ordinates of the other given point be x", y", z". Now, since it passes through this second point, we must have AKALTTICAL GEOMETET OP THEBB DIMENSIONS. 89 a!"=as"+a, C!) y"^bz"+li. M Subtracting (1) from (3) and (8) from (4), we oHtain x'-x'^=a{z'-z")i . „_x'-x" 2/'-2/"=S(^'-s"); •■■*=fe?- (10) Substituting these values of a and b in (5) and (6), we have s — z and y-y'= ^-^'' {e-z'), (12) — z — z \ which are the equations of a straight line in space pass- ing through two given points. Peoposition IV. ' (134.) Through a given point without a straight line in space, to draw a straight line parallel to the given line. Let the equations of the given straight line be and a;', y\ z' the co-ordinates of the given point. The equations of the required straight line will be of the form 83— a;'=A(3— s'), y-y'=B(2_2'), where A and B are quantities to be determined. Since the straight lines are to be parallel, the planes which project them respectively upon the planes of £cg and yz must also be parallel ; hence the intersections of these parallel planes with the co-ordinate planes must be par- allel, or, in other words, the projections of the straight lines must be parallel. .-. A=a, andB=:5, which give for the equations of the required line a:— «'=«(«— s'), 90 ANALYTICAL GEOMETEY. Proposition V. (135.) Given the equations of two straight lines in space, to determine the relation which must exiit between the constants, in order that the two lines may intersect, and to find the co-ordinates of the point of intersection. Let x=az-\-a, CV\ y=bz+l3, hS x^a'z+a', (3) y=b'z+fi', (4) be the equations of the two given straight lines. If these straight lines intersect each other in space, their equations must hold good together at the point of intersection. Therefore, eliminating x, y, z, we find the relation („_„') (J_6')^(/3_^') («_«'); and, unless this equation of condition be satisfied, the lin«s do not intersect ; if it holds good, then the point of intersection has for its co-ordinates and a^a' b-V a;=- b-V a— a Peoposition VI. (136.) Griven the equations of two straight lines in space which intersect, to find the angle contained between them. Let the equations of the given lines be x=az->ra, y=bz+(i, (1\ x=a,z+a„ yz=zb,z+(i^, (2) A being the origin of co-ordi- nates. Through A draw two straight lines, AP', AP" re- spectively, parallel to the given Y straight lines. Then the angle P'AP" will be equal to the required angle. Put y,^=^W, a;2'=«,V, ^2'=^^; and a;,a;2=aa,a,S2, y^y^-hh,z^^. Adding, we have ^■?^-y?+^?—^?{o? -f 53 +1) =.r\ xi+y^+zl=z^^{a^+b;^+l)=rli__ .•.r=3^Vl+«^+5^ and »•^=^2-^/l-|-a,^^-V• Substituting these in (7), we obtain / ^ l+aa,+bb, V cos. V= , ,- VT+a2+52 V 1 -f a/ + 5^2 (13T.) In Eq. (A), let V=90° ; then cos. V=0, and which is the equation of condition for two straight lines in space to be perpendicular to each other. (138.) Again, if we make V=:0, cos. V=l, and the two straight lines become jpafallelto each other, and equa- tion (A) becomes l+aa^+bbi ■ Vl+a2+6Vl+a/-fJ/~ ' which, being cleared of fractions, squared, and reduced, becomes {a,-ay+{b,-hY+{ab-aPY=0. (A) 92 ANALYTICAL GEOMETRY. Since each, of the terms is a square, and on that ac- count essentially positive, the. equation can only be satis- fied when the terms are separately equal to zero. These conditions give The projections of the lines on each of the co-ordinate planes are therefore parallel to each other. Peoposition VII. (139.) To find the angle which either of these straight lines in space makes with each of the axes of co-ordinates. Let the angle which the line AP' makes with the axis of X be denoted by X, the angle which it makes with the axis of 2/ be denoted by Y, and the angle which it makes with the axis of z be denoted by Z. Then Bviti x=ir. cos. X J :. COS. X=-. r y=r. cos. T; .: COS. Y^y r' z—r. COS. Z; .: COS. Z=? ^2_a.2_|.j^2^g2_ z-^{l+a^+b% or r=z-\/l-\-a'^-\-b'^; X .: COS. X= — ■ — . 2-v/l + a2+62 But x=zae; a .: COS. X= ■ — Vl+a^+h^ b Similarly, cos. Y=— f======, yl+a^+o^ COS. Z=— 7==^=. Vl+a2+62 (140.) Since a;2+2/Hs^=»"^(oos.2 X+C0S.2 Y-(-cos.2 Z), and a;2-|-2/2-|_g2_^2^ .-. cos.2 X+ C0S.2 Y-f C0S.2 Z = 1 ; which shows that the sum of the squares of the cosines of the angles which a straight line in space forms with the three axes is equal to unity. OF THE PLANE. 93 Proposition Vlll. (141.) To find the angle included between two lines in space given by their equations in terms of the angles which each of the straight lines makes with the axes of co-ordinates. Let the angles which r makes with the axes AX, AY, AZ, be denoted by X, Y, Z, and the angles which r, makes with the same axes be denoted by X,, Y„ Z, respectively. Then, as before, ' ^ cos.V=^i^±M2±fi^ rr, Bat x,=r. cos. X, and X2=r, cos. X^, y=r. cos. Y, yj^r, cos. Y,, e^=r. cos. Z, 22='"; °os. Z^. .-. cos. Y=:cos. X cos. X^+cos. Y cos. Y,+ COS. Z cos. Z . That is (142.) ITie cosine of the angle included between two lines in space is equal to the sum of the rectangles of the cosines of the angles which the lines in space form with the co-ordinate axes. Of the Plane. (143.) TTie equation of a plane is an equation which expresses the relations between the co-ordinates of every point of the plane. Pecfposition IX. (144.) To find the equation of a plane. A straight line is said to be perpendicular to a plane when it is perpendicular to all the straight lines in the plane which pass through the point in which it meets the plane. Hence a plane may be generated by the mo- tion of a straight line round a point in another atraight line to which it is perpendicular. Let a;=as+a, y=5»+/3, (1) be the equations of a given line, and a', 2/', a' 94 ANALTTICAL GBOMETET. the co-ordinates of the point in this hne through which the generating line passes. Then its equations will be x—x'=a{z—z'), (2) y-y'=.h{z-z'). (3) And the equations of the perpendicular passing through the given point wUl be l'^b,{z-z!i. (5) (6) y-y From (4) we have a^ — x—x z—z" From (5) we have i,=- — %. h) But the equation of condition for two lines to be at right angles to each other is l-fTaa,+65,=0. (8) Substituting the values of a,, 5,, found in (6) and (7), we obtain «-«' . z y-y'_n l + a. % + bA :.z-]rax+by—{z'+ax'-{-by')=0. But a, b, x', y', z' a,re known quantities, and we may therefore represent s'+acc'+Sy' by a single letter, c. Doing this, we have g+acc+Sy— -0=0. (145.) The equation of the plane, which is an equation of the first degree cohtairiing three variables. (146.) The, lines in which a plane intersects the co-or- dinate planes are called the traces of the plane. We can find these traces by combining the equation of the plane with each of the equations of the co-ordinate planes. * ' At every point in the plane xz, y is equal to zero ; if, then, we make y=0 in the equation of the plane, we shall have x^-l.z+1 a a for the equation of the trace BC on the plane xz. In this equation — is the 5 a OF THE PLANE. 95 tangent of the angle which the trace BC makes with the axis of Z, and - is the distance AB. a Similarly, y=—\z+^ o 6' is the equation of the trace DC on the plane yz. (147.) Now, when the line which was perpendicular to the generating line was projected on the plane xz, that projection made with the axis of z an angle whose tan- gent was ay and we find that the trace BC of the plane on the same co-ordinate plane makes with the same axis, z, an angle whose tangent is — -. Hence the product of these two tangents is equal to minus unity, and there- fore the projection of the line is perpendicular to the trace. The same may he shown for the other traces. (148.) Therefore, if a line in space be perpendicular to a plane, the projections of the line will be respectively perpendicular to the traces of the plane. (149.) The most general form of the equation of a plane is Ax+By+Qz+Ti^O. Peoposition X. (150.) To find thi equation of a plane which shall pass through three gi^en points. Let k', y', z'; x", y", z"; and x'", y'", z'", be the co-ordinates of the three given points. The equation will be of the form z=Ax+By+G. (1) And, since it passes through a point whose co-ordinates are x', y', »', these co-ordinates must satisfy (Eq. 1). .•.s'=Aa!'+By'+C. (2) Subtracting (1) from (2), z'-z=A{x-x')+B{y-y'), (3) which is the equation of a plane passing through one given point. The constants A and B being arbitrary, the problem will be indeterminate, as it should be, for any number of planes may be passed through one given point. Let the plane pass through the second point whose 96 ANALTTICAl GEOMETET., co-ordinates are a", y", »", then these must satisfy the equation, and we shall have a"=Aa;"+B2/"+C. (4) We may eliminate two of the constants by (1), (2), (4). One, howeygi', will still remain, and the problem will be indetermmate, for any number of planes may be passed through two given points. Finally, let the plane pass through the third point also, the co-ordinates being a;'", y"', z , and we shall have z"'=Ax"'+By"'+G. (5) The three constants may then be eliminated between equations (1), (2), (4), (5), and the result wiUbe the equa- tion of the only plane that can pass through the three given points. IVoblems. 1. Find the equation of a plane which shall pass through the three points whose co-ordinates are x' = 1, y' =-2, z' = 2; x" = 0, y" = 4, a" =—5; x"'=-2, y"'=. 1, g"'= 0. If we substitute the given co-ordinates of the points in the general equation s=Aa;-f-By+C, we shall have the three equationsj 2= A— 2B-fO, (1) —5= 4B+0, (2) 0=— 2A+B+0, (3) from which we have A= — , B=— — , C= — 5' 15' 15 Substituting these in the general equation, we shall have 15z= — 9x—lQy+l, the equation of the required plane. 2. Find the equation of a plane which shall pass through the three points whose co-ordinates are aj' = 2, y' =-1, z' = 3; x" = 3, y" =—2, z" = 5: x"'=-3, y"'= 0, z"'=-2. 3. Find the equation of a plane which shall pass through the three points whose co-ordinates are OF THE PLANE. 97 x' = 2, y' =-1, S' = 3; x" — 3, 2/"=— 2, 3"= 5; a!"'=-7, y"'= 4, s"'=:-l. Proposition XI. (151.) To find the equations of the intersection of two planes. Let the equations of the two planes be z=Ax+By+C, s=A'a!+B'2/+C\ Then, since these equations hold good together for the straight line which is their common intersection, if we eliminate z, we shall have the equation of the projection of that intersection on the plane xy ,• that is,. CA-A')£t;+(B-B')2/+C-C'=0. We may find, in a manner entirely similar, the equa- tion of the projection of the intersection on the co-ordi- nate plane yz, and also on the co-ordinate plane xz. Peoposition XII. (152.) To find the conditions which will cause a straight line and a plane to be parallel or coincide. Let the equation of the straight line be x=az-\-a., 2/=:Ss+j3, and the equation of the plane z=Ax+'By^Q. In the equation of the plane, subh titute for x and y their values az-\-a and bz-\-(i, and we shall have {Aa+Bb-l)z+{Aa+Bfi+C)=0; (A) • „_ Aa+B/3-fC "^~ Aa+B&— 1* Hence, if the straight line and plane have only one point in common, that is, if the line pierces the plane, the co-ordinates of the common point will satisfy the equations of the line and plane, and we would be able to determine its co-ordinates ; but if the straight line be altogether in the given plane, the equation of condition (A) must hold good, whatever may be the value of z. Hence the two parts of the equation must be independent of each other, and we shall have E 98 A2IALYTICAL GEOMETEY. Aa+B6— 1=0, and Aa+B/3+C=0, the. equations of condition required. If the straight line be merely parallel to the given plane, if we move them in a direction parallel to their original position until they reach the origin, the plane and straight line will coincide ; hence the above equa- tions must be satisfied on the supposition that a, and /3, and C are each equal to zero/ therefore Aa+Bb—1—0 will be the equation of condition which must he satisfied in order that a straight line and a plane may be parallel. Peoposition XIII. (153.) To find the conditions which will cause a straight line to be perpendicular to a plane. If a straight line be perpendicular to a plane, the pro- jections of the line and the traces of the plane will be respectively perpendicular to each other. Let the equations of the given plane and straight line be z—Ax+By-if-G, ■' (V\ X—az+a, (21 y^bz+p. (3) Eq. 2 is the equation of the projection of the given straight line on the co-ordinate plane 852, and (3) is the projection on yz. 1 C The trace of the plane on xz is x=—s— — (a\ A ■ A" ^ ' ■ 1 C The trace of the plane on yz is y^—z . fg\ But, since the projections of the line must be perpen- dicular to the traces of the plane, we shall have ax^+l=0; .•. A= —a, and B=—b, which are the required conditions. Peoposition XIV. (154.) To draw from a givenpoint a line perpendicu- lar to a given plane, and to find the length of the perpen- dicular. OF THE PLANE. 99 Let the equation of the plane be z=Ax+By+G, (1) and let tc', y\ s' be the co-ordinates of the given point. Then the equations of the required line must be of the form £B— £»;'=:«(«— s'), And, since it is perpendicular to the plane, a=— A, 5=— B, and therefore the equations of the. perpendicular are a— a;'+A(«— s')=0, 2/-2/'+B(2-»')=0. If we denote the co-ordinates of the point in which the line pierces the plane by »", y", s", equations (1), (2), (3) will become s"=Aa!"-|-B2/"+C, (4) x-x"=-K{z-z"\ (5) y-y"=-B(s-s"). (6) Combining (1) and (4), we have s-s"=A(a;-a;")+B(y-2/")- W K to the second member of (7) we add C and subtract its equal (s"— Ace"— By"), it will take the form g-3"=A(a;-a!")4-B(y-2/") + C-s"-fAa;"+By'. (8) Now, equations (5), (6), (8) will hold good together. .•.g-g"=-A2(3-s")-B2(s-s")-(-C-8"+Aa;"+By", or (s-s") (l+A2+B2) = C-s"+Aa;"+B2/". . „ „„_ C-g"+Aa!"-i-By" . l+A2-fB2 . ,, ,,"- B(C^i."-fA«"+By") ••^ ^ - 1+A2+B2 and ,_,„^-A(C-."+^A."+By") And the distance Y)=^/ {x-'siiy-\-(y-y"Y^{z-z"f\ z-Ax"~By"-G "^~ -v/l+AHBa ' the distance required. Peoposition XV. (155.) To find theangh included between two planes. Let the equations of the planes be 100 ANALYTICAL GBOMETEY. 2=Aa;+By+C, (1) z=A,x+B,y+G, (2) (156.) If we let fall from the origin two straight lines perpendicular on these planes, the angles contained by the straight lines will be equal to the angle contained by the planes. Let the equations to these straight lines be The angle between them is given by the equation cos. Y:= , 2— , ' . (3) Vl+a2+6Vl+a/+6/ But, in order that the straight lines may be perpen- dicular to the given planes, we must have A.— —a, B=— 5, A,=:— a„ B,= — 6,. Substituting these values, we have the cosine of the angle between the two planes COS V= l+AA.+BB. Vl+A2+BVl+A,2+B/ '' ' If we wish to find the angle which any plane mates with the co-ordinate planes, we have only to suppose that one of the above planes assumes in succession the position of the diflferent co-ordinate planes : thus, that (Eq. 2) is the plane oixy; then its equation becomes, by making s=0, A,a!+B,y=::0. And, since this is true for all values of x and y, we shall have A,=0,B,=0. -Designating the angles formed by the planes by V, we have cos.V''=-7= • (5) If we designate by V" and V" the angles which the given plane makes with the planes xz, yz, we shall obtfiin °°^-^'-Vl+l+B^' (^) OF THE PLANE. 101 If we square the three . values of cos. V, cos. Y", COS. Y'", and add them together, we find COS.2V'+COS.2V" + COS.2V'"=1. (16'7.) That is, the sum of the squares of the cosines of the three angles which a plane forms with the three co-ordinate planes is equal to radius square or unity. (158.) If we now suppose the first plane to coincide in succession with each of the co-ordinate planes, and des- ignate by V], Vj, V3 the angles formed by the second plane with the co-ordinate planes, we shall have COS. "Vi=-7 . , " (8) cos. V5=— ;===, (9) cos.V3=-7=====. (10) If we multiply (5) by (8), (6) by (9), (1) by (10), mem- ber by member, add the results, and compare the equa- tion with (4), we shall find that cos.V=cos.V cos.Vj-f cos.V'cos.Vj-f cos.V" cos.Vg, which expresses the cosine of the angle included between two planes in terms- of the angles which the planes form with the co-ordinate axes. DIFFERENTIAL CALCULUS. CHAPTER I. DEFINITIONS AND INTEODtTCTOET EBMAEKS. — DIFFER- ENTIATION OF ALGEBEAIC FUNCTIONS. DIFFEEENTIAL COEFFICIENT OF A FUNCTION OF A FUNCTION. 1. In the Differential Calculus we shall meet with quantities which always retain the same values during the same investigation, while others are subject to cer- tain laws of change, by which they may take, in succes- sion, an infinite number of different values without chang- ing the form of the expression into which they enter. 2. The former of these are called constant quantities, and are represented by the first letters of the alphabet. The latter are called variable quantities, and are repre- sented by the final letters of the alphabet. 3. Some variable quantities are confined within certain limits, while others vary without limit : thus, the cosine of an arc commencing at zero degrees is equal to the ra- dius, but as the arc increases the cosine diminishes ; when the arc becomes 90° the cosine is equal to zero/ and after the arc passes that point the cosine becomes negative, reaching its greatest negative value at 180°, being at that point equal to minus the radius. At 270° it again becomes equal to zero; and when the arc is more than 2Y0°, the cosine becomes positive, and continues positive till its value becomes zero again. On the other hand, the tangent commencing at zero degrees is equal to zero, but as the arc increases the tan- gent increases also. When the arc becomes 90° the tan- gent becomes infinite, and after the arc passes that point the tangent becomes negative, and diminishes in value till the arc becomes 180°: here the tangent is equal to zero. Passing this point, the tangent again becomes pos- 104 DIFPEEEHTIAI, CALCULUS. itive, etc. Hence we see that the cosine varies from plus radius to minus radius, and the tangent varies fvom plus infinity to minus infinity. 4. It is important for the student to understand what is meant by the term limit, or limiting value, and pass- ing to the limit. If a regular polygon be inscribed in a circle, and we inscribe another polygon having double the number of sides, the perimeter of the second polygon will approach more nearly in value to the circumference of the circle than that of the first ; and if we continue the process of inscribing polygons, each having double the number of sides of the other, the perimeters of these polygons will approach nearer and nearer to the circum- ference of the circle, and the last may be made to dijffer from it by less than any assignable quantity. Hence the circle is said to be the limit of the inscribed polygon. 6. If we have the series TV+Toir+ io^ou + stc., to inr Jlmty,'we readily perceive that the sum of the series ap- proaches in value to -J-, but can never equal it while the number of terms is finite. We therefore say that ^ is the limit of that series. 6. If" u—ax, the product of a and x becomes less and less as x dimin- isljes in value. Hence M = when X becomes equal to zero ; or, the limit of ax is zero when the variable x is zero. 1. If «=-, X the quotient becomes greater as x diminishes in value. Hence w=infinity when X becomes equal to zero. We may illustrate this in the following manner : As division is a repeated subtraction of equal quantities from a given sum, suppose we have ten dollars : we say, " two would go into ten five times 4" or, we could go five times and take two dollars at each time. Now, how often could we go and take nothing ? Evidently an infinite number of times. The truth of the proposition may be proved thus; Divide :; — - ; we shall have INTEODUCTOEY EEMAEKS. 105 Y^—=l-\-x+x^+x^+etc., to infinity; and making x=l, Q=l + 1 + 1 + 1+ etc., to infinity, which is an infinite quantity. a •■0 • 8. Since the value of a fraction diminishes as the de- nominator increases, it follows that, as the denominator approaches in value toward infinity, the value of the frac- tion approaches zero ; hence the limit of a X is zero when x is equal to infinity. 9. As the denominator of a fraction becomes nearer and nearer in value to the numerator, the fraction itself approximates in value to unity ; and when the denomi- nator becomes equal to the numerator, the value is unity. That IS, —=^a°=l, when «=m. 10. By the term limit, therefore, we mean that value toward which a variable quantity is approaching, hut which it never reaches so long as the number of terms is finite. We shall illustrate the foregoing by a few examples. Ex. 1. Fin d^ the limit of the value of the fraction 2x+5 first, when x=0 When 03=0, fraction is f. Again, when nominator by x. 3a;+6' , and then when a5=a). 2a; and dx both become zero, and the a; =06, divide both numerator and de- and we have 5 ^+x 6 3+- E 2 106 DIFFERENTIAL CALCULUS. 5 6 Now, when x=cd, - and - each become zero, and the ' ^ X X fraction is f . Ex. 2. What is the limit to which the ratio of 2xh+h^ to h approaches, as h diminishes and ultimately becomes equal to zero ? That is, to find the value of h when A=0.' By dividing both numerator and denominator by A, we have 2a;+A r~' and when A=0, the ratio is 29; : 1. Ex. 3. What is the limit of 3a;— 7 7a!— 8 when a;=0 and a;=infinity? \\. As the arc of a circle diminishes, and ultimately/ becomes equal to zero, the ratio of the sine, tangent, or chord of the arc to the arc itself is a ratio of equality. Let B'B be the side of a regular polygon inscribed in a circle, and D'D the corresponding side of the regular polygon circumscribed' about the circle. Assuming the radius of the circle equal to unity, we shall have BE=sin. BCA; .-. B'B=2 sin. BCA. DA=tan. BCA ; .-. D'D =2 tan. BCA. Now if n=the number of sides of the polygon, the angle BCA will be equal to -, and B'B =2 sin.-, D'D =2 tan.-; therefore the perimeter of the inscribed polygon will be "■ 2n sin. -, w LIMITING VALUE OP A PEACTION. 107 and the perimeter of the ch-cumscribing polygon will be In tail. -. n Now if we call p the perimeter of the inscribed and P the perimeter of the circumscribed polygon, we shall have "■ sin. - p n tc "p— =cos. -; tan.— n and as n, the number of sides, increases indefinitely, the angle subtended by one of these sides may become less than any assignable angle whatever, and therefore the IT cos. - becomes ultimately cos. 0=1. P ^ .•.p=l, orp=P. Hence an wth part of the perimeter of the polygon is equal to the wth part of the circumference of the circle. That is, TT 27r TT 2 sin. -= — —2 tan. -, n n TV , TT TT IT TT TV IT or sm. — =-, tan. -=;- : also chd. -=-. 71 rv n n' n n ^ "■ Put -=0. n sin. tan. 6 chd. 9 Then -^=1,-^=1,-^=1. 12. From what precedes, it will be understood that by the term limit, or limiting value, we mean a fixed quan- tity to which a variable quantity continually approaches, by making certain suppositions, and which may be made to approach so near to it that the difference shall be less than any assignable quantity. These suppositions are, generally, that one of the quantities becomes indefinitely great or indefinitely small ; that is, becomes infinite or nothing, or that some of the variables become equal to constant quantities. 13. When we speak of the limiting value of a fraction, we do not mean the limit of the numerator divided by 108 DIFFEEENTIAl CAICITLUS. tJie limit of the denominator, but the limit of the quotient resulting from actually dividing the numerator by the denominator. Thus, if we wish to find the limit of the a? — afi fraction , when the variable x approaches the value of a, and ultimately becomes equal to it, we would not say a^ -a^ a— a 0' for this is not the limit we are seeking, a^—x'' («+£») {a—x) But — a+x; and when x=a, the value or limit is 2a. 14. If from the point A two secants, AB, AF, be drawn, cutting the circle, then (Geom., B. H., Theor. XXI.) ABxAD=AFxAE. " Put AE:=a;, and EF=A/ then AF=a!+A, and ABxAD=a!(a;+A). Wow let AF move to the right till it takes the position of AG-, a tangent to the circle; then EF, or A, becomes ■ zero, and AB x AD^te^ . or AB -.xy.x: AD. Hence, by this application of the theory of limits we demonstrate the property, that if from a point without a circle a tangent and, secant he drawn, the tangent loill he a mean proportional hetween the secant and its ex- ternal segment (Geom., B. II., Theor. XXI., Cor. 1). Examples, a^-^a? Ex. 4. What is the limiting value of— — — when x=a? Ans. 3a2. CB+1 Ex. 5. What is the limiting value of „ .. when a;= od? Ans. ^. hx-V-h^ Ex. 6. What is the limiting value of — j; — when A=:0? A Ans. X. INCREASE OF VAEIABLE QUANTITIES. 10& ^ .^, . , ,. . . , „ Zx^h+ZxhP' + h^ Ex. 1. What IS the hmiting value of r TvhenA=0? Ans. Zx^. ax Ex. 8. What is the limit of the value of — ; — when x a-\-x becomes infinite ? B . Ex. 9. What is the limiting value of xv 2Aa;+CB2 when X becomes infinite ? -V2M+x^^~x\pEj^\. A A V a; 2A Now, when x becomes infinite, the quantity 0; B therefore the limiting value=x- ^"-'^ 15. If a variable quantity increase uniformly, then other quantities depending on this and constant quanti- ties may either increase uniformly, or according to any variable law whatever. 1°. Thus, if a variable quantity, x, increase uniformly, then 2a!, 3a;, or any given number of times x, will in- crease uniformly. 2°. Let x increase uniformly by one, then its successive values will be x+1, x-\-2, a!+3, etc. Or, if we take ax and add a constant quantity to it, the values will go on increasing uniformly. Thus, ax-\-h, ax+2b, ax+Sb, etc. 16. But it most frequently happen^ that the quantities we have to consider do not increase uniformly, while -the independent variable increases at a uniform rate. For, let X increase uniformly by one, so as to become cc-f 1, a!+2, aj-|-3, etc., then the values of the squares of these quantities do not increase uniformly. By squaring these quantitieSj the successive values will be (a;+l)^=a;=+2a:+l, (a;+2}2=:a;2+4a;+4, (a;+3)2=a!2+6a;+9, etc., etc., in which the successive differences are no DIPPEEBNTIAI, CALCULUS. 2a!+l, 2a:+3, 2a!+5, etc., ■which go on constantly increasing. 17. When quantities are so connected that the vahie of each is dependent upon that of the others, each is said to be a function of th^ others (Alg., Art. 169). Thus the area of a square depending on the length of its side is said to be & function of the side. The algebraic expressions , a+bx^ ax\ Va^+x^, ^^ , etc., depending for their value on that which we assign to x, are all functions of x. By the term function of x, then, we mean any alge- braic expression into which x enters in combination with constant quantities. If the expression be put equal to a single letter, as u= Va^+x% X is called the independent variable, and u, or its equal, the dependent variable. 18. The relation between a function and its variable is generally expressed thus : u=f(x), u=f'(x), u-,p{x), where each expression is read, " m is equal to a function of a;." 19. Injunctions are either algebraic or transcendental; these, again, are either explicit or implicit, increasing or decreasing. 20. An algebraic function is one in which the relation between it and its variable is expressed by the sum or difference, product or quotient, roots or powers of the variables, the roots or powers being constant quantities. Thus, .. w= (Sai^+ca;^— ace)*, are algebraic functions. 21. A transcendental function is one in which the re- lation between the function and its variable can not be expressed in the usual algebraic terms. Thus, DEFINITIONS OF FUNCTIONS. Ill M = sin. X, t«=tan. X, M=sec. X, which are called circular functions ; or i«=log. X, which is a logarithmic function ; or u=a', which is called an exponential function. 22. An explicit function is one in which the value of the function is directly expressed in terms of the variable, as u=^m,x-\-n. 23. An implicit function is one in which the value of the function is not directly expressed in terms of its variable, as au'^+2t(x=bx\ 24. An increasing function is one which increases as its variable increases, and decreases as its variable de- creases, as u=ax^+l>x, u=sm. X. 25. A decreasing function is one which decreases as its variable increases, and increases as its variable de- creases, as ^ u=—, x' M = COS. X. 26. The mutual dependence of one variable on another is generally expressed under the form /(a',.3/)=0, in which x and y are functions of each other. 27. Every equation containing two or more quantities expresses a relation among those quantities. If, there- fore, we make any one of these quantities dependent on the others, the quantity thus made dependent may be called z, function of the others. Take, for instance, the equation of the circle when the origin of co-ordinates is at the centre. As R is constant, we may make y to depend on cc, or we may make x dependent on y . Thu s, y-riz-^ W—x^ or a;=±-\/R'^— 2/^. 112 DIFPEEENTIAL CALCULUS. It is evident that x and y both vary froin H-R to — R, and that for every value we assign to the one between these limits, there will be two values of the other numer- ically equal, with contrary signs. 28. When one or more quantities vary, any function of these quantities varies also, the quantity and manner of this variation being dependent on the nature of the function. This kind of dependent variation is the first idea the student has to master, and he will be greatly facilitated in his progress if he can acquire that knowl- edge from analytical relations alone. Let us take y=2x — 5 ; then, supposing x to vary through all magnitudes from — 00 to +00, y will also vary through all magnitudes from — CO to +03. By taking values of x that have very small differences, the mind can easily acquire the idea of the variation of X and y when passing through magnitudes which have insensible differences. 29. A variable quantity, in changing its sign, must pass through zero or infinity^ Thus, in the equation y=2a;-5, while X varies by insensible differences from 1 to 3, y varies from —3 to +1, and must, during this variation, pass through zero. This takes place when x—2.b. 30. The converse of this proposition, however, does not follow. That is, a variable function, in passing through zero or infinity, does not necessarily change its sign. For, if we have the function y={2x-5y, it could never change its sign ; but while x varies from — CO to +C0, y will decrease from +oo to 0, then in- crease from to +00. The value of cb that makes y=Q is, as above, equal 2.5. 31. The whole theory of the variation of variables and their functions constitutes the science called the Dif- ferential Calculus. 32. One great object of the Differential Calculus, and an object which is continually occm-ring, is to find the limit of the ratio of the increment of the function to that of its variable. 33. This limiting ratio of the increment of the function DIFFEEBNTIATION OF FUNCTIONS. 113 to that of its variable is called the differential coefficient, because it is the coeflSoient of the differential of the in- dependent variable. 34. Thus, if /(k) denotes any function of x, and we give to the independent variable x any arbitrary incre- ment, A, we shall have /(a;+A) as the same function of (a;+/i), and the increment of the function is evidently /0«+A)-/(ce). Now the ratio of the increment of the function to that of the variable is f{x^K)-f{x) h ' Then, if we let h diminish without limit, this ratio is the differential coefficient of /{x) with respect to its varia- ble X. 35. The symbol used to represent the differential co- efficient off(x) is d.fjx) dx ' or, if we put Uz=f{x), then the symbol is du dx' The numerator of the fraction (1) represents, as we have said before, the increment of the function ; the de- nominator is the increment of the variable. Hence the fraction expresses the ratio of the increment of the func- tion to that of the variable ; and when we pass to the limit by making h=0, the numerator becomes the dif- ferential of the function, which we represent by df(x), and the denominator becomes the differential of the va- riable, which we represent by dx. In both cases the letter c? is a mere symbol, denoting " the differential of." 36. Let u—f{^—ax. (1) Now, if we give to x an increment. A, u will change its value, and its new value may be expressed by u' . We shall then have v:—f{x+h)—a{x^K). (2) And, by subtracting (1) from (2), lit DIFFERENTIAL CALCULUS. v! —u =f{x + A) — /(«) = ah. Ox; by dividing by h, u'—u f{x+h)-f{x) _ h - h "*• f{x-^K\—f{x) Here the member v '■ — , which we have also u'—u represented by — -, — , expresses the ratio of the incre- ment of the function to that of its variable. Hence, if we pass to the limit by making A=0, we shall have du df(x\ The first of these forms, being more easily printed, will frequently be used instead of the second. Multiplying both members by dx, we shall have du—adx. (4) du df{^ 37. In (3) the first member, 3- or , , is the symbol of the difierential coeflScient of u with respect to a;, and the second member, a, is the difierential coefficient itself. 38. Again: suppose f{x)^=ax^, and let x become x+h; then f{x+h)=alx+h)\ =zaa?+ 2axh+ah? ; an d f{x-{-h) —/{x) ^'i.axh+ah?, f(x+h)-f(x) or ^v -r ^ J\ ' =2ax+ah. Here the first member expresses the ratio of the incre- ment of the function to that of its variable. Therefore, by passing to the limit, we have df(x) which is the difierential coefficient of ax\ Hence the differential of a constant multiplied hy a variable functio7i is equal to the constant into the differ- ential of the function. 39. Let /(«')=i' and give to x an increment, h / then DIFFEKENTIATION OF FUNCTIONS. 115 ah -/(c«+A)-/(.)=-^^:p^, f[x-\-K)-f{x) a ^"^ h a;(a;+A)' and passing to the limit df(x) _ a dx ~ a!^' a which is the differential coefficient of -. X 40. The differential of a constant connected hy the sign plus or minus to a vanable is always zero. Suppose ii=.<3?aa^ and let x become x-\-h, then u will become m'/ .-. M'=(x+A)2±a=a!2+2a;/t+A2d=a, and m'— w=2a;/i+A2, u' —u or -^=2x+A; du dx 41. From this we perceive that the differential of a variable functioti and the differential of the same func- tion increased or diminished by a constant quantity are both equal. 42. The differential of the sum or difference of any number of functions depending on the same variable is equal to the sum or difference of their differentials taken separately. Suppose u=f{x)+f'{x)-f"{x), where /(«),/' (a;), and /"(a;) are different functions of a;, and give to x an increment, h; then u will become u'. .: u'=f{x+h) +f'{x+h) -f"{x+h, u' -u=f{x+h) -fix) +f'{x+h) -fix) - f'\x+h)+f"{x). And, by dividing by h, u'-u fjx+h) -fix) _. f'{x+h) -f ix) _ h ~ h + h f"ix+h)-f"ix) h 116 DIFFEEENTIAL CALCULUS. The first member of this equation expresses the ratio of the increment of the whole function to that of its va- riable, and the terms in the second member express the ratio of the increment of each function, respectively, to that of its variable. Therefore, by passing to the limit, ■we have du dfjx) df'{x) _ df"{x) dx~ dx '^ dx dx ' or du = df{x) + df {x)—df" (x) . 43. The differential of the product of two functions depending on the same variable is equal to the first into the differential of the second plus the second into the dif- ferential of the first. Let u=f{x)f'{x), _ (1) each being difierent functions of x, and give to x an in- crement. Ay then u'=f(x+hY'{x+h\ f{x+hy"{x+h)-f{x)f'{x\. Subtract and add the quantity /"(a;)/'' (a-j-A) from and and u'—u—f{x+h)f'{x+h)—f{x\ to the second member of (3), we shall have u' -u=f{x+h)f{x+h) -f{x)f'{x) -f{x)f'{x+h) + f{x)f'{x+h) ; or, by factoring, u'—u— f{x+h){f{x+h)-f{x)] +f{x){f{x+h)-f{x)}. Now divide by h, Passing to the limit, we have ^^ -p'l \ ^f^ ft \ ^Vi^ doT^f ^^' • ~dx^J^^> ■ ~^r' or du=f'{x) . df{x) -\-f{x) . df'{x). If we divide this last equation by the first, member by member, we shall have du_df(x) df(x) 4:4. Hence the differential of the product of two func- tions divided hy their product is equal to the sum of the quotients which arises by dividing the differential of each function by its function. This holds good for the product of any number of functions. DIFFBEENTIATION OF FUNCTIONS. 117 Let u—wys, where w, y, z are each functions of x. Put wy=v ; then M=t)«. Then, by the preceding, du dv dz u~ V ' z'' dv dw dy V ~~ w ~^ y' dv Substitute for — its value, we have du dw dy dz Clearing this of fractions, asd taking wyz for the value of u, we shall have du—d. {wyz) =::yzdw+ wzdy + wydz. 45. Hence the differential of the product of any nuni- ber of functions is equal to the sum of the products of the differential of each by the product of all the others. 46. The differential of a fraction is equal to the de- nominator into the differential of the numerator minus the numerator into the differential of the denominator^ divided by the square of the denominator. f{x) Suppose ''"/>)' and let x become x+h. fix+h) Then u'=^^^^^ f'(x+hy and u -u-jrjp^^-jrj^, f'{x)f{x+h)-f{x)f'{x^h) f'Wi^+h) ■ . If we add to and subtract from the numerator of this fraction the quantity /(a;}/" (a;), we shall have f{x+hY{^)-f(x)f{^+h)+fW{x)-f{xY{x) " -«= f^{x)f'{x+h) • Put the numerator into fab-tors, , f{x) \A«^+h) -fix) ] -f{x) \f'{x+h) -fix) \ 118 DIFFEKBNTIAL CALCULUS. Divide both members by A, u'-u J ^^'- h -J^^l • h h - f'W{^+h) Here the first member expresses the ratio of the in- crement of the function to that of the variable, and, pass- ing to the limit, f{x+h)-f{x) dj\x) j^ becomes -^-, f'{x+h)--f'(x) , df\x) and •■^-^ h becomes -=^^, and the denominator of the fraction, which is f'{x)f'{x+h), becomes {fxY; "dx~ {f'xY f'{x)df{x)-f{x)df{x) or au- {f'{x)Y 4'7. Turning to Art. 39, we see that the differential co- efficient of - is — -2- Now — is a decreasing function, and its differential coefficient is negative. We may easily show that the differential coeffUiient of every de- creasing function is always negative. Let M=/'(a;), f{x) being a decreasing fuij^tion. Then let x become x+h. Then ▼ ll.'=zf(x+h\ and W— M=/(a^A)— /(aj), u'-u f(^h)-f{x) *''' h ="~^"1 • Now as w>m', the second member must be essentially negative ; du_ df{x) " dx dx ' " 48. 27ie differential of any constant power of a varia- ble is equal to the exponent of the power into the variable raised to a power less by unity than the primitive expo- nent, and that product by the differential of the variable. DIFFEEENTIATION OF FUNCTIONS. 119 Let «(=»;'', and give to x an increment,'A ; we shall then have , u'=(x+h)\ and u'—u^^x+hY—x", Now the difference between the same powers of any two quantities is always divisible by the difference of those quantities (Alg., Art. 144). .-. u'—ii={x+h—x\ yix+h)''-^+ {x-\-hy-^x+. . , .a;"-'}; or, dividing by h, ^^-^= \x+hY-^+{x+hY-'x+ .... SB"-'] , where the second member contains n terms. Hence, passing to the limit, du .•. du^nx'^~^dx. 49. The differential of any constant power of a func- tion is equal to the exponent multiplied by the function raised to a power less by unity than the primitive expo- nent, and then by tlie differential of th^ function. Let M=(/(a;))°, where y(a;) denotes any function of a;, and n is a positive whole num.ber. Let a; become £8+ ^. Then u'={f{x+h)Y, and M'-M=(/(a!+«)»-(/(a;))°. As in the preceding article, separate the second mem- ber into two factors (Alg., Art. 144). .•. u' — M= l/(^+'^)-/W 1 \f{'^+hf-'+fW{x+hY-'+. . ./w-i: -r^. .1. 1 T U'—U Dividing by h, — r — = ■^"'^^^"•^^"'^ \f{x+hY-'+f{x)f{x+hY-'+.. .f{xY-'\ . The first mem'Ser expresses the ratio of the increment of the^owe?" of th^ifunction to the increment of the va- riable, and therefore, in passing to the limit, becomes the differential coefficient of the power of the function. The expression — ^^ r- — '■ expresses the ratio of the increment oi ^e first power of the function to that 120 DIFFKBENTIAL CALCULUS. of the variable, and hence becomes the differential co- efficient of the function. And as there are n terras in the second factor, we shall have, by making A=0, w(/lCa;))"'~' as the sum of those terms. .•.^=:«.(/Ca;))°-'..-^, or du^n . {f(x)y-^d . f{x). We have thus proved the rak for the differentiation of a positive integral power of a function. Again: let w=(/Ca;))-°, where n is integral ; then 1 or «(^a;))°=l. The first member is the product of two variable func- tions, and (by Art. 43) we have ud(fCx))'+(f{x))'du=0, the dif. of 1 being =0. But the differential of (f(x)Y=n{fCx'))-^^df{x) ; im{fax-)Y-'df(x) = -n{JXx^^'df{x), by substituting for ii its value {f(.x))~°. m Again: let t«=(/lC.¥))°; -Aen M»=(/(a;))°'.- AnUy Art. 49, ^^ m^-^duz=m{/(.x))'°-'d/{x), ■■■^^=~-{A^)f^~'d.J{x). If !!!=i, then the differential of (/(x))* will be That is, the differential of a radical fiinction of the second degree is eqiml to the differential of the functi2 du x{4:a^+x^) Ex.79. M^aa;"— 5a;'"-'+ca;°'-''. „ «+« Ex. 80. u= — r-=. Ex. 81. u—~. X Ex. 82. M= Ex. 83. U—Zxy. Ex.84. M=:(a+a;2)". Ex. 85. M=:a£C°+'- Ex. 86. «<=^- Ex. 87. M=(a£c''+a;"")3. Ex. 88. u,— j. a(a+a;)° Ex. 89. w=— . yz Ex.90. M= ^ „ ^ ■- o Ex. 91. u=3x^y. 2a;* Ex. 92. M=-5 ^. Ex. 93. M=(a!3+a) (Sai^+J). Ex. 94. u={a+x) {b+x) {o+x). Ex.95. M= y x+ V 1 —x^ „ a;^ Ex.96.«=^j^^. 128 DIFFEEEOTIAL CAICULUS. Ex.9?. u={x^+{a+x^)i]^. Ex. 98. M= 3. {i-x^y Ex.99. M Ex, 100. w: 85-\/ax3_a^ 3a Ex.101. U-- —. (x—a)^ Ex.102. u=— — —Y-. (a— X* Ex.103. M=(a2_£e2)2. 6 , Ex.104. u=-V2ax—x^. Ex.105. u={2n-l)x"'+'^-{2n+l)x^' Ex. 106. u=ax^'—bx''+c. „ ^ a+a; Ex.107. «=^.^ Ex. 108. ^=-3+— . a^ ' as j^ ^M Ex.109. M=^s:^. (?M 2wia'°a!'"-' •'• dx~ {x'^+ar)^' „ 3a!2-a2 Ex.110. «=p:p^. du 12x{a^—a?) ■'•dx~ (a^+x^)* • a+5bx Ex.111. u= , I . ^5 - a+3bx^ Ex.112. M=(^^j^,)3- DIFFBEBNTIATIOK OF FUNCTIONS. 129 du_ 125^a;3 Ex.113. u=afc^—bx^+cx '—ex '^. Ex.114, u={a+hx^+cx^i. Ex.115. u=i{a+x) (a—xy. du a—3x "dx~2Va—x Ex.116. M=(a2+a!2)(a2-£B2)*. du x{a'^—3x^) '■ dx~ -y/a^—x^ ' Ex.117. M=(2a2+3a!2) (a^-K^p. dx Ex.118.-w=(2V«+Vk) W"-+Vx)}- du_ 1 3Va+2Vx Ex.119. u={bx—3a)(a+bx)^. du 4b^x "^^ 3(a+6a;)^' Ex.120. w=(3Sa!— 2a)(a+5a5F- du Ex.121. w=(5Sa;2— 2a)(a+6«2p. .■.—^3h¥xHa-^lx^Y. dx Ex.122. u={8a^—4ahx+3b^x^) {a+bx)^. du ISS^a;^ --i4i^xVa+bx. Ex.123. M= " dx X iy/a+bx Va -bx^' du a " dx~ (a-bx^Y F2 130 DIFFEEEJSTIAL CALCULUS. Ex.124. M=:^^£!=^ X Ex.125. M= dx cB^-y/a^—aj^' 3 x^ Va—x ^'*' 2{a-xy Ex.126. 11— - , . c?M 4ac— 62 "^^ 2(a+Ja;+ca;2)"^ Ex. 127. «=^=l)_fe!±l)! SB+l ^^<_ 2(g!°+a;*— a;3+l) "^ ■ (a!+ 1)2(3!*+ 1)*" Ex.128, u—i du 2x "^"^ (l-a;2)(l-a;*)^' Ex.129. «=:^^I±Z±:v^5^ Vl+x^— Vl_a;2" ^dx''^dx^\.2^dx^\.'i.S^' ' 134 DIFFERENTIAL CALCULUS. J^x y)^u _y+___ ____+, etc., in which u represents the value of the function when y=o. Let u' be a function of a!+y, which we will suppose to be developed into a series, and arranged according to the powers of y, so that we have M'=/(a;+2/)=A„+Ai2/+A2y2+A32/3+,etc., (1) in which Af,, Aj, Aj, etc., are independent of y, but functions of as, and dependent on all the constants which enter into the function. We are required to find such values for A^, Aj, etc., as shall render the development true for all possible values that may be attributed to x and y. By differentiating (1) under the supposition that x is variable and y constant, we have -5-=-j-°+-T-'y+-5-y +-T-^y +, etc ax ax ax ax ax And differentiating the same under the supposition that y varies and x is constant, we have fit/ ^=A,+2A^+3Ariy^+iA,y^+, etc. dy But, by Art. 54, du' _du' dx dy Hence we must have A,+2A2y+3A32/3+4A42/^+, etc. And since the coefficients are independent of y, and the equality exists whatever be the value of y, we shall have, by the Theory of Indeterminate Coefficients CAlff., Art. 144), ^ ® dAi _ dx dA dx etc. = etc. TAYLOES THEOEEM. 135 N"ow, make 2/=0 in equation (1), andf{x+y) will be- come /"(a;), which we may denote by u. Hence Ao=u. Substituting this value of Aq in (2), we have A =—. dx Substituting this value of Aj in (3), we have 1 d^u A,=. »2- In like manner, 1.2 dx^ A 1 d^u 1.2.3 dx' Finally, substituting these values of A^, Aj, etc., in (1), we have tatloe's theoeem: «'=/(a.+y)=^.+ff y+f^ f-+pL ^+, etc. dx dx^ 1 . 2 dx^ 1.2.3 By making -—-——-—, we would obtain the develop- ocx o/y ment oifix—y). ' 56. Taylor's Theorem may be applied to the develop- ment of the second state of any function of the form ■u-f{x), when X becomes a;+A. For, by substituting A for y, we have , , du I , d'^u h? , d^u h^ , . u=u+—- h+—— +-T-, +, etc., dx dx^ 1.2 dx^ 1.2.3 ' , die r , d^u li?- , d^u h? , .^ or w — M= — h-\-— -F-:r-;; +, etc. ; dx dx^ 1.2 dx^ 1.2.3 il-u^da (d^ _J_ d^ _h_ ^^\j^ h dx \dx^ 1.2 dx^ 1.2. 3 ) It is evident that h may be made so small that the term /^'J_+^_^+,etc.V \dx^ 1.2 &3 1.2.3 / shall be less than any assignable quantity, and there- fore less than — , dx 136 DIFPEEESTTIAL CALCULUS. or — >^(-:; +-^— ; 4-5610.1. cfo'^ \c?a;2 1.2 &J3 1,2.3 ' ) Or, if we multiply both members of tbis inequality by h, we shall have ^ h>(^ JL+^ -H- + , etcV dx \dx^l.2 dx^ 1.2.3 J 57. Hence, if toe have a series expressed in the ascend- ing powers of the variable, we may assign to that varia- ble so small a value that the first term shall he greater than the sum of all the remaining terms, and the sign of the series wiU depend upon the sign of its first term. 58. In the foregoing demonstration of Taylor's The- orem, we have supposed all the functions to be continu- ous; but if the function to be developed be infinite for values of the variable lying between certain limits, the demonstration is not valid. 59. A function of a variable is said to be c07itinuous between certain limits of the variable when the function changes gradually as the variable passes from one limit to the other ; that is, an indefinitely small change in the variable causes an indefinitely small change in the func- tion. Hence a continuous function can never become infinite between the limits for which it is continuous, inasmuch as an indefinitely small change in the variable would then cause a change in the function not indefin- -itely small. Maclauein's Theoeem. 60. If u represents a function of x which can be de- veloped in a series of positive ascending powers of that variable, then we shall have for that development M=/(a:)=U+U'a;+U"-^+U"'_^-f-, etc., in which TJ, XT', U", etc., represent the values that the function of x and its successive differential coefficients respectively assume when x is made equal to sero. Let u=f{x), and put ^t^Ao-fAiaj+AjtB^-l-Ajaj^-f , etc. ; (1) then ^=Ai+2A2a!+3A3a!2+4Aia;3+,etc., (2) DEVELOPMENT OP PUHCTIONS, ETC. 137 ^=2A2+2.3A3a!4-3.4Aia!2+,etc., (3) |^=2.3A3+2.3.4A^9)+,etc. (4) If in (1), 42), (3), (4), etc., we make a;=0, then M=:Ao=TJ, ^"-A -TT' g=.2A,=U";.-.A,=iU". ^=2.3A3=U"': .-.As^J-U'". Substituting these values of Aq, Aj, etc., in (1), we have Maolaxtein's Theoeem. t.=/(x)=U+U'c«+U"^+U"'^+, etc. Development op Functions op two oe moeb Vaei- ables when each ebceivbs an Incebmbnt. 61. In Art. 50 we have defined successive differential co^kients; that is, that the second differential coefficient of a function is the differential coefficient of the differ- ential coefficient of that function. The differential co- efficient of the second differential coefficient is called the third differential coefficient of that function, and so on. We shall now explain another notation. If we have a function of two variables, as «=/(«, 2/)i (1) we may suppose one of them to remain constant, and differentiate the function with respect to the other. Thus, if we suppose y to remain constant and x to vary, the differential coefficient will be £=/'(«=, 2/); (2) and if we suppose x to remain constant and y to vary, the differential coefficient will be %=f\^,y)- (3) 138 DIPFEEENTIAL CALCULUS. The differential coefficients in (2) and (3) are called partial differential coefficients. That in (2) is the par- tial differential coefficient of m regarded as a function of X, and that in (3) is the partial differential coefficient of M regarded as a function of y. 62. If we multiply both members of (2) by (&;, and both members of (3) by dy^ we shall have -^dx=f'{x, y)dx, (4) and -;^dy=f"{x, y)dy. (5) du du The expressions -j-dx^ndi -f-dy are called partial differentials ; that in (4) is the partial differential of u as a function of x, and that in (5) is the partial differen- tial of M,as a function of y. 63. If we differentiate (2) under the supposition that X is variable and y constant, we shall have -^=f"'{x,y), or ^=f"'{o=,y); (6) and if we differentiate (2) under the supposition that ce is constant and y remains variable, we shall have d(p) -^=rix,y); dhc The first member of (6) denotes that the function ic has been differentiated twice, both with respect to x; and the first member of (7) denotes that the same func- tion u has been differentiated twice, once with respect to x, anfl once with respect to y. If we differentiate (7) again, regarding x as the vari- able, we shall have ^2^=/'(a',2/); (8) DEVELOPMENT OF FUNCTIONS, ETC. 139 and, regarding y as the variable, ^^=/"(«',2/). (9) la (8) the symbol of the diiferential coefEcient denotes that the function has been differentiated three times: twice with respect to x, and once with respect to y. In (9) the symbol of the differential coefficient denotes that the function has been differentiated three times; once with respect to a;, and twice with respect to y. And generally denotes that the function u has been differentiated m-\-n times ; m times with respect to a, and n times with re- spect to y. 64. Let u=f{x, y), and give to x an increment, h, and let y be considered constant ; then, by Taylor's Theorem, „, , ^ duh d'^u h^ 'dhi h? , ^ f{x+h, y)^u+^^ i+^TT2+d^ 170+' ^^°-' (^) du d'^u n • n ■■ -I ■■ where u, -r-, -^, etc., are functions of x and y, and de- pendent on the constants that enter thef{x, y). Each of these. expressions must be developed by the same theorem, by giving to y an increment, 7c. Then ■ du d^ii .P dhi It? , u will become u+^Ic+^^ 1:2+%^ 17273+' "*°-' du du dht dhc h^ ^^"^ ^ ^'11 1.^°°™^ ^+&^^-+^^ l72 + ' «*"■' -. <^^*' •■,■, , <^^*' ^^^ ■, ^*^ ^^ and ^ will become ^+^^^+^5^2 0+ , etc., d^u . d^u d^u , d^u I? Substituting these values of m, -5-, etc., in equation (1), we shall have 140 DIFFERENTIAL CALCULUS. du d^u d^u hj^ +^^+^^^* +^^ 172 +' ^^'^•' d^u A^ d^u h^k (2) +^17273 + ' ^^«- The general term being dafdy° '(1.2.. .m)(l .2...w)" If in (1) we had made y to vary first, by giving to it an increment, k, we would have had, by Taylor's Theorem, du d^u ! ^""^ ^^—~^' TKAUSCENDENTAL FUNCTIONS. 147 Substituting these, we have , dx dx x.z.y X log. X log.^ a' du 1 *"^ dx~x log. X log.2 a;' Ex. 145. If M=log.* X, du 1 then dx~x log. £B log.^ X log.3 a;' From which it is evident that if M==log.° x, du_ i dx~x log. X log.2 a; (log.)°~'fl3" / a: \ du w Ex, , 146. u=l.\ ,-^- =\. dx x{a?'+x'^)' du 1 «*+a;* (?M a* Ex. 147. «=Z.(a,+ Vl+^). .-. ^=:;7Y^ Ex. 148. u=l Ex. 149. u=l. Ex. 150. M=Z. a^~x^ x^(a—x) X du 1 \ x+^/\+x^' " ^'^ a: Vi+aj^' a; (?M a ■«— V«^— a:^' "*^* aiVa^ -a;'' /»•+ V*"^— ?/\ du Ir ( ^ Ex. 151. M=Z.|^-— ^^===1 .-. ^= -— ^=^ (V'f+a;+ -s/r—xX ,— 7=)- V^+a;— yr— a;/ «a; xVf^ — a;'^ 70. To find the differential of an exponential /unction. Let u=^a', where a is a constant quantity. J'aking the logarithms of both members in the Na- perian system, we shall have log. u=x log. «/ and, differentiating by (Art. 69), -—=&;. log. «, 148 DIFFEEENTIAL CALCULUS. or du=u. log. adx. Substituting for u its value a', we obtain d.a'^a^ log. adx. That is, the differential of a constant raised to a va- riable eaeponent is equal to the quantity itself into the Naperian logarithm of the constant into the differential of its exponent. Taking the successive differential coefficients, we have du ^=a'log.a, ^=a-(log.an ^=«'(l0g.«)3, d^u ^=a^{log.ay. Applying Maclaurin's Theorem, we shall have, by making a;=0, U=l, U'=log. a, U"=(log. ay, U"'=(Iog. 0)% U"=(log.«)*; .-. a'= 1 +a;.log.a+i(a!.log. a)2+— (a; log. «)'+, etc. (A) If we make x=l, we shall have a=l+log. a+i(log. ay+j-^{log. ay+, etc. (B) In (Eq. A), by changing a into the Naperian base e, we have e'=l+x+ix^+-^-^^+,etc., because the log. e=l. 11. Now if a!=l, we obtain 6=1 + 1+^+^7273+ 1.2.3. 1+' etc.=2.|1828+, which is the base of the Naperian system of logarithms. 72. As the logarithms of the same number, taken in two different systems, are to each other as the moduli of those systems, therefore. As log. e in the Naperian system : log. e in the common system :: modulus of the Naperian system : modulus of the common system. TEANSCENDENTAL FUNCTIOKS, 149 But the log. of e in the Nap. system is unity, and in the common system it is 0.434294 ; hence 1 : 0.434294 :: 1 : 0.434294 =mod. com. sys. Again : to find the Naperian logarithm of ten, we say, As the modulus of the common system : the modulus of the Naperian system ;: the log. of 10 in the common system : the log. of 10 in the Naperian system. That is, as 0.434294 ; 1 :: 1 : 2.302585 =log. 10 in the Naperian system. 73. Let n—x^. Taking the logarithm of both members, we have log. u=y log. «/ and diflFerentiating, du dx —^\og.xdy+y-, or dm—u log. xdy-\-uy— ; .•. log. £B^=a;^ log. X dy+yx'~' . dx. Hence, to differentiate a variable root raised to a va- riable exponent, differentiate first under the supposition that the root is constant, then that the exponent is con- stant, and take the sum, of these differentials. Ex. 153. Let u-x^'', (1) where x, y, and z are variables. Put w=3/' ; (2) then dw—zy'^ldy-^-y^ log. ydz. (3) But / M=a!"; .•. (?M=ioa3'"~Wa;+a;' log. xdw. (4) Substitute in (4) the values of w and dw, and reduce, we obtain 7 -i^ J , zl.xdy dx\ Ex. 154, Let u=a?^. Ex. 155. Let u=a^. du „ 150 diffeeentiXl calculus. Ex.156. LetM=a'°s'. du a'°g- ' ■ log, a " dx~ X 1 Ex. 157. Let M=a:'. Ex.158. Leti«=(to)°. ■■&!~ X CIECUXAK FUNCTIONS. 74. yAe differential coefficient of the sine of an arc, radius being (1), is equal to the cosine of the arc. Let M=sin. x, and give to x an increment, h / then. t«'=sin. (x-\-K\, an d ?«' — M = sin . (a;-(- A) — sin . a;. But, by (Trig., Eq. 11), sin. (a3+A)— sin. x=2 sin. \hcos. {x-\-\h) ; .•. m'— M=2 sin. JA . COS. (a+^A), m'— i< 2.sin. ^A and — T — = -, . COS. (a+^A), Divide the numerator and denominator of the fraction) in the second member by 2, u' — u sin. ^A -^=-p- COS. (x+iA). Passing to the limit, du d.sm.x or (?. sin. a!=cos. (?a;. '75. TAe differential coefficient of the cosine of an arc, radius being (1), is negative, and equal to the sine of the arc. Let M=cos. x; then m'=:Cos. {x+h\ tf'— M=COS. (CB+A)— COS.tB. But, by (Trig., Eq. 14), COS. (cB+A)— COS. x=2 sin. (a!+^A) sin. (— JA) = — 2 sin. ^h sin. (as+lA). CIRCUXAE FUNCTIONS. 151 .-. i«'— M=— 2 sin. ^h sin.-{x-^^h), u' — tc — 2 sin. ^h -JT^ 1 sin. {x+ih), — sin. JA . = — P — Sin. {x+^h) ; ^'^ • jr -J .: ^= — sm. X, or a. cos. x= —sin. x ax. Because versin. a!=l— cos. x, the diff. of versin. x= — diff. of COS. a;=sin. x d.x. 16. The differential coefficient of the tangent of an arc, the radius being (1), is equal to the square of the secant, or={l+ta,ii.^) . Let «=:tan. x, w'^tan. (cB+A), «'— w=tan. (a;-i-A)— tan. !«, sin. {x+h) sin. x ~cos. (a!+A) COS. x' sin. A ~cos. (as+A) COS. a;' sin. A u'—u A A cos. (a!+A) COS. x du 1 :5F-= 5— =sec.^ a!=l+tan.^ x, ax cos.^aj ' ' [^. du dx 1+x+x^' Ex. 164. If M=cos.-' {xVl-x')- . <^^— (— l+2ce^) • • dx~ Vl—x^+x*{l—x^y CIECTTLAE rUlfCTIOKS. 15? Ex.165. IfM=tan.-' -, dy du •'~ dx dx~ y^-{-x^ ' 92. Methods of expansion are often adopted in special cases, of which we shall give some examples. To expand sin.-' x in powers of a;. Assume sin;-' x^^A^+AiX+A^x^+f etc. (1) Differentiating both members of this equation, and di- viding through by dx, we have ^ - =Ai+2Aaa!-)-3A3a!^+, etc. (2) V 1 — * But, by the Binomial Theorem, 1 / 1.3 1.3.5 :^^=l+i«=H2-74«'*+2:476«'»+,etC. (3) Comparing the coefficients in (2) and (3), we have, by first making x=0 in (1), AfltO, A, = l, A2=0, ■A.3=j-^, etc.; . , x^ 3a^ .-. sin.-' "'=«'+273+ 2.4.5 +» etc. 2. To express an arc in terms of its tangent : Assume taTi.~^x^AQ+A^x+A2X^+A^x^-\-,etc. (1) Differentiating. both members, and dividing by dx, we * 1 have Y73p=Ai+2Aaa:+3A3a;2+, etc. (2) But Yr^=l—x'^+x^—x^+x^+,eto. (3) Equating coefficients of like powers of x in (2) and (3), we have Ai=l, A2=0, A3=— J, etc. And putting x=0 in (1), we get Ag=0 ; .". tan. x^ x^ x' =«— — +-g— y+, etc., =:X\l "+i"-y+?-'^^«- From this we can readily calculate the length of an arc of the circle ; for if we call the arc whose tangent is x 158 DIFFEEBNTIAL CALCULUS. 30°, its tangent x -will be equal to --p=.5'7lr35, and these being substituted in the above, we shall have arc 30°=tan.-'— p V3 1_/ _J_ J_ _]_ 1 \ ~-v/3 V~3 . 3+5 . S^"" 7 . 33+9 . 3*~' ^*°7' =.523598'7=:the length of an arc of 30°. If we multiply this by 6, we shall have the semicir- cumference to radius unity=3. 14159. 93. By means of Maclauvin's Theorem we are enabled to find the value of the principal functions of an arc in terms of the arc itself. Thus : Let «=/(«)= sin. a:. (1) If we make £c=0, then U=0. Differentiating (1), we have du ^ = cos. a;, whence, ifa;=0, U' = 1, and -^=:— sin. £b, whence, if a;=0, U" = 0, and ^^=— cos. a;, whence, if £6=0, 11'"=— '1, d'^u and 'J~i— sin. £», whence, if a!=:0, U"'= 0, and Zj^~ cos. a;, whence, if £6=0, U' = 1^ 94. To develop cosine x in terms of a:. Let u=f{x)= COS. X. (1) If we make a;=0, cos. a!=l ; .-. U=l. Differentiating (1) successively, and making x in each differential coefficient equal to zero, we have ^^ -^=-sm.x; .-.U = 0. ^=-cos.a;; .-.U --1. d^ii ^= sin. a:; .-.U'''^ 0. IMPLICIT FUNCTIONS. 159 -^,=oos.x; ■U"=l. etc., etc. ; .•.cos.a;=l-Y^+j--27^-,etc. . The formulas for the sine and cosine of an arc are very convenient in calculating tables, especially -when the jwc — is small. Implicit Functions. 95. When the relation between a function and its va- riable is expressed by the equation /(a:,2/)-0, y may be called an implicit tunction of x, or x an implicit function of y. For if the equation f{x, 2/)=0 can be solved with respect to either of them, the result may be or a!=/'(y)- Biit, as it is often dilEcult to solve the given equation, it becomes necessary to obtain a rule for determining dy , . . the value of -3- which does not require this labor. Put u=f{x,y)=0, (1) and let x become cc+A, and y become y+k, so that ,• f{x+h^y+k):=0. (2) Hence /(aJ+A, y+k) -f{x, y) = 0. (3) If we subtract from, and add to the first member of (3) the quantity /■(£!:+ A, y), we shall have f{x+h, y+k)-f{x+h, y)+f{x+h, y)-f{x, y)=0. (4) Let us divide by A, and at the same time multiply the k first fraction by t, and we shall have f(x+h,y+7c)-f{x+h,y) k f{x+h,y)-f{x,y) k •A+ A -O-g) (Eq. 5) being always true, remains true when A and k are diminished indefinitely. ^ f(x+h,y+k)—f(x+h,y) The limit of -^^ k ^'*'"''^ ^® *^® difier^ntial coefficient of/(c«!+A, y) formed on the suppo- 160 DIFFEEENTIAIi CAXCULUS. sition that y alone varies if h remained constant ; but as h diminishes without limit when k diminishes without limit, the limit of the expression is the differential co- efficient off{x, y), or u with respect to y formed on the supposition that y alone varies. It may be represented by ( j^l. The limit oij- is -5-. The limit f^''+^^y)-f('''y\ ^ten h=0, is the dif- ferential coefficient of f{x, y), or u with respect to x formed on the supposition that x alone varies, and the ^du\ limit may be denoted by I j~ !• Therefore /du\ dy ldv\ , . ^°^ £=-1^- (^) \dy) 96. Hence the differential coefficient ofy regarded as a function of x is equal to the ratio of the partial dif- ferential coefficients of u regarded as a function of x, and u regarded as a function ofy, taken with an oppo- site sign. Examples. Ex.166. Let/(9;,2/)=M=2/2+a;2— E,2_o; (1) dy ' dx' (du\ y \dy) Differentiating the second time, we obtain dx^ 2/3 y- APPLICATION OP THE CALCTILUS TO CUEVES. 161 Ex.167. LetM=2/2+2a:y+a!2— a2=0; then J^= — 1. ax then Ej then Ex.168. LetM=2/2— 2ma;y+a!3— a2=0; dy my—x dx y—mx' Ex.169. liQt u=x^+Zaxy-\-y^=.0; dy _ x^+ay dx~~ax+y^^ d^y_ Icfixy di^~ (ax-\-y'''Y' CHAPTER III. APPLICATION OF THE DIPPEEENTIAL CALCDXUS TO THE THEOEY OP CUEVES. 97. We shall now apply the Calculus to Geometry, and first show how an equation may be freed from its con- stants. For instance, if we take the equation of a straight line, y^ax^hy (1) and differentiate it, we shall have i=«- (2) Differentiating again, we find — tED. J. J y^/ '• y. f^. X / ! °^ TR~f?a!' T B. .-. TR=2/ • jj^— subtangent. 105. Hence, to find the subtangent to any point of a curve, we have the following Rule. Differentiate the equation of the curve, and find the value ofy-^. Examples. Ex. 175. Find the subtangent of the curve whose equa- 3 „ , x{2a—a^ S»^tan.=:^^. Ex. 176. Find the subtangent of the curve whose equa- tion is y'^— Zaxy -\-x^=0. laxy—'x? Subtan.= 5-. ay—s? Ex. 177, Find the subtangent of the curve whose equa- tion is xy^=a^{a—x). 2(aa!— CB?) Subtan.= — ^^ -^ ^ a 106. To determine the length of the subnormal to any point of a curve. ^ In the adjoining figure, RN is lit 5^^ the subnormal, and it is required /^^iV to find its length. In the right- yV j \ angled triangle RPN, we have y^ / \ \ RN ^ / : \ ^p=tan.NPR. T R N 166 DIFFERENTIAL CALCULUS. But the angle NPR is equal to NTP ; RISr dy •■•RF=^5 andasRP=y, dy .: RN= 2/-^= subnormal. 107. Hence, to find the subnormal to any point of a curve referred to rectangular axes, we have the following Rule. Differentiate the equation of the curve, and find the dy value of y-^. Ex. 178. Find the subnormal to any point of the com- mon parabola. The equation is y'^=2px. By differentiating, we obtain dx~y ' dy The subnormal is equal to half the parameter. 108. To determine tlie length of the tangent to any point of a curve referred to rectangular axes. '• In the right-angled triangle PRT, we have ^^^^ PT'=PR'-ffR^, ^/^ dx^ yy \ ^2» y^/ \ ._y2^y% 109. To determine the length of the normal to any point of a curve referred to rectangular axes. In the adjoining figure, PN is the normal^and we are required to ^P/«^ find its length. y/i\ Inthe right-angled triangle RPN, // jX y'^difi jT f i J — APPLICATION OF THE CALCULUS TO CURVES. 16 V ■•■™=''nA^- We shall now apply these formulas to lines of the sec- ond order embraced in the equation Ex. 179. y^^mx-\-'rKi?. (1) By differentiating, we obtain dx 2y dy^ m+lnx^ ^ ' dy m+2nx m+2nx and;:7-= — ;r- — = — . „ =tan. of the angle which dx 1y 2Vmx+nx^ \ ^ the tangent line makes with the axis of x. Multiplying both members of the last equation by y, we have dy ni+2nx y^— o =the subnormal. Also, multiplying (2) by y, dx 2y^ 2{mx-\-nx^) y ^=^;^:+ 2^x = m+2nx =>"^tange nt. ^•^^ 2/l/^=\A^^r^^^^T^^gf ^tangent; ako y\/l+-^=z V»ia!-t-«a3^+i(»i+2wa;)2=normal. If we attribute proper values to m and n, the formulas above will be applicable to each of the conic sections. In the case of the parabola, n=0 ; hence we have for the tangent of the angle which a tangent line makes with the axis of x, dy m dx~~2y'' ydx -5— = 2a;= subtangent, y\J da?- 1 + — = ■y/mx-\- 483^ = tangent, dy m y-i/l -)-^=\/>Ma!-f-— =normal. 168 DIBTEEENTIAIi CALCULUS. Ex. 180. In the case of the ellipse, take the equation AV+B2a!2=AzB''. (1) By differentiating, we have -^=— -^=the tangent of the angle which a tangent line makes with the axis of x. If we place this equal to zero, we shall have £8 = 0, and from (1), y=±B, which are the co-ordinates of the points at which a tan- gent line is parallel to the axis of x. If we put -T- equal to infinity, the denominator be- comes equal to zero; that is, and »= ±A. The tangent line to either of those points is then per- pendicular to the axis of a;. dx A^yZ Agam: ^^=-3^. = =snbtangent, X " ' dy B^x * and y^=— -^=subnormal. 110. We perceive that the value of the subtangent on the axis of x does not comprise B, and therefore it will be the same whatever be the length of the minor axis ; consequently, if any other ellipse, or a circle, be described on the same major axis, the tangent at the point in which the ordinate to the point of tangency meets this new el- lipse or circle will cut the axis of x m the same point as the tangent to the first ellipse. 111. In the circle, A becomes equal to B ; hence ' the subtangent=— — , J,, Rw ^he tangent = — ^ * % a; , ithe subn^i-mal == —x, jth^qoj^al V = ^R. APPLICATION OP THE CALCULUS TO CUEVES. 169 112. 7b determine the equations of tangent and nor- mal lines to plane curves. The equation of a straight line passing through a given point is (Analytics, Art. 10) y-y'=a{x-x'), (1) where a denotes the tangent of the angle which the line makes with the axis of x when referred to rectangular co-ordinates. We have found that the first differential coefficient of y regarded as a function of x represents the-tangent of that angle, Art. 102. Hence the equation of a tangent line to a curve at any point is dy y-y =^('«-'«')- And since the normal is perpendicular to the tangent at the point of tangency, we shall have dx for tJie equation of the normal. To find the equation of a tangent line to any curve referred to rectangular axes, we have the following RULS. Differentiate the equation of the curve, determine the dy value of -J-, and substitute that value in the equatimi dy y-y =^(«'-"'')- Meamples. Ex. 181. Determine the equation of a tangent line to the ellipse at the point whose co-ordinates are x' and y'. The equation of the ellipse referred to its centre is AV+B2a!2=A2B2. By differentiation, we have ^_ B^a; dx~~ A^y^ II 170 DIFrERENTIAIi CAlCtTLUS. .•.2/-y'=-_^(a'-a!'), which, by reduction, becomes A2»/2/'+B2a!a!'=A2B2, the equation required; A?y and 2/-y-gi^(a!-a:'). the equation of the normal. Ex. 182. Determine the equation of a tangent line to the circle at a point whose co-ordinates are sc', y'. The equation of the circle is 2/2+a!2=R% (1) and, by differentiating, we have dy_ X dx~ y' Substituting this in the equation dy we have y—y'=—y{^—^')^ which, by reduction, becomes y2/'+asB'=R2 for the equation of the tangent line to the circle. Ex. 183. Find the equation of a tangent line to the parabola at a point whos^ co-ordinates are x', y'. The equation of the parabola is y'^=.1px. And, by differentiatmg, we haye . c?fc y' Substitute this in the equation dy and we have y—y'='Z^P~^'^t or y^—yy'=px—px'. But y'^=.'2,px; :. '2,px—yy'-=px—px', ■•■ yy'=p{x+x'), the equation of the tangent line to th^ parabola. ASYMPTOTES. 171 115. We may also free the terms of an equation from their exponents. Ex. 184. Let P-^Q, (1) where P and Q are functions of a;. Then, by differentiating, we have and multiplying both members by P, we obtain wP°(?P=P(?Q ; and substituting for P° its value, then wQc?P=P^ 2/=aaf +50!°-'+, etc.+A+-+, etc., and then make a!= oo, the eqilation of the curve at an in- finite distance becomes 2/=aa!°+5a3°-'+, eto.+A, which is also the equation of an asymptotic curve of n dimensions. If w=l, then 2/=aa;+5, which is also the equation of a rectilineal asymptote, Alt. 112. If M=2, then «/=a£B2+6a;+c, which is the equation of a common parabolic asymptote. Ex. 190. Let the equation of the curve be a;3=(a;— a)2/2; a? then w^^ , or 2/=± T. {^—ay- As X approaches the value of a, both y and -j- increase also without limit, and is the equation of a rectilineal asymptote. If we put y in the form and expand by the Binomial Theorem, we have 2/=d=«[l+^+^+,etc.J. (1) SINGULAR POINTS OF CURVES. 175 Hence y=±{x+la) are the equations of two rectilineal asymptotes. Again : from (1) we have y=:t{x+-+- which are the equations of two asymptotic curves of the second order. Ex.191. Find the equation of the asymptote to the curve whose equation is y^=:x^{2a—x). Ans. y=:—x-\-%a. Ex. 192. Find the equations of the asymptote to the curve 2/^(a;— 2a)=a;'— a^. Ans. x=2a, and y=±(x-\-a). Ex. 193. Find the equation of the asymptote to the curve y^{ay+bx)=a'y^+b'x\ Ansi 2/= — x+2a. Ex. 194. Find the equation of the asymptote to the curve iii?y^=:a'^{x^—y'^). . .■ Ans. y=.±.a. SiNGULAE Points op Curves. 119. In a curve whose nature is expressed in a gen- eral manner by the equation or /(a!,2/)=0, it is evident that if we attribute different values to the principal variable, we shall generally obtain different values for the function and its differential coefficients; and when any of these functions attains a value attended with some peculiarity either in its value or its form, the point in the curve corresponding to this will be distin- guished by something peculiar in its character. Points of this description are usually denominated Singular Points, the principal of which are characterized by the circumstances explained and exemplified in the follow- ing articles : 120. To determine the position of a curve at any point with reject to its tangent. 176 DIFFEEENTIAL CALCULUS. Let PPT" be any curve, and let x and y be the co-or- dinates of the point P. Give to X the several increments, RR', R'R", each equal to h, and draw the ordinates R'P', R"P". Join PP', and produce it to B ; also join P'P" ; draw PD and P'D' parallel to AR". Then A+ H P'R'=/(a:-fA) R' R" 1.2 y+dx' +, etc., (1) and P"R"=y-f^- .P'R'-PR=P'D=;^/i+ ^2"+. etc.; dy d'^yh^ I'^dx^ Also, P"R"- dy d'^y SA^ -P'R'=P"D'=jA+J^+,etc.; (2) (3) (4) . p"D'-P'D=:P"B=§A2+, etc. (5) Kow it is manifest that if the curve lie above its tan- gent at P', or be convex to the axis of «, P"D' is greater than BD', and therefore we must have the second differ- ential co^icient positive. But if the curve lie below its tangent at P', or be con- cave to the axis of a, D'P" would evidently be less than BD', and therefore the second differential coefficient would be negative. Also, since by the continual diminution of h the first d^y term -^-j . h^ will become greater than the sum of all the succeeding terms of the series, it follows that the alge- braic sign of this series will be the same as that of its first term, and therefore, according as a curve is convex or concave to the axis of abscissas, the second difieren- /d^y\ tial coefficient ( ^2) will be positive or negative. SINGULAR POINTS OF CUETES. 1^7 Hence, conversely, a curve will be convex or concave to the axis of abscissas according as the second differen- tial coefficient of the ordinate is positive or negative. Or, more generally, A curve will be convex or concave to the axis of ab- scissas according as the ordinate and its second differen- tial coefficient have the same or different algebraic signs. 121. A singular point of a curve is a point which has some peculiarity which the other points do not generally have ; as, for instance, 1. The point at which a tangent line is either parallel or perpendicular to either axis. 2. The point at which the curve changes from being convex to concave to the axis of x, or from being con- cave to convex. The point at which this change of curv- ature takes place is called a point of inflection. 3. A multiple point; that is, a point at which two or more branches of a curve intersect each other. 4:. A cusp; that is, a point at which- two or more branches of a curve terminate and have a common tan- gent. If the branches lie on different sides of the tan- gent, it is called a cusp of the^fs^ order ; if on the same side, a cusp of the second order. 5. An isolated point; that is, a point whose co-ordi- nates satisfy the equation of the curve, although the point is entirely detached from every other point of the curve. The first of these has been discussed in Art. 103. 122. In order to determine whether a proposed curve is convex or concave toward the axis of a;, differentiate the equation of the curve twice ; then, if the ordinate and second differential coefficient have the same sign {whatever be the value attributed to the variable), the curve is convex; if they have different signs, the curve is concave (Art. 120). Ex. 195. As a first example, let us take the equation of a circle referred to the centre as the origin of rectan- gular axes, 2/2_jj,2_3.3_ . . dy X Difierentiating, we have ^="~^5 d'^y a;2+y2 _ R2 H2 irs DIFFEEENTIAL CALCULUS. When y is positive, the second differential coefficient is negative ; and when y is negative, the second differ- ential coefficient is positive. Hence the curve is entire- ly concave toward the axis of cb. Ex. 196. Again: let us take the general equation of the circle (2/-/3)'+(a'-«)'=R'; ••.2/=/3 And differentiating, d'^y _ dx^' V'R2_(a;-a)2 X—a and -{X- R2 ■aY If the upper sign be used in the value of y, the corre- sponding part of the curve is concave toward the axis of a;/ but if the lower sign be used, it is convex. Ex. 197. Let the curve be the common parabola y'^=:2px. Ex. 198. Determine whether the ellipse is concave or convex toward the axis of x. 123. To determine whether a proposed curve has a point of inflection. Differentiate the equation of the curve twice; then, if the second differential coefficient changes its sign {by at- tributing different values to the variable), the point at which this change takes place is a point of inflection. Therefore, Art. 29, there must be one value of the sec- ond differential coefficient equal to zero or infinity, and the roots of the equations d^y d^y will give the abscissas of the point of inflection. Ex. 199. Determine whether the curve whose equation is y=:a+c{x—bY has a point of inflection. Differentiating, we have T ■ %=H^-bY, ^=6c(x-a). BINGULAE POINTS 01" CUETBS. 179 When a! =5, the first differential coeiEcient is zero, and the tangent is parallel to the axis of x at the point whose abscissa is b. Substituting this value of x in the equa- tion of the curve, we get y=a. Examining the second differential coefficient, we find that when x is less than b, the second differential coeffi- cient is negative, and the curve is concave up to that point; when a;>&, the second differential coefficient is positive, and the curve is convex ; hence there is an in- flection of the curve at the point whose co-ordinates are x=b, y=a. Ex. 200. Determine whether the curve whose equation is i/—b—c{x—ay, has a point of inflection. Ex. 201. Determine whether the curv© whose equation is 2/=a!-f-36a;2+2a33— a;* has a point of inflection. It has two points of inflection. 124. To determine whether a proposed curve has a multiple point. If, when a certain value is assigned to the abscissa sfe, we find only one value for the ordinate y, but more than dy one value for -j-, it is evident that the curve has two or dy more branches intersecting at that point; for -r- ex- presses the tangent of the angle which a tangent line to the curve makes with the axis ofx; hence there must be more than one rectilineal tangent at the corresponding point, which is therefore a multiple point, and its degree of multiplicity is expressed by the number of real values Ex. 202. Let the equation of the curve proposed be y=±x{a^—x^)K (1) Differentiating, we have dy_ a''—2x^ ^'^ (a2-a;2)*' For every value of x in (1) we have two values of 180 DIFPEEENTIAL CALCULUS. y numerically equal with opposite signs ; hence the curve has two branches symmetrical with regard to the axis of a;. Also, when a!=:±a, y=Q ; that is, the curve cuts the axis of a; at a distance equal to a to the right, and a to the left of the origin. When 85=0, 3/=0; hence the two branches of the curve intersect at the origin of co-ordinates. The origin is therefore a multiple point. Ex. 203. Let the equation of the curve proposed be y=b±: {x—a) Vx. 125. To ascertain whether a proposed curve has a cusp. If P be a point of a curve at which PR is a tangent, then, if the ordinates P'R' and P"R", drawn to the consecutive points on each side of the point P, be both less or both greater than the ordinate at P, it is evident that P will be the point at which the two branches of the curve will unite ; or, in other words, be a cusp of the first order. Ex.204. Let {y—ay—(x-bY=0, be the equation of a curve. It is required to determine whether it has a cusp. Transposing and extracting the root, we have y—a=(x—by; dy_ 2 R'RR" and dx^' 3{x-by 2 ^i^ R'E.K." 9{x-by If we make x=b, the first differ- ential coefficient will become in- finite, and the tangent will be per- pendicular to the axis of x at the point whose co-ordi- nates are x=b, yz=a. In regard to the second differential coefficient, it will be negative for every value otx, since the factor (x—b) is raised to an even power. Hence the curve will be concave toward the axis of x. SINGULAE POINTS OF CUKTES. 181 If we make x—h+h or a;=6— A, in either case we shall have and therefore y will be less for x^h than for any other value of X, either greater or less than h. Hence the value a;=6 is the abscissa of the least ordinate ; and there is a cusp of the first order at the point whose co- ordinates are x—b, y—a. Ex.205. Again: let (y— a)3+(a;— 5)2=0. Then y-az=-{x-b)^, l" dy 2 and d'^y dx^~ Z{x—b) 2 R'RR" 9(a;-&)^ As before, x:=b makes the first differential coefficient equal to in- finity. The second differential coefficient will be 2^ositwe for every value of x either greater or less than b. The curvtf will therefore be convex toward the axis of x. If we make Xz=:b-\-h, or a;=6— A, we shall have, in either case, y=a-f?; and hence y will be greater for x=:b than for any other value of a; either greater or less than b. Hence the value x=b' is the abscissa of the greatest ordinate. The branches PP' and PP" are not considered as parts of a continuous curve, for the general relations between x and y which determine each of those parts are entirely broken at the point P, where x=zb. Ex.206. Let {y—aY={x—bY; (1) then and y=:a±{x—b)'^, c?V_ 3 4(a!-5)2 If we put the first differential coefficient equal to zero, 182 DIFFERENTIAL CALCULUS. we shall have a; =5, and substituting this value of a; in the equation of the curve, we find y=:a. Hence the point whose co-ordinates are x=h,y=ia, is a point of the curve at which a tangent line is parallel to the axis of x. Now for a;=:5— Athe ordinates are imaginary, and for a;=5+A the values of d'^y 3 Hence there is a cusp at P, where the upper branch, PP', which is convex towarqjthe axis 3 of X, is determined by + — 7=, and the lower branch, PP", is concave, determined by 3 If for X in (Eq. 1) we substitute 5+ A, we shall have 3 y=a±Ji^; and giving to A a succession of values, which shall have small differences, we may trace the curve. Ex. 207. Let {y—x') —x^ be the equation of the curve ; 6 then 2/=a!2±a;^, (2) and ^=2±ii.x^. If we make x=0, y=Q; hence both branches of the curve pass through the origin. Putting the first differ- ential coefficient equal to zero, we have 3 2a:±|a5^=0. One value of x is eero/ this shows that the axis of x is a tangent to the curve at the origin. Examining the second differential coefficient, we see that the upper i branch, corresponding to 2+-J^% is continually convex toward the axis of x, while the lower branch is convex from the origin to the point whose abscissa is ■^^, and concave beyond that point. If a!=l, then y=0, and the lower branch cuts the axis of a; at that point. SINGULAR POINTS OP CUEVES. 183 Ex. 208. Let («— y^) =2/° be the equation. 5 Then x=y'^±y^ ; "^ o.. , 5,, -J 22/±f2/ _ . dy -=co. 3 we have 2y ±42/^= ; «/=0, and a;=0. Hence the axis of y is tangent to both branches of the curve at the origin of co-ordinates. If y be made neg- ative, X becomes imaginary, which shows that the curve does not pass below the axis of a;. In the equation 5 5 _ if y be less than unity, y'^ will be less than y', and x will have two positive values, represented by RP and HP'. Therefore the point A is a cusp of the second or- der. This example and the one preceding it will give the student an idea of cusps of the second order. 126. To determine whether a proposed curve has an isolated point. If, when a certain value is given to the abscissa x, the value of the ordinate y remains possible at the same time that one or more of its diiferential coefficients become imaginary, the corresponding point of the curve is called an isolated or conjugate point. Hence, to determine the positions of such points, we must ascertain what possible values of the co-ordinates satisfy the equation of the curve at the same time that they render one or more of the diiferential coefficients imaginary.- Ex. 209. Let us take the curve whose equation is a2/2=a;(a;-|-5)2; (1) .•.y=±(a;+5)y/^, (2) :r^=±-^ 3-v/a+-7= ■ (3) 184 DIFFEEENTIAI, CALCULUS. Now if x=—b in (1), we shall have y=0, and the corresponding value ot dx~'^\/ a will cause the values preceding and succeeding this to be imag- inary ; so that, if A be the origin of the co-ordinates, and AP be taken =5, the point P belongs to the curve, but it is entirely de- tached from the rest of the fig- ure, making the point an isolated point. Ex. 210. Let ay'^=x^—hx^ be the equation of the proposed curve. Then, if a!=0, 2/=0. The co-oydinates of the origin satisfy the equa- tion, and we might at first suppose that the curve passes through that point. Solving the equation for y, we have ^ (x-h\^ y-- Now for all values of as less than 5, y will be imagin- ary; and when x=.h, 3/=0. Hence the curve cuts the axis of a; at a distance equal to h from the origin of co- ordinates. For every value of x greater than 6, y will have two values numerically equal with contrary signs. Difierentiating, we have dy 3a!— 25 dx" dy and makmg a!=;0, dx' 'l'\/a(x—h) ' 25 2-\/— «&' an imagmary expression. Hence the origin of co-ordinates is an isolated point, and its co-ordinates satisfy the equation of the curve ; but there is no point of the curve which is consecutive with it. Transcendental Curves. 127. Curves may be divided into two general classes, algebraic and transcendental. When the relation be- THE LOGARITHMIC CUBVB. 185 tween the abscissa and ordinate of a curve can be ex- pressed in algebraic terras, the curve is called an algebraic curve ; but when this relation can not be expressed vi'ith- out the aid of transcendental terms, the curve is then called a transcendental curve. The Logaeithmic Cuevb, 128. The logarithmic curve takes its name from the property that, when referred to rectangular axes, one of the co-ordinates is equal to the logarithm of the other. Take the equation «'=2// then (Alg., Art. 153) x is equal to the logarithm of y, taken in a system whose base is a. In the diagram, let Abe the origin of rect- angular axes. Then, if a;=:0, 2/=l=:AB; if a;=l, y^a; if a;=2, y^za"^; etc., etc. If a;=— 1, y=a~'= Ifa;=— 2, y=ar'= etc., etc. If a, the base of the system, is greater than unity, the logarithms of all numbers which are less than unity will be negative. Hence these are represented by the parts of the line AD to the left of the origin. When y—Q,x must be infinite and negative, as «-'=0, or ^ ^,=0. Taking the equation a;=log. y, (1) and differentiating, putting »i=modulus of the system, we shall obtain dx_m dy~ y'' ^ ■' dx or y\ dy -m. (3) 186 DIFFBEBNTIAL CALCULUS. Equation (2) may be expressed f=^; (4) ax m' ^ ' and as this represents the tangent of the angle which a tangent line makes with the axis of a;, therefore the tan- gent will be parallel to the axis of x when y=0; but when y—0, x=aD. The axis of x is therefore an asymp- tote to the curve. 129. (Eq. 3) represents the subtangent, which is equal to the modulus of the system of logarithms from which the curve is constructed. In the Naperian system, m=l, and the subtangent will be equal to AB. The Cycloid. 130. A cycloid is a curve which is described by a point in the circumference of d circle while the circle is moving along a straight line. The circle is called the generating circle, and the point which describes the cycloid is called the generating point. E A K Let EPN be the generating circle, A the origin of co- ordinates. Let us suppose that the point P coincided with the point A, and that it has described the arc AP while the circle was rolling along the straight line AM. Then, if A'R=x, and RP=y=NH, and ;'r=radius of the generating circle, we shall have PH= ■\/2ry-y^=RTS. (Geom., B. IV., Theor. IX.) Now AR=AN-RN ; but A]Sr=aro PN, and HN is the versine of the arc PN". SPIRALS. 18T tangent .: £i!=versin.~' y— ■\/2ry—y'', (1) ■which is the transcendental equation of the cycloid. By differentiating (1), we obtain V^ry—y^' ^ ' which is the differential equation of the cycloid. 131. To find the normal, we have = -y/^ry— normal. dy , Subnormal y'^^^v'^i"y—y\ rr~d^ y\/2ry V dy^ ■Viry-y'i Buhtangent ^ y^^ ^-—^^. ' If we differentiate the equation dx— , -, ■y/'iry—y^ regarding dx as constant, we obtain, after a little reduc- rdy^ tion, ^'2/=-2;^3^- Hence the cycloid is concave toward the axis of 03. Spieals. 132. A spiral is a curve generated by a point which moves along a straight line according to some prescribed law, the straight line having at the same time a uniform angular motion. jj The fixed point P, about .-••■" T""--,., which the straight line re- .••'' / volves, is called the pole of the spiral. That part of the spiral which is form- % ed while the moving line makes one revolution, is called a spire; and the dis- tance, PM or PA, which 188 DEPFEEEin'IAL CALCULUS. the point has traveled while the line has been in motion, is called the radius vector. 133. From the manner in -which the spiral is gener- ated, its equation will be '•=/(e), where r is the radius vector, and 9 the arc of the meas- uring circle. Spiral op Archimedes. 134. If, while the line PD revolves uniformly about P, a point also moves uniformly along the line PD, that point will describe a spiral, which is the spiral of Ar- chimedes. From the definition, the radii vectores are proportional to the measuring arcs estimated from A. That is, PM : PA : : arc AD : cir. ADE. Calling PM the radius vector=r, " PA the radius of the circle ADE=R, and Q the measuring arc estimated from A, we shall have »-:R::0:27rR; _B_ •'•^~'2ir- 1 If we put 2^^"*^' then r=ad, the equation of the ^iral of Archimedes. LOGAEITHMIC SpIEAL. 135. If, while the straight line is revolving about the pole, a point is also moving along this line in such a man- ner that the logarithm of the radius vector is constantly proportional to the measuring arcs, the point will de- scribe a spiral which is called the logarithmic spiral. Hence, by the definition, 6= log. r is the equation of the logarithmic spiral. Hyperholic or Reciprocal Spiral. 186. If, while the straight line is revolving about the pole, a point is moving along this line in such a manner that the radii vectores shall be inversely proportional to THE CISSOID. 189 the corresponding measuring arcs, the point will describe a spiral which is called the hyperbolic or reciprocal spiral. E From the definition, we have PM : PB : : arc AB : cir. ABE. Calling PM unity, we shall have 1 : PB : : arc AB : cir. ABE, or 1 : >• : : 9 : : 27r ; 27r .•.j-=y=27re-'. Putting 2ir=a, we have r=aB' "6' or r6=a, which is the equation of the reciprocal spiral. This spiral is called the hyperbolic spiral from the analogy which its equation bears to that of the hyperbola when referred to its asymptotes. The Cissoid. isr. If AQB is a semi- circle, and we take any equal parts,CM and CN, and draw the ordinates, MR, NQ ; join AQ, cut- ting MR in P ; then the locus of P will be the curve called the Cis- soid. 190 DIFFBBBNTIAL CAIiCULtJS. Let A be the origin of rectangular axes. Put AB=a=the diameter of the circle, a;=AM, y=MP. Then MP':AM'::QN':AN', or X' t2 ::AN.]SrB:AN ; ::]SrB:AN, vQN^=A]Sr.NB, X ■.a-x; .•.NB=AM; x" (1) "" a—x! the equation of the cissoid. Differentiating (1), we have dx 2(ax—x'^) „ , , ^d^= Sa-2x ' ^°^" *^" ^^^ia^ffent. To find the polar equation of the cissoid: Take A as the pole, and join QB. Put AP=?-, and the angle QAB=e, and AB=2ay then AQ=2acos. 0, and Q]Sr_ BH AM ycos.e- ^"~sin. fl""tan. 6 sin. 0~tan. sin. e^tan. d sin. fl' ' ' Equating (1) and (2), we have r . cos. d : ^—. — s= 2a cos. ; tan. sin. ' 2« sin.2 •'• ''~ cos.0 ' the polar equation required. 139. 2b determine the differential of the length of an arc referred to rectangular axes. Let AR=a! and RP=y be the co-ordinates of the point P ; AP the arc=:Z. Then, if we give to x an incre- ment, RR'=A, AP'=s', we shall have z'-z arcPF h -' PD • But, by (Art. 11), the arc and chord are ultimately in a ratio of equality. Passing to the limit, we have dz ,. . chd.PP' ^=limitof-p5-, liSKGTHS AND ABEAS. 191 =limit of ' cos. 6— tan. cos. O^+r sin. 6 tan. cos. ^'Ta—i' sin. 0+tan. 8 sin. 6;^+»' cos. tan. 9 ■^ . sin.^e But sin. tan. 6= s", and cos. tan. e=sin. 0. COS. 0' Making these substitutions, we have de tan.PDC=»-^. (5) Now, in the triangle PDT, we have PT „^^ dd ^=tan.PDT=.^; .-. PT=PD tan. PDT, JO = r^-jr= su Dtangent. 196 DIFFBEBNTIAL CALCULUS. 146. For the suhtangent of any curve referred to polar co-ordinates is a line drawn from the pole perpendicular to a radius vector, and limited by a tangent drawn through the extremity of the radius vector. Or, in other words, the subtangent is the projection of the tangent on a line drawn through the pole, perpendicular to the ror dius vector, which passes through the point of contact. 147. Since the arc and tangent are coincident at the point of contact, the angle in which the radius vector cuts the arc of the spiral is also the angle PDC. If we denote this angle by 0, we shall have 7-dd tan.0=-^, or tan.2 0=-^p-. sin.2 sin.2 1— cos.*0 But tan.^ (4= T~:^^ = — 5~r=^ 9~; — ; T cos.^0 1— sin.^^ COS.2 ' r-^dd^ ( rH(F\ sin.'' 0: dr"^ ' rdQ :. sm. 0= , — , . ^ Wdr^^rHQ^- In like manner, we may find dr COS. <4= — ■ =. ^ ■yJdr'^^rMW' 148. If we draw PB perpendicular to the tangent, we shall have PB=PD sin. d»= . ^, ^ ^/dr^^■\■rM&' ^„ „^ rdr DB=PD COS. d.=: ■ ==. 149. It is sometimes convenient to know the relation between the radius vector of a spiral and the perpendic- ular let fall upon the tangent ; hence, if we call the per- pendicular PB=/?, we shall have in all curves referred to polar co-ordinates. NOEMALS. — ASYMPTOTES. 197 Also or c7e P dr- dr dd~ P Normals. 150. To draw a norrnal to a spiral, and find the length of the subnormal. Let DG, perpendicular to the tangent at the point of tangency, meet the line TPG in G, drawn perpendicular to the radius vector PD ; then PG is called the polar sicbnormal, and DG the normal. Then in the triangle GPD, we shall have PG=PD tan. GDP=PD cot. PDT, dr _dr ~^rdd~de' And the normal DG= VpD2+PG% ■v^ >-2+: Asymptotes. 151. To ascertain whether a spiral admits of an asymptote, and to determine its position. In the equation r=f{e), find the value of 6 in terms of r. Then, if this value and that of the subtangent, 198 DrPFEEENTIAI, CALC]p,US. which is —3—, do not become infinite when the radius vector is indefinitely increased, the spiral has an asymp- tote, and its position may be determined. For, if PR be the di- rection of the radius vec- tor when infinite, and the subtangent PT be drawn perpendicular to it, then TS, parallel to PR, will evidently be the asymp- tote. Ex. 211. Let us take the hyperbolic spiral r— e- .: e=-. r By differentiating, we have dd a dr dr ■——a. a, constant quantity whatever be the value of r, and con- sequently when r is infinite. Hence, if PA "be the line E"" from which 9 is measured, it is evidently the direction of the radius vector when' infinite ; and if a line, PT, were drawn perpendicular to PA and equal to a, then a line, ASYMPTOTES. 199 TS, drawn through the point T parallel to PA would he the asymptote required. a Ex.212. Let '*~'7fl' a2 then '^'^- r'= 6' and e=-?. If r^oo, we shall have 0=0, and the subtangent r'^dQ_ 2a2 dr ~ r '' which is also equal to zero when r— co ; hence the line PX, from which the angle d is estimated, is the asymp- tote of the spiral. Ex. 213. Let us take the Cissoid of Diodes. Its polar equation is 2a sin.^ 6 }•=- (1) COS. d where 2a is the diam- eter of the generating cltcIo Differentiate; then ^ M C" dr 4a sin. 6 cos.^. 0+20! sin.' N or But dd' dd_ dr' C0S.2 6 C0S.2 9 "4a sin. cos.^ d+2a sin.^ 0' 4a2 sin.* d f.2 .:r- C0S.2 d dd 2a sin.3 Q dr'~2—sm.'^ 0' (2) Ifr= : o) in equation (1), we shall have COS. 0=0; .•. 0=47r; and in equation (2), the subtangent becomes =2a=AB. Hence a line drawn perpendicular to the diameter of the circle at B will be an asymptote to the Cissoid. M 200 DIFFEEENTIAL CAICULTJS. OSCULATOET ClECLE. — RADIUS OF CuETATUEE. 152. The deviation of a curve from a tangent line is called the curvature of that curve ; and of several curves, the one which deviates the most rapidly from its tan- gent is said to have the greatest curvature. Let AC be a tangent to the ABC curve DBE at the point B, and __.^ BM a normal at the same point ; -"/^ / then will AC be a tangent to d ■: every circle passing through B, ••. and having its centre in the normal BM. It is evident that the centre of a circle may be taken so near to B that its circumference shall lie within the curve DBE; and it is also evident that the centre may be taken so remote from B that its circumference shall lie between the curve DBE and the tangent AC. Then every circumference described with a greater radius will lie without the curve DBE. Hence there are two classes of tangent circles, the one lying within the curve, the other lying without it; the former having a greater curvature than the curve, the latter a less curvature. Of all the circles that may be described from a centre in the line BM, there is one which coincides most intimate- ly with the curve DBE, and is therefore called the oscu- latory circle, or circle of curvature, and its radius is called the radius of curvature of the curve. 153. The curvature of different circumferences is meas- ured by the angle formed by two radii drawn through the extremities of an arc of given length, and this curv- attcre varies inversely as their radii. Let r and r' denote the radii of two circles, a the length of a given arc measured on the circumference of each, e the angle formed by the two radii drawn through the extremities of the arc in the first circle, and 6' the angle formed by the corresponding radii of the second. Then, since the angles are proportionjil to the arcs which measure them, we shall have the following : 180° a Q:^ „ (dx^+dy'^Y From these the radius and co-ordinates of the centre of curvature may be expressed in terms of the co-ordi- nates of the proposed curve. In order that the radius of curvature should always be positive, we have only to take the positive sign when the second differential coeflScient is positive, and negative when the second differential coefficient is negative. Ex. 21.4. Let us take the common parabola whose equation is y^^2px/ dy _p - dy'^ ^^ ••dx'~'^'^^^ds^^'^^' d'^y _ p^ ' ' dx^ ~ y^' Substituting these in (7), taking the negative sign, since the second differential coefficient is negaave, we have 3 K^+^)r p^dx^ yZ which readily reduces to the radius of curvature at any point of the parabola. (1.) If we make y=0, we shall have the radius of curv- ature at the vertex of the parabola. .•.'R=p= half the parameter. (2.) If we make" y=p, we shall have R=22}V2, which is the radius of curvature at the extremity of the lotus rectum. 204 DIFFEEBNTIAL CALCULUS. 157. To determine the radius of curvature at any point of an ellipse. Ex. 215. The equation of the ellipse referred to its cen- tre and axes is Differentiating, we obtain dy b'^x dy^ h^x'- dx~ cC-y'' " dx^^ a'^y'-' cPy_ ¥ dx^ d^y''" 3 ,idx^+dy^)i=^2^t±^^^I!^. a^y^ ¥dx^ and dxd^y—— „ 3 . As the second differential coefficient of y is negative, we take the negative value of R, which makes the radius of curvature positive. . ^_ {dx^+dy'^y dxd'^y _ _ {a'^yi+¥x^ydx^ ^ _ b*dx^ a^y^ ' a^y^ ' _ {a^y^+¥x^)^ ~ a^b^ ' which is a general expression for the radius of curvature at any point of the ellipse. If we take it at the extremity of the transverse axis, x=:a and y=0 ; .: R=:— . If we take it at the extremity of the conjugate axis, x=0 and y—b/ then R=-t. 158. To determine the radius of curvature for any pro- posed curve, we have the following ; , Rule. Differentiate the equation of the curve twice; find the values of dy and d'^y, and stSstitute them in the expres- sion EADIUS OF CUBVATUKE. 205 ~~ - dxd'^y If the radius of curvature for a particular point of the. curve he required, then substitute for x and y the co- ordinates of that point. 159. We will now find the radius of curvature for lines of the second order. The general equation of these lines is y^=mx-j-nx\ (1) Differentiating, we have (m+2nx)dx , , <^y- 2y. ■' <2) (m+^nxYdx^ and dx^+dy^=dx^+ t--^ , { 42/2-1- (ni-\- 2nxy}dx^ — 4y^ Differentiating (2) the second time, Inym? — {m-\- 2nx) dxdy d^y=^ -^p •-, [4:ny'^—{ni+2nxY}dx^ m^dx^ ' 4y^ 4y^ Substituting these values in the equation (4) (5) dxd'^y ' 3 \i(mx+ Jia;^) + (m -f- 2n.xY\^ we have R= gwi^ ' (^) which is a general expression for the radius of curvature in lines of the second order for any abscissa x. If we take it at the extremity of the transverse axis, a;=0, and we have that is, in lines of the second order, the radius of curva- ture at the vertex of the transverse axis is equal to half tlie parameter of thai axis. If it be required to find the radius of curvature at the vertex of the conjugate axis of an ellipse, make 206 DIFFEBENTIAIi CALCULUS. 2^2 m- '*=-«' x—a. Then, by a little reduction, we shall have as in the pveceding problem. In case of the parabola', in which n=0, we have 3 andifa;=0, R=Jm, which is the radius of curvature at the vertex of the parabola. 160. To determine the radius of curvature at any point of the cycloid. From the equation we have dx^ '^Iry—y'^ y'^'dy^ R=-- Iry — y'^' Substitute this value of (&^ and d^y in the expression {dx^+dyf)^ dxd'^y ' and reduce, we shall obtain 'R='2.-\/iry. Hence the radius of curvature is double the normal (Art. 131). 161. To determine the ra- dius of curvature at any poirit of a spiral. Let PHD be a spiral, P the pole, DB a tangent, and ^ DG a normal at the point of tangency D. Let C be the centre, and CD the radius of curvature corresponding to the point D. Draw PB and BADIUS OF CUEVATUEE. 207 PG respectively perpendicular to DB and DG. Then, considering the point D as describing the circle, the points C and P being fixed, we shall have CD and CP constant. Put PD^r^radius vector. DG=PB=^, CD=p=radius of curvature. Then (Geom., B. I, Theorem XXXVII.), PC'=:PD^+CD'-2CD . DG, (1) or _ _ PC^=}-2+p^— 2p/>. (2) Differentiating, we obtain Q=rdr—^dp ; (3) dr •'■ 1"='^=''^'^*^* o/ curvature. (4) Also from (2) and (4) we have — —2 dr"^ dr VG =r-^+r^,--2pr^; ••■^^=\^^^?^- (^) Cor. Join FE ; then the triangles DPG, DEF are sim- ilar, and we have DP:DG:: DE :DP, or r : p :: 2r^:2i^=DF, which is the chord of curvature at the point D. Ex. 216. Let us take a^ an example the logarithmic spiral. By Art. 135, its equation is log. r—B. Hence, by difierentiating, we obtain dr r dd~m'' and by Art. 152, p— -J^" .-.p-. Vi+ rw- hence p=— yT+w*^- dr And the chord of curvature =2^^= 2r. 208 DIFFEEENTIAL CALCULUS. EVOLUTES AND l2SrT0LUTES. 162. If a thread fixed around a plane curve be un- wound, and kept perfectly tense, its extremity will de- scribe another curve in that plane. The curve from which the thread is unwound is called the Eoolute of the curve that is formed by the extremity of the moving thread; and this latter curve is called Xh& Involute. Thus, in the diagram, when the thread kept tense is in the position P"C", its extremity has described the arc of a curve, APP'P", the nature of which will depend on the properties ofthe curve ACC'C". From the manner in which the involute is generated from the evolute, we may conclude that the portion of the thread which is disengaged from the evolute is a tangent to it ; and if a tangent line be drawn to the involute at P, P', or P", the line PC, P'C, or P"C" would be perpendicular to that tangent at the point of tangency; hence either of these lines may be considered a radius, and the point C, or C, etc., the centre of a circle of curvature at the correspond- ing point P, or P', etc. We may therefore define the evolute of a curve as the locus .ofthe centres of the cir- cles of curvature at the different points of that curve. It is farther evident that the difference of two radii of curvature is equal to the arc ofthe evolute intercepted between them, and that a tangent to the evolute is a nor- mal to the involute. 163. To determine the equation of the evolute of a curve. We have seen. Art. 156, that the circle of curvature is characterized by the equations dx^-^-dy^ ETOLTJTE OF A CURVE. 209 ''-"=-;^(y-^) (2) dx \ dP'y ) dy dx If we combine these with the equation of the involute curve, we shall obtain an equation from which x and y may be eliminated. 164. As an illustration, let us take the common pa- rabola as the involute, and thence deduce the equation of its evolute. Ex. 217. The equation of the parabola is y'^='2px. (1) Differentiating twice, we shall have dy_p dx~ y'' d'''y _ p^ Whence 2/^=(i'^/3)^, and x—az y'}-{-]>^ If we substitute 2px for y'^, we shall have X — a=— 2a;— ^, or X=z^{a—p). Substituting these in (1), we obtain (iJ=/3)*=§^(a-p); ■the equation of the evolute. If we make /3=0, then a=p ; hence the evolute meets the axis of a; at a distance from the origin equal to half the parameter. If we remove the origin from A to C, the equation becomes /3^ = 2?^°^ which is called the semi-cubical parabola. It passes through the origin C because, when a=0, ft—Q. 210 DIFFEEENTIAL CALCULUS. The curve consists of two branches symmetrically sit- uated with respect to the axis of x, and lies entirely to the right of the origin, because every positive value of. a gives two equal values of /3 with contrary signs, and every negative value of a makes ^ impossible. The branch ACM corre- sponds to the part AP of the involute, and the branch ACM' to the part AP'. The straight lin_e MQ has the same position as an inextensible thread wrap- ped along ACM, and disen- gaged from it as in the dia- gram, at the same time that it is kept constantly tense. 165. Hence, to determine the evolute of a curve, we have the following Rule. From the equation of the involute, or given curve, find the values of dy and substitute them in the equations dx^-\-dy'^ 2/-/3=- dhj Combine these equations with the given equation, and eliminate x and y ; the resulting equation will be the equation of the evolute. The only diflSculty the student will have to encounter is the elimination of x and y. . We may here remark that the number of cases in which this elimination can be readily effected is very small in comparison with the number of curves presented to our notice. Ex. 218. To determine the evolute of the ellipse. The equation of the ellipse referred to its centre and axes is cPy^-{-b^x^—a^^. EVOLUTE OF AN ELLIPSE. 211 xyuicicuuu dy V^x dx"' a^y ' dy^ ¥x^ ■■ dx^~a^y-i'> and d^y ¥ dx^ a^y^' Then dx^+dy^ y p- a^y , (a'>'y^+¥x^)y ~ a?¥ ' {a'y^+¥x^)y ■■■P-y- a^^ , (1) and x—a——~^(y—B), dx {a*y^+¥x^)x {a^y^+b*x^)x ••"='^ S^P ■ (2) From the equation of the ellipse, we have d^y^—a'^b'^—b^x^ or b^x^=a^'^—a'^y'^. If we multiply the first of these equations by d^ and the second by b\ we shall have a*y^=a*b'^—d^b''x\ and b*x^=d^b^—d^b^y\ Adding these two equations member by member, there results "^ a V + 5**^ = a*5^ + d^b^ — aWx^ — a^S V, (3) =5=(a*+a252_a2a;2— «22/2). (4) Putting for a^jz j^g value a^y'^+b^x\ we have a*y^+b*x'^=b^{a*+a^y^+b^x^—a^x^—a^y% =b^(a^-c^x^); vc^^a^-b^ (5) Again : taking (Eq. 3), we shall have a^y^+b^x'—d^iaW+b^—b^x^—b^y^), —d^ld^y^+b^x'^+¥—b^x^—b^y^), =a2(5*+cV)- (6) Substituting these values o£a*y'^-{-b*x^ in (Eqs. 1 and 2), we have 212 DIFFEEBNTIAL CALCULUS. Finding the values of y and x from (7) and (8) in terms of a and /3, and substituting them in the equation of the ellipse, we obtain (5/3)^+(aa)*=c^, the equation of the evolute. If a=0, then ;3= ± tt ; hence the curve meets the axis of y in two points equidistant from the origin. If /3=0, then a— ±— ; hence the curve meets the axis of X in two points equidistant from the origin also. Ex. 219. To determine the radius vector in the spiral of Archimedes when the radius of curvature is equal to the chord of curvature. The equation to the curve is (Art. 134) r=a9; dr. But $-^^^'-^^' (Art. 149); .;—!- i-=a, or »■* — p''r':=aV; P dr dr dr p{r'''+aP) whence ^^2{2r^-p-^- But ;^='-=+a^; . p2—-— . Qj. p-. 3 dT {r'^A-cP\^ Hence 0=?'---— V „ — i— =:rad. of curvature. ^ dp 2(r2+2a2) Now the chord of cnrvature=2»-^-= -, „ , „ „( . ^dp 2(»'2+2a2) EVOLTJTE OF THE EECTAKGULAE HYPEEBOLA. 213 Comparing this value of the chord of curvature with the value of the radius of curvatui-e, it appears that the radius of curvature is equal to the chord if (r^+a^)^=2r{r^+a'), or {r^+a')i=2r, from which we determine the radius vector a r——7=: V3 Ex. 220. To determine the evolute of the rectangular hyperbola referred to its asymptotes from the equation xy=a\ differentiation we obtain dx~ y_ a2_ ' X x^' dy-^ a* ••dx^~ ''x*' d-h, 2a? - x^ ' .■.y-.(i= x^ a^ ' 2a^ 2a;' X—a = X a* 2 2x3' /3= Zcfl x^ ' 2x +2a2' a~ _3a; , a* ■ 2 2a:3' (1) (2) or /3=^+5^« (3) By addition and subtraction we have _ a* Sgg 3a; x^ "+^~ 2^+ "2i"+T+ 2a2' a/a^ a x x^ aia a;y ~2\S+«/ ' _a/a^ a x x^ ~''""2V«5-3-+3--^j, _a/a aiV ~2\^~a} ' •••(«+^)*=(f)*^l)' (4) (5) (6) 214 DIFFERENTIAL CAlCHLtTS. or and or 0) (8) Whence (a+/3)^-(a-/3)*=4p\ =(4a)^, the equation of the evolute. Ex. 221. To determine the radius of curvature of an ellipse in terms of the angle which the normal makes ■with the transverse axis. The normal PN =yy 1 + Put the angle PN'R=e; PR y then sm. e=p^=pjj; , sin. e= 1 4 1+. But B ,_ the equation of the ellipse ,^_ " dx Ba; and and Hence difl djf^. A-v/A2_£b2', ^^'^dx^) ~(a2. A2(A2-cB?)" 3 A2-£B2 » e^aiA's AB )' R (A2-aj2)^ dPy ds? e^x^Y AB (A) But sin. 2 e VANISHING FEACTIONS. 215 1 A2— a;2 .-. A2 sin.2 d—e^x^ sin.^ e—A^—x\ , Ag(l-sin.ze) ~ 1— e^ sin.2^6 ' ^ ,_ AV(1— sin.^e) ^^~ l-e2sin.2 ' 1 — e^ sin.2 a 1 — e^ sin.^ Q' Substituting this in Eq. A and reducing, we have K— 3. (l-e2sin.2ep B2 A2 If 6=0 or 180°, R = a; i^ = 90°, R=-g, as in Ex. 215, CHAPTEE IV. ON TANISHING FBACnOlTS. — MAXIMA AND MINIMA OF FUNCTIONS OF ONE AND TWO VAEIABLES. CHANGING THE INDEPENDENT VARIABLE. TANGENT PLANES AND NORMAL LINES TO CUEVED SUBFACES.- — EADITJS OF CUEVAXXTEE OP CURVES OF DOUBLE CUEVATUEE. 166. It was shown in Algebra (Art. 87) that the ex- pression - is not always a symbol of indetermination, but that it sometimes denotes the presence of k common factor, which being canceled, the true value of the frac- tion may be obtained. We shall now proceed to find the value of any fraction whose numerator and denom- inator both vanish for some particular value of the va- riable. ■ -Pix-ar ^®* Q(a!-a)° be the form of the fraction, in which P and Q are func- tions of «, which do not reduce to zero when x—a. We shall have, if we substitute at first, 216 DIFFEEENTIAL CALCULUS. P(a;— a)" Q(a;— a)°~0' The value of the fraction, however, may be zero, finite or infinite, depending upon the values of m and ri. Thus, if m be greater than n, F(x— g)"-" If TO be less than w, P P.O. If OT be equal to n, then P o . Q=a finite quantity. Put X equal to the numerator, and X' equal to the de- nominator of the prpposed'fractiop, each being such a function. of x that for some particular value of x they both reduce to zero ; then give to x an increment, A, and develop by Taylor's Theorem, gpd we have _ (?X cZ^X A2 drsx A3 -^ +'^ ^+"^ 172+.;^^ n2T3+' ^^°- ■^ + <& ^+ (?a;2 1 . 2+ (?a;3 1.2. 3+' ^^^• If we substitute for x its value a, both X and X' re- duce to zero ; then, dividing both numerator and denom- inator by A, we have ^ ^ _A_ ^ A^ dx'^ dx^ 1 . 2+ c?a:3 1 . 2 . 3+' ^'^°- <^X' (^^X' A £3^ A^ c?a;+c?a!2 i.2+(?a;3 1.2.3+'^*°- (^X Now, making A=0, this fraction reduces to -^=7 ; and W-A. if, by substituting for x its value a, we find that each term of this fraction becomes 0, we must expunge these terms, and divide again by A both numerator and denominator, until we obtain a fraction of which the numerator and denominator do not both reduce to zero. Hence we ob- tain the following VANISHING FBACTIONS. 21^ Rule. I^nd the differential coefficients of the numerator and denominator separately, and if one or the other of these do not redicce to zero lohen the value ofx is substituted, the ratio of these differential coefficients will be the value of the fraction ; but if both reduce to zero, we must dif- ferentiate again, until we find a differential coefficient which does not reduce to zero when the value ofx is sub- stituted for x; then the ratio of these differential coeffi- cients will be the value of the vanishing fraction, Examples. Ex. 222. Find the value of the fraction tan. x—x ^3^j^^whena:=0. If we substitute for x its value =0, we shall have tan. 0—0 0— sin. 0~0' Put M=tan. x—x; du then ^~ (^ + *^°- "') — 1 5 du and when x=0, -^=0, and du' then -^=1—008. a; / , , du' and when x=0, "dx^^' Diflferentiating the second time, we have ■^—2 tan. a!(l+tan.2 x) ; d^u and when x=0, 'da^~^ ' and when x=0, -j^=o. Differentiating the third time, we have K 218 DIFFEEBNTIAJL CALCULUS. ^=2(l+tan.2 xy+4: tan.^ x (l+tan.^ a;) ; _ , d^u and ■when x=0, dt^~^' dV "^3-= COS. a;/ . , dV and when a;=0, "^5""^' dhi ■ ^ -=-=2=value of the fraction. dx^ Ex. 223. Find the value of the fraction — — r when a;=l. X — 1 _ , du 1 Put u=\og.x; .•.^=-; - , du and when a;=l, Z"~-^' du' du dx •'• ^~^~^^^ value of the fraction. dx Ex. 224. Find the value of the fraction jg 2 when 33=1. x^—1 Ex. 225, Find the value of the fraction ^_2 when x=l. Ex. 226. Find the value of the fraction a"— 3!° log. a-log. X ^^^^ «'=«• Ans. i. Am-l- Ane. nop. YANISHING PKACTIONS. 219 Ex. 22^. Find the value of the fraction 1 a; „„^ ,_^ when £0=1. cot. ^irx Ans. -. IT 167. In the last article the numerator and denomina- tor of the fraction were developed by means of Taylor's Theorem; but when this development can not be made by that Theorem, recourse must be had to the common al- gebraic methods of obtaining their expansions. Thus, if "-/'(a;)' where /(as) and/' (a) each become=0 by assigning to x the particular value a, let a-f A be substituted for x; and suppose that by the operations of Algebra we obtain , AA'+BA^+, etc. ^ =A'A"'+B'A«'-f , etc' then, if a be greater than a', we shall have, by dividing both terms by h°-', ^_ Ahr-'+Bh^—'+, etc. "^ ~ A'-fB'A*'-'+, etc. • Now if A=0, we shall get M = 0. Again : if a=a', we obtain, by dividing by A% , A-f-BA^-'-<-, etc. ** ~A'+B'hf-'+, etc' where, if A=0, we have A ^ . u=-^i=a finite quantity. Also, if a be less than a', then, by dividing by A", we obtain , A-f BA»-+, etc ^ -A'A"'--f B'A^-+, etc' which, on the supposition that /t=0, gives the corre- sponding value of w=oo. Hence the particular value of the fraction will be zero, finite or infinite, according as the least exponent of the development of the numerator is greater than, equal to, or less than that of the denominator. We may remark 220 DIFFEEENTIAL CALCULUS. that this method is applicable in all cases, and may fre- quently be used with advantage even in those examples in which Art. 166 applies. Ex. 229. Find the value of the fraction r— — - — —TV when 83=1. log. x—x+1 If we substitute unity for x, this fraction becomes 0* Since the value of x that makes this fraction evanes- cent is unity instead of a, we put 1+A for x/ then (l+A)'+^-(l+A) log. (1-1- A) -A ■ Developing the terms of the fraction, we have ^H 1(1+^)^^+1 etc. Divide by h^ ; then l+i{l+h)h+, etc. -i+P-iAHj etc.- Now if A be made equal to 0, the value of the fraction is —2. This example may also be solved by the first Rule by differentiating twice. Ex. 230. Find the value of the fraction ^!=5^^±^whena,=a. Ex. 231. Find the value of the fraction Ans. 0. i ^ when x=a. {x-ay 3 Ans. (2a)^. 168. The method of determming the value of a frac- tion whose numerator and denominator become zero for some particular value of the variable, leads us to obtain the value of any fraction whose numerator and denom- inator become infinite for some particular value of the variable. VANISHIKG FEACTIONS. 221 'Letf{x) and /'(a;) denote two different functions of X, such that the fraction (jQ when x—a^ then it is obvious that f\x)- 1 -Q- /H So that if we determine by either of the preceding meth- ods the value of this last fraction, the value of the pro- posed fraction will also be obtained. Ex. 232. Required the value of *^°- Ta , when a;=a. a-'(a;2— a2)-'a;2 This may be put under the form «B ■ . cot. ;;- ^ 4a 2a Ans. — — . It 169. In like manner we may find the true value of the product of two functions, one of which becomes zero, the other infinite for some particular value of the variable. For, let /(a;) and /'(a) be the functions. Then, if by substituting the value a for x in each of these functions, we have /(«) =0 and /'(a;) = co, or/(a;) /'(«) =0 x oo, (1) their product may be put in the form -i- «■ (2) 170. Again: the true value of the difference of two functions may be obtained in the case in which the sub- stitution of a particular value for the variable causes each of them to become infinite. Thus, if /(a;) = co , and f{^ = oo, then /(a;)— /'(a;) = 00 — 00 (1) may be put in the form 222 DIPFEEENTIAL CALCULUS. 1 1 f{x)-f{x) ^^^ 1 0' Ex. 231. Find the value ot the function a{l—x) tan. — when x=^l. Here the first factor a(l —x) becomes zero, and the second infinite when X—\; but (Art. 169) ira; a{\—x) a(l-a;)tan.Y= ;^. cot.-2- Put w=a(l— ce); Put M'=cot. l^x; du' -kI „ TrtcN t^^° ^=-2(l+''°t-'TJ5 dx a du' nl TTtcX ' -^..2^1 + cot.^yj la and when a!=l, = — . ■K Ex. 232. Find the value of the function 1 X -•, , when £B=1. log. X log. X This becomes co — oo. But, by (Art. 1'72), 1 X _ 1— a; iogrK~Io^~log. a;~0 ''^^^^ ^-^• du Put w=l— a;/ then -^= — 1. <7m' 1 .'. ^=:— l=value required. dx, VANISHING FRACTIONS. 223 Miscellaneous Examples. Ex. 233. Find the value of the fraction log. a: when £0=1. Ans. ir Ex. 234. Find the value of the fraction e'_l_?(l+{B) when a;=0, and e the base of the ITaperian log. Ans. 1. Ex. 235. Find the value of the fraction a°— a!° , when a;=a. log. (a)°-log. {xf Ex. 236. Find the value of the fraction e"'— 1 -r-. 7r~, — \ when 83=0. e log. (1+a;) Ex, 237. Find the value of the fraction 2 1 -5 — 7 r when £6=1. a;'-— 1 £8—1 Ex. 238. Find the value of the fraction a— (a^— a;^)* „, „ 5^ — '— when 83=0. :. a'. Ans. 2. Ans, —\. Ans. ^. Ex. 239. Find the value of the fraction (a+2£);)*-(3a)* Ans. iVe. Ex. 240. Find the value of the fraction aie^+l-e^'-a! , jr- ^-^ when a;=0. e — 1 Ans. —1. 224 diffeeential calcttlus. On the Maxima and Minima oe Functions op one Vaeiablb. 171. Definition. — A maximum value of a variable function is that value which is greater than the value which immediately precedes or immediately follows it. And the minimum value of a variable function is less than that which immediately precedes or immediately follows it. If «=/(»), and let x become x+h, and x—h, then the correspond- ing values of the function will be u=f{x), (1) u'=f{x+h), (2) u"=fix-h). (3) Then, by Taylor's Theorem, we shall have ^ du dHi h^ d^u- h^ ■ u ^f{x+h)^u+-^ji+^ 171+^ t:27§+' ^*''- (^) and du d^u l? d^u h^ u =/(«'-/0=W-^A+^T:2-^ 17273 + ' ^*°- (^> Now, if M is a maximum, we must have from the def- inition u' and ■«" both less than u/ or, if m is a minimum, we must have u' and u" both greater than w, however small h may be assumed. By (Art. 66), h can be taken du of such magnitude that -j-h shall be greater than the sum of all the terras which follow it. Now, in equation (4), u is less than u', and in (5) u is greater than u", which can not be the case if m is a maximum or a mini- mum ; and therefore, when it is either a maximum or a minimum, du_ dx~~ When the first differential coefficient vanishes, d^u h^ dHi h^ '^ ="+^2 T72+&? r7271+' ^t°-' d'^u h^ d^u h^ and u =M+^ T72-^T7273+' ^t°-' which, on the principle that h can be taken so small that MAXIMA AlfD MINIMA OP PUNCTIONS. 225 ^5- ^—5 snail be greater than the sum of all the succeed- ing terms, u' and m" are both greater than u when -^ is positive, and both less than u when -^-j- is negative. That is, u will be a maximum when the values of x, ob- tained from the equation ^=0, renders the second dif- ferential coefficient negative, and a minimum when these values make the second differential coefficient /(osjiiw. It may happen that the roots of the equation du when substituted in the second differential coefficient, cause that coefficient to vanish. The signs of the devel- opments will then depend on the signs of the third dif- ferential coefficients ; and as these signs are different, we must have d'^u The conditions of maximum and minimum will then be, that if the values of x obtained from the equation du being substituted in the fourth differential coefficient, make that coefficient negative, it will be a maximum, and if they make it positive, a minimutn. Hence we de- rive the following Rule. Differentiate the function ; put the first differential co^icieiit equal to zero, and find the roots of that equa- tion. Then substitute these roots in succession in the suc- ceeding differential coefficients, stopping at the first thai does not vanish. If this is of an odd order, the values that we have used will not render the function either a maxim,um or a minimum ,' but if it is of an even order and negative, the function wiU be a maximum, if posi- tive, a minimum. K 2 226 DIFFEEBNTIAL CALCULUS. 172. Before we apply the preceding theory to exam- ples, it is proper to state that if the given function has a constant factor, that factor may be omitted. If a function is a maximum or a minimum, its square, cube, or any other power must obviously be a maximum or minimum also. Hence, when the function is under a radical sign, that sign may be removed. The rational function may, however, become a maximum or a mini- mum for more values of the variable than the original root ; indeed, all values of the variable which render the rational function negative, will render every even root of it imaginary ; such values, therefore, do not belong to that root. Mcamples. Ex. 241. Let u—x{a—x)^ ; to find the value of a; which will render u a maximum or a minimum. Differentiating, we have du ^={a-xy-2x{a-x), and ^=63!— 4a. Now, if M be a maximum or a minimum, we must have du_ or (a— a;)^— 2a!(a— a;)=0; .*. x—^a, or x=a. Substitute ^a for a; in the second differential coefficient ; d^u And substituting a for x in the same, we have dht The value x=^a renders u a maximum, and x=a makes it a minimum. Ex. 242. Find the value of d which will render the function M=sin. 6— vers, d a maximum. du -3a=cos. e— sm. 9, d^u and -^——sm.B—oos.d. MAXIMA AND MINIMA OF FUNCTIONS. 227 Putting the first differential coefficient equal to 0, COS. e— sin. 0=0; .-. sin. e=cos. 0=sin. (90°— 6) ; .•.0=9O°-e; .-.0=45°. Substituting the value of 6 in the second differential coefficient, = — V2. The second differential coefficient being negative, proves that m is a maximum. Ex. 243. Find the values of x that will render u a max- imum or minimum in the equation M=a3*— 16a;3+88a!2— 192a;+150. Differentiate; then ^—4x^—4Bx^+l'76x—lQ2. Equating this mth zero, we have 4x^—4:8x'^+l16x—192=zO, K^— 12a;24- 44a;— 48=0, an equation of the third degree, whose roots are 2, 4, and 6. If we differentiate again, we shall have -^=12x^-96x+ll6. Substituting 2 for x in this second differential coeffi- cient, the result will become 4-32, which shows that u is a minimum; substitu- ting 4 for X, we have — 1 6 : this being negative, indicates a maximum. Again : substituting 6 for X, we obtain +32, a min- imum. The adjoining dia- gram represents the form of the curve. There are two minima values cor- responding to the abscis- sas 2 and 6, and one max- imum value corresponding to the abscissa 4. 228 DIFPEEENTIAL CALCTJLtrS. Ex. 244. Find the values of x wMcli render a maximum or a minimum. Ans. +a,a max. —a, a min, Ex. 245. Find the value of x which will render (CB — 1)" a maximum or a minimum, m and n being positive whole numbers, and w>m. n+m Ans. x= —=a, max. n — m But if w +< d\i dhi / d^u Y dx'^'^ dy^ \dxdy) /dhiy \dx'^) -k^> and therefore will not change its algebraic sign what- ever values are given to h and k, provided dhi dh( / dhi y dx'^ ' dy^^\dxdy) ' The truth of the proposition is therefore manifest. Mcamj^les. Ex.271. Let M=!B*+2/*— 4aa;2/^ to determine x and y when u is a maximum or minimum. ■-z-=4x^—4:ay^=0; .:x^=ay^; du - „ „ — = 42/3 — 8axy = ; .-.y^— 2ax. From which we find x=z±aV2, and y—a-s/i. Differentiating again, we have -— =12332= 24a2, dv? ' 234 DIFFERENTIA!, CALCULUS. — = 12yz— 8aa3=i6aV2. dHi dhj, d'^u ■ ■ ffe^ ■ dy^ ^dxdy' , Since the algebraic sign of y-^ and -5-5 is positive, the values of x and y as above give u a minimum. Ex.272. Let M=a{sin. cc+sin. y+sin. (x+y)}; find the values of x and y when m is a maximum. ^=a{cos.a;+cos. (a;+y)}=0, (1) du ^=a{cos.2/+cos.(a:+2/)|=0; (2) .-. x-y. Substituting y for x in Eqs. (1) or (2), we have cos. a;+cos. 2a:=0; .-. cos. a;+2 cos.^a;— 1=0. Whence cos. a!=J; .-. a;=60°=w. dHi ^=«{— sm.a;— sin. (cc+y)} = — «[sin. 60°+sin. 120°}, =.—a^/^•, dhc _ ^=a{-sm.y-sin. {x+y)]=-aV3. dhi d^u d^u dx^ ' dy'^ -^dxdy ' ,. , 1 • • .^^" -.dhc . and smce the algebraic sign of ^-^ and ^-j is negative, therefore i«=-|a-v/3=a maximtim. Ex. 2V3. A'cistern in the form of a rectangular paral- lelopipedon is to contain a certain quantity of water; determine its form so that the interior surface shall be a minimum. Ans. The base must be a squarof and the depth =half the length or breadth. Ex. 274. In a given circle inscribe a triangle whose perimeter shall be a maximum. Ans. The triangle is equilateral. CHANGING THE INDEPENDENT TAEIABLE. 235 Ex. 275. If a square be inscribed in a given circle, a circle in that square, again a square in that circle, and so on continually, prove that the sum of all the inscribed squares is equal to the area of the square circumscribing the given circle. Ex. 276. If the greatest rectangle be inscribed in an ellipse, the greatest ellipse in that rectangle, again the greatest rectangle in that ellipse, and so on continually, prove that the sum of all the inscribed rectangles is equal to the area of a parallelogram which circumscribes the given ellipse. Ex. 277. The centres of two spheres are at the extrem- ities of a given straight line, on which a circle is de- scribed ; find a point in the circumference of the circle from which the greatest portion of spherical surface is visible. ^ Ex. 278. Of all the ellipses that can be inscribed in a rhombus whose diagonals are 2a and 2b, prove that the a greatest is that whose transverse axis is — ^ and conju- . . b gate axis is —i=. To Change the Independent Variable. 175. If we change an equation between x and y to the form x=f{y), y is called the independent and x the de- pendent variable. Let it be required to change the differential coefficient found on the supposition that x=:f{y) into another in which both y and x are considered functions of a third variable, t. • Let a:=/(y), (1) and _ y^f\t). • (2) Then, by virtue of these two equations, we must have x=f"{t). (3) Now, if h and k denote the contemporaneous incre- ments oit and y, we shall have from (2) and (3), dy d'^y li? d^y h^ ^^tt^'^W 17^+"^ 17273+' "'°-' (*) rzJ, or \—cos.Y = — {x^h+y^k+z,t). (A) Again : {x,+hY+{y,+hY+{z,+T)'^=l. Actually squaring, and reducing as above, we have a!,A+y,A+g/=-i(A2+A;H^^). (B) But 1 —COS. V=2 sin.2 ^Y, when the arc is indefinitely diminished. Substituting in (A) the value of 1 —cos. V, and that of x^h+y ik-{-z^l, we obtain The radius of curvature, which we will denote by p, is ds ds But, by Art. 91, dx dy ds "''=Ts^ y-=d^^ ^°^ ^'=ds- Moreover, when the normals become consecutive, -=(ty. -(?)■ -(sr ■■'^ //d^-xy Id^yV /d^zY'' or p= Vid^'xY+id'yY+id^Y ELEMENTS OF THE INTEGRAL CALCULUS. 1'79. The Integral Calculus explains the method of finding the function from which a given differential has been deduced. In the Differential Calculus we have a system of rules by means of which we can deduce from any given func- tion a second function, which is called the differential of the former ; but in the Integral Calculus the rules are not so direct ; they are deduced by reversing the process by which we obtained the differential from the function. The terms integral and integration are taken from the infinitesimal calculus. In the infancy of the science, dif- ferentials were considered as infinitely small quantities; hence the original functions from which these differen- tials were deduced were called the sums of the infinitely small elements ; and the process by which these original functions were obtained from their differentials was called the summation or integration of the indefinitely small component parts, and the operation was expressed by the character / prefixed to the differential. Thus, the integral of stfcfo; was written fx^dx. This character is still retained by most writers upon the subject. We shall first take up those differentials which are functions of a single variable, and show how they may be integrated. 180. Let us find the integral of xl'-'^dx. We have seen (Art. 48) that d.a^=nii?~^dx; ,, d.x"" .■.af-^dx= . n INTEGRALS^- 241 Now as the numerator of the fraction in the second member is only the indication of a differential, its inte- gral is evidently the function a3°. Therefore SB" from which we derive the following Rule. To integrate a monomial of the form x''~'dx, add unity to the exponent of the variable, divide hy the ex- ponent so increased, and drop the differential of the variable. Mcamples. dx Ex. 1. Find the integral of — j. x^ Bringing the denominator into the numerator by changing the sign of its exponent, we have ^_ -f , £B* Then, by the rule above given, f^=fx-^dx=^_ ^Sx^ . x^ 181. In (Art. 41) it was shown that the differential of a variable function and the differential of the same func- tion increased or diminished by a constant quantity were both equal ; hence the same differential may answer for several integral functions differing from each other in the value of the constant term. In integrating a differential, therefore, we must always add a constant quantity to the first integral that we ob- tain, and then find such a value for this constant as will characterize the particular integral required. The integral first obtained is called the indefinite in- tegral, but after it has been corrected by the method just ' named it is called the definite integral. rdx X , . , .-. _/ —g=Zx^-{-c, G bemg the correction. x^ Ex. 2. Integrate ax^dx. 242 INTEGEAL CALCULUS. 182. By (Art. 38), the differential of a constant multi- plied by a variable function is equal to the constant into the differential of the function. Hence we may conclude that the integral of a constant into a variable function is equal to the constant into the integral of the differen- tial. Therefore, if the expression to be integrated has a constant factor, it may be placed in front of the sign of the integral. Hence fax^dx—afx^dx, x' =\ax'-\-c. 183. There is one case in which the rule (Art. 180) does not apply : that is, when the exponent of the vari- able is equal to minus unity. Thus, by the rule, ,, a;" 1 /a;-Wa;=-Q-=-=a>, which is not true. We can, however, take a factor from the numerator and place it in the denominator by chang- ing the sign of its exponent. dx :. x^dx= — . X Now we know from (Art. 69) that d.log.x=:—, . taken in the Naperian system ; /dx -=l0g.CB + C. 77ie integral of every fraction whose numerator is the differential of its denominator is the Naperian logarithm of the denominator. Integrate the following : Exs. (3.) fxhx. (4.) /^. (5.) r^. y^ -' bx^ (6.) fWdx. (7.) fx^dx. (8.) fSa^dx. (9.) f^x^dx. r^dx , , pdx (lO-)./-^- (11-) /i^^^^a'- (12.) y^- {\^:) JaxHx. (14.)/«.3,.. (,5.)/^^. (16.)/^. (1.)/^^. INTEGRALS. 243 (IB.)/^. (19.) j^. (20.) J ^. (21.)/-^. (22-) yn^- 184. Since the differential of the sum or difference of any number of functions is equal to the sum or difference of their differentials, it follows that the integral of the sum or difference of any number of differentials must be equal to the sum or difference of their integrals taken separately. Thus, if Ex.23. du^^a^xdx—Qax^dx+dx, then u—ia^fxdx—Qafx^dx-^fdx, u=a'^x^—2ax^+x+c. _, , , _, x^dx , % , adx l!iX. 24. du~xdx— -Ybx^dx r, u—fxdx — J x^dx-\-bJ x^dx—a I — ^. 185. If we have a polynomial of the form {a+bx+cx'+, etc.)°(fe, where n is & positive whole number, we can integrate it by first raising it to the power indicated by w, multiply- ing each term by dx, and then integrating each term separately. Thus, if we have /(a +bx+ cx^) ^dx— d^Jdx + 2 abfxdx + (2 ac + b'^)Jy?dx + 1bcjQis^dx+c^fx!^dx; and integrating each term seiparately, we have f{a+bx+cx^Ydx=a'^x+abx^+l{1aG+b'^)x^ +ibcx''+\c^x^+C. 186. Many expressions may be transfonued into mo- nomials by the introduction of an auxiliary variable, and then integrated by the first rule. Let du= {a+bx'ymx'^-'^dx. Put a+ba^=zv; then nbaf~^=dv/ , dv nb' Substituting these in the given expression, we have mv^dv 244 INTEGEAL CALCULUS. m V ,p+i 'nhJ nb' p+V and, replacing the value of v, Hence, to integrate a binomial differential when the exponent of the variable without the parenthesis is less by unity than the exponent of the variable within, we derive the following Rule. Multiply the binomial, with its primitive exponent in- creased by unity, by the constant factor, if there be one ; then divide this result by the continued product of the new exponent, the coefficient, and the exponent of the va- riable within the parenthesis. Madmples. Ex. 25. Integrate t?M=(a2+6£B3)~2a!2cfe. Ans. u=\{a^-\-6a^)^-^c. . Ex. 26. If c7M=(a-a!)-'6cfo;, *^^° ^=4(i=^*+''- 3 Ex. 27. If du={a\-hx'^yxdx, 5 then u=(2±Mll. 66 Ex.28. If du=ax^dx{a^-\-6x^)~^, then u^^^^+^+c. 27 Ex. 29. If c?M=3(2— 3a;=)~^a!*-\-cxfdx can he integrated when either m or n is a positive whole number. For, when n is positive, we can raise the binomial to / 246 INTEGEAL CALCULUS. the power indicated by n, and then multiply every term by ax'^dx, and integrate each term separately. And when n is fractional or negative, m being positive and entire, put b-\-cx=v; v—b then x= , and dx= — ; , a/v-bY , a r/v—bY , -'■''^'cj \-r) ^"^": Ex.32. Let du=2xdx{l—3x) '■'. Put 1— 3a;=uy 1-v then a;=— 5— , and dx— —^dv/ .: du— — f (1 — ■u)«'~^(?w, --Iv'^dv-Y^-o^dv; or ^^(l_3a:)l_4(i_3a,)i+c. Ex. 33. Integrate du—bx^{a—x)^dx. Ans. u=^ab{a—x)^—\ab(a—x)^—^(a—^-\-c. Ex. 34. If du— , —, u—^/d?^■\-x^^-\-0. Ex. 35. If (7m- Va^— x" w=— Va2— x2+C. Ex. 36. If du= r, INTEGRATION OF CIECTJLAE FUNCTIONS. 24'? xdx Ex. 37. If du- i«=(a2-a;2)~i+C. Ex. 38. If du={a—x){2ax—x'^Ydx. Put 2ax—x'''=z; :. (a—x)dx—^dz, and {a—x){2ax—x^)^=^z^dz/ .-. f{a-x) {2ax-x^yidx=^2EEz:^+C. o -^ _„ , (a+x)dx Ex. 39. If du= , ^ ^=, V2ax+x^ u — V2ax+x'^-\- C. _ ^„ , V^ax—x^ , Ex. 40. If (?w=: ;= .dx, Vx ^t=-^2a-x)^+C. Ex.41. If du=(a+bx+cx^)^{b+2cx)dx, M=|(a+5c(;+ca;2)^+C. Ex.42. If du= bx^dx 3c8*+7' M=T^log. (3a!«+7)+C. Integeation op the Diffeeentials of Cieculae Functions. 189. If X denotes the arc whose sign is u, we have, by (Art. 84), du dx— , ; VI— m2' f du . :.x=: I ■ =sin.~' M+C. If we let M=0, then £B, or the arc=0+C, and if m=1, then a;, or the arc=|7r+C. Now, taking the difference between these two, we shall eliminate the constant C ; then i"' du Vi -ifi ^ £ 248 INTEGRAL CALCULUS. f du The expression I . is called the. integral taken between the limits: t<=0, and m:=1. Ex. 43. Sdx= I — ,~ — =cos.-' M+C. Ex. 44. / . =versin.-' ti+C. Ex. 45. J j-p^=tan.-' M+C. Ex. 46. j -j^p^=cot.-' M+C. — 7=== sec.-' w+C, „ r du Ex. 48. / . „ — cosec.-' u+C. J uyw'—X _, T du du Ex. 49. Integrate Va^- Put ^=^^; u ^ du _ then ~=^'v and — =dv, a ' a Substituting these in the given expression, we shall have f dv I , „ =sin.~' w=siu.~' -+C. u /— — OAAJ. U Dili. Vl— -yz a _, _ du Ex. 50. Integrate u =cos.-' -+C. a ' Ex.61. Integrate ^jq^. C dx r dx INTEGRATION OF CIECULAE FUNCTIONS. 249 Pat %=V^; Substituting, / dx 1 /• and dx=adv. Substituting, / tfe! _ r dv V2ax—x^ J V^v—v^ versin."' v. =versin. -+C. In these integrals the radius of the circle is unity ; but as it is frequently desirable to integrate differentials in which the radius is r instead of unity, we shall exhibit a few of those which occur most frequently to that radius. Thus, Ex.53. / . - =sin.-' M+C. Ex.54. / — . ,- ==cos.-' M+C. r Rdu . , / , „ ==Yersin.~' m+C. Ex.55. , ,__ V2Rm— M^ '• J Rz+m2^ Ex. 56. J j^2^M2 =tan.-' i«+C. Ex. 57. Integrate (?M=sin. a; cos. a; . dx. Put sin. »;=:«/ then, by differentiating, cos. xdx=dv. Substituting for sin. x and cos. xdx their yalues, we have L2 250 INTBGEAL CALCULUS. 1 or '^=^''-' 1 . =- sin.°a!+C. Ex. 58. If du=sm.? X COS. xdx, then ■ M=i sin.^ x+C. Ex. 59. If du —sm? X COS. xdx, 3 then ^< = f sin.^ as + C. Ex. 60. Integrate. cZm= (tan." e+tan.°+'' d)dQ, =tan.° e(l+tan.2 B)dB. Put tan. ft=wy then, by differentiating, we have (l+tan.2e)cZ9=(?w, and substituting, du—v'dv; :. u=—r-7 tan.°+' 0+C. n-\- 1 Ex. 61. If (?M=(tan.5e+tan.'' e)c?0, then M=|- tan.^ e+C. Ex. 62. Integrate du=sva. 2xdx. By (Trig., Prob. XVI.), sin. 2a;=2 sin. x cos. x; ' .: du—2 sin. x cos. cc^a;. M=sih;?£e+C. Ex.63. If du={cot.^e+_cot.^ d)de, u=—icot.^8+G. Ex. 64. Integrate , Put Vx'^dzo^—v; then aj^ia^—^a. hence, by differentiating, xdx=vdv. Putting this into a proportion, x:v:: dv : dx, or, by composition, x+v:v:: dx-\-dv : dx; dx+dv dx dx ■• x+v ~ V ~"-v/a;2±a2' /dx , , \ ■ =:log. (x4-v), or, substituting for v its value, :log. («+ VV±a^). INTEGRATION OF CIECULAE FUNCTIONS. 251 Ex. 65. Integrate , :. Va;2±2aa3 Put V'a;2±2aa;=uy then x'^±2ax=v'^, xdza—Vv^+c^ ; vdv .: dx— I — ^=. _, dx dv Hence . -= , =, •va5^±2aaj Vv^+a^ and, by Ex. (64), /^Sl^^log. («+ V^S^T^) ; •/^ Ex. 66. Integrate ^q;^. dx , log.{Vx^dz2ax+x±a} +C. / 2dx _ r Idx 4:-\-X^~J x^' 1+4 Put J=-y2; X .: -^=v, and ^dx=dv. Substitute, and we have dv I Ex. 67. If 3c?a; =tan.-' «+C, tan.-' r+C. '""1+40:2' M=| tan.-' 2a;+C, ^ ,„ , xdx Ex, 68. If (?M= «= — ;rr sin."' a;2\/_+C. 2-v/* V a^ 252 INTEGEAL CALCULUS. Ex.69. If Ex. 10. If du M= — 7=-- tan.-' x^\/-+C. sVab V a 'Va-bx"-' n- n'\/b „ , mdx Ex. 71. If du-= -^ sin.- £c°\/-+C. vo V ffl M=sin.~' mx+G. 2a; 4-1 Ex. ?2. If (?e<=-r-; rrr.dx, x^+x+1 ' w=log. (a32+a3+l) + C. Integeation bt Seeies. 190. Every expression of the form f{x) dx that can be developed in the powers of x may be inte- grated by a series. For if we suppose that f{x)=A„+A^x+A2X^+A^3+, etc., then f(x)dx=Agdx+A-^xdx+A2X^dx+A^^dx+, etc.; •■• ff{«i)dx=A^x+iAiX^+iA^^+iA^*+, etc. We have simply to develop the function of x into a series, multiply each term of the series by dx, and inte- grate each term separately. Ex. 'TS. Let us integrate , —dx^x^—l) ^. If we develop (x'^—l) *, we shall have (x^-l)~i=x-'+ix-^+—x-'+h^^ar''+, etc.; ^ ' ^ 2.4 2.4.6 ' dx , , „ 1-3 „ 1.3.5 .-. -i=^=j=«' dx+ix-^dx+j^x-^dx+j-j-^x-^dx+, etc.; therefore r dx 1 1.3_ 1.3.5 J v'S^HT" ^^•'"■~4£(;2~32:b*~ 2.4.6. 6a;6~'®*°''+ ^- INTEGEATION OF DIFFBEENTIAL BINOMIALS. 253 lUTEGEATION OF DIFFERENTIAL BiNOMIALS. 191. Every binomial differential can be represented under the general form p in which m and n are whole numbers, and n positive. For, if m and n were fractional, and the binomial of the form we might substitute for x another variable whose ex- ponent should be equal to the least common multiple of the denominators of the exponents of x. We should then have an expression in which the exponents are whole numbers. Thus, put x=v^; then dx=Qv^dv, 1 1,E p and x^dx{a+hxV'^—&v''dv{a+bv^)\ Again : if w be negative, or H a!°'-'Ja;(a4-5a!-°)'J, 1 we may put x=-; th en dx=— ir^dv ; and substituting, we have E E a!°'-'c?a;(a+ Sa;-")' = —v-'^-^dv {a+bv^Y, an expression in which the exponents of the variable are whole numbers. If the variable x is in both terms of the binomial, as p we may take a;' from under the vinculum, and we shall have X ' dx{a+ba?-'')\ where only one of the terms within the parenthesis con- tains the variable. 192. Suppose, then, that it be required to find the in- tegral of p du=x'^-'^dx{a+ba?Y. 254 INTEGEAL CALCULUS. Put a-\-baf=v'' ; then baf=v'^—a, m m and b°x'"— («' — a) ° ; ni an expression which is easily integrated when — is a whole number. Hence every binomial differential has an exact inte- gral when the exponent of the variable without the pa- renthesis increased by unity is exactly divisible by the exponent of the variable within, 193. Again: let us make a-^-bx^—vV; then af=a(«p— 5)-'; m m .-. a;'°=a°(w''— 6)~°; qa° ■y'~'(?u ^ (v^-by''' ■ p £ E _E and {a+bx°)'^={v'^x''Y=v^a'i{Vi—b) "; mp £ aa" ' «p+'-'c?y .-. du=af^'dii:(a-i-bx°)'>=—- . '■ s-;— , {v'^—by 1- m ■ p , which can be integrated when — +" is a whole number. Hence every binomial differential can be integrated when the exponent of the variable without the' parenthesis, increased by unity and divided by the exponent of the variable within, and the quotient, added to the exponent of the parenthesis, is an entire number, or zero. Examples. Ex. 74. Integrate du=:xdx{a-\-}Ki?Y. Ans. u=- — ^^^+C. 86 INTEGKATION OF DIFFERENTIAL BINOMIALS. 255 2xdx Ex. 15. Integrate ^"= (3.2 i 1)2- Ans. u— dx Ex. 76. Integrate au-= r. x^a'-xY' f^ =ar^dx(ofl-x^)~^. x^a^-x') ^ Put '\/a'^—x^=vx; :. oP'—x^—v^x^^ and a;^=Y+^ a :.x—- avdv and f?a;= — 3- Again: ^=-^, and -,- ^^ ; .-.a; - ^4 • Substituting these in the original expression, we have ,-.,,(,2_,2)-i^_(l±!:!)!.^H^.^. « (l+«2)^ ^'*' But K a • • vx~ av ' .-. ar^dxia'-x^fi = -1(1 + v^)dv; .: far*dx{a^-x')'^=-\{v+iv^). (Xi And substituting for v its value ^ — —■, we have . -4^ / , 2N-i (a2+2a!2)(a^-«!)*4.r 256 INTEGRAL CALCULUS. Ex. I'l. Integrate du= , — ==. Put Vct'^ —x'^v; :.a? — aj^^w^, or x^z=a^ — ifl; :. £B*=a*— 2aV+uS x^dx — — a?vdv + v^dv ; a?dx :. , = — a^dv + w^ou. Va?' — x^ Integrating and putting for v its value ■y/aP'—x^, we shall have f x^ U= I —7= '+0- ■y/aF'—x'^ 3 st^dx Ex. 78. Integrate du=—^-T^2^=x^dx{cfi-\-x'')~^. Put a2+a;2='y2; and a;^=«^— Sw^a^+Su^a-t—a^; .-. a;5(fe = («' — 2a2w3 + a*w) f?« / /x^dx „ , , A?w ^^:^.=P'dv-2c^vdv + a^J -. Integrating and putting Va^+a;^ for v, we have r afidx (a2+a;2)(a;2— 3a2) , "^ J ^^+^-2== 4 ^+«' log.V^^+^+C. Integeation of Ratiokal and Ieeational Feac- TIONS. 194. Every rational fraction may be resolved into par- tial fractions by the method of Indeterminate Coeffi- cients (Algebra, Art. 144). ^ Ex. Y9. Integrate c?m=— 5 5- (Algebra, Art. 144), „ . , —-^, — ; — r+r? r : ^ ° ' ■" a2_a;2 2{a+x) ' 2(a— a;)' _ and ax= — 1_: Vs 2 •■■^^ V ,/ ^ (2cc+l)y ^(^-«)-2-j 1(1+^' b—a tan.-' g, *-ffl , /2a!+l\ ^ /(a+5a;)(& a:— 1 ^3i)p+^+I)=K«+5) log. ^^q:^+ 195. Integeatiok of Ieeational Feactions. Ex. 89. Integrate du=- Vl+aj+iB^' Here l+x+x^=l+{\+xY; _ /* -^-x^)^+~ fdx{r^-x^)'K 2 2j But (Art. 189), fdx(r^-x^)~^- f , dx [X , =:sin.~' I- .•.Sdx{f-x^)^=\r^-x^Y-{-'^- sin.-' (-) + C. 2 Ex. 97. Integrate du—dx{a^-Vx'^)^. Ans. t«==-(a^-|-a;^)*+— / dx{a?-{-x'') '•* But (Ex. 64), fdx(a^+x^)'^= /-— =2==log. (x+Va^+x^); J ya^+x^ vC 1 €t^ — ^^^^^ .-. M=r-(a2-l-a!2)5^— log. (a;+ -v/«^+a3^) + C. 198. We will now obtain two formulas: one for di- minishing the exponent of the variable outside the pa- renthesis when that exponent is negative, the other for diminishing the exponent of the parenthesis when that exponent is negative also. Take Formula A, J'x^-'^dx{a+bafy= af^''(a+bx'y+' a{m—n) f , , , — z,/ , \ TF — T\ I x^-'-'dxia+bx'y. o{pn-\-m) o{pn+m)J ^ ' ' Reducing, we have /a;"-°-'(fe(a -^-l^)^— a;n— ■(Q5+5a;»)p+i b(pn+m) f , , , , ^ / V , X / of-^dxia+bx")^; a{m—n) a{m—n) J \ ' j > or, substituting — m+w for m, we have Formula C, fx-^-^dx{a+bQg'y— ^—^ — ^^ — I x-'^+'-'dxia+bx'y, am am J \ ' j ' in which the negative sign has been attributed to the exponent m,. Ex. 98. If du=x-^dx{l+x'^)~i, then — m — 1 = — 5; .-. — m=:— 4, and n=2, a=l, 5=1, p=—^ ; 264 INTBGKAL CALCULUS. 1 .-. /a:-=&(l+a;2)"^= -T~^ ifii^dx{l+x^) 2. Apply Formula C again to fx~^dx(l+x^)~^. Here — m— 1 = — 3; therefore — to=— 2, the other letters being the same as before, ifx-^dx{l+x^yi = i^-^^M^-ifar^dx{l+xr'^'j. We have fx~^dx{l+x^)~^^ or, in another form, r dx I , , to integrate. T^ i ^ dv Put x=-; :.dx= s-. Substituting these, we have .•./a;-'&{l+CB2)~^ = -(.+«¥[i-S]-|i4.og.(i±^±?)+c. 199. To find a formula for diminishing the exponent of the parenthesis when that exponent is negative. Take Formula B, f3r-^dx{a+ia?y= '^ , — '-+ r— f ar-'dxfa+bx")^' ; .-. {pn+ni)fx''-^dx{a+hx''Y= x'"{a+bx''Y+anpfx'^-^dx{a+ba?)^'^ ; And substituting — ^+ 1 forjo, we have Formula D, foir-^dx{a + 5a!°)-p = .x^(a+b3f)-^+^ m+n—np C — —1 Ti — — — 7 TT / :e°'-'t?£B(a+5£c°)-''+'. na\p—\) na\p—\)J ^ ^"-^ j When p—l, this formula is inapplicable. INTEGEATION OF LOGAEITHMIC FUNCTIONS. 265 200. Let it be required to obtain a formula for inte- grating V2aa;— a;2 ^ ■' This may be put under the form Applying Formula A to this, we have :. fx'^^&{2a-xy^z^ m-l ^ Now, passing the fractional' powers of x within the parenthesis, we obtain Formula JE, f x'°dx _ x'°-\2ax—x^)^ V2ax—x^~ ^ (2m—l)a r ai^-Wce J 1 »^ J ■\/2ax—x^' We shall have, if m is a whole number and positive, after m integrations, , — versm.-' - (Art. 189). V2ax—x^ « ^ "' Integeation of Logaeithmic Functions. 201. "We can integrate but few of these forms by any general process. An approximation, however, may al- ways be had by the method of series ; but we should never resort to the method of series until all our efforts to obtain an exact integral fail. Let it be required to integrate f{x)dx(\og.xY. If, in the formula fudv—uv—fvdu, (1) we make M=(log. sb)", and dv=f{x)dx, or v=ff{x)dx, and substitute these in (1), we shall have M>rmiila F, M 266 INTBGEAL CAiCULUS. ff{x)clx (log. a;)°= . (log- x)'ff{x)dx-ff{x)dxfcl . (log. x)'. (2) If w is a positive whole number, the successive appli- cation of this formula will finally reduce the integration of the proposed form to that of an algebraic function ; then the proposed expression will be integrable, provided we can integrate, in succession, the algebraic functions which enter into it. Ex. 99. Let us take, as an example, X log. xdx , , du^ , ^ ^. (1) In this expression, we have w=l. Put M=log. X, xdx and dv = , = ; VaP'+x^ dx then du= — , x ' /xdx — === ■v/^2+i5 (Art. 187). Substituting these in Formula F, we have To integrate the second term, -/ ^''±t.dx. we shall multiply it by . , VaP+x^ ,.,.„■■ . f{cfl+x^)dx which will reduce it to I . =, J x-VaP'+x^ /dx C xdx x^/cfi^ J Va'+x^ I o =-yJa^-\-x^ (Art. 186). The other can be integrated in a manner entirely sim- ilar to that of Ex. 98 by assuming a;=- ; but we shall do it differently. Thus : mTEGKATION OF LOGAEITHMIO FUNCTIONS. 267 dx In the expression . , put -^/a^+x^^v/ .:a^+x^=v% and a;=V«J^— «^, vdv and xdx=vdv ; .: dx= , . dx dv by dividing the second member by vVv^—cc'—vx. / dx C ^'" _ ^ /v+ay x-\/a^-\-x^~'J v^-a^~~a-i °^' \ x ) (Art. 194). __1 v+a a' °' a; ' «■ ^' a; ' ■'■ « / / n , 2 = -« • log. -51 — 31—!—. Substituting these in (1), we have /x los. xdx . —;===-= VV+^ log. X+ a log. V^+^±^- V¥+^^+C. X 202. A very useful case of the above formula is that in which f{x) =a;°', the form being .pof (log. x)°dx. In the expression fudv=uv—fvdu, (1) let w= (log. a;)", and dv=x'°dx; dx itf""*"' then du=n{log. a;)"-'—, and v=-^^. Substitute these in (1), then /af (log. x)'dx= af'*'* n C 268 INTEGEAl CALCULUS. Again : /a!'°(log. xy-^dx^ and /a!°(log. xy-^dxzzz ^;^(log. x)^'-^^^J a:-(log. «!)■'-'^+y^+|-log. (.y+Vp^+y^)+C. If we estimate the arc from the vertex of the parabola, we shall have, if a=0, y=0; P .•.0=2-log.25+C; .•.C=-ii9log.^, BECTIPICATION OF PLANE CUEVBS. 277 and, consequently, s=^VpH^+ 2 log- \ ^ ^j- The value of the arc for a given ordinate can only be found approximately. Ex. 113. Find the length of the cubical parabola By differentiating and squaring, we have 9 dy^=^-7p'xdx^ ; ._^/i+V^7+c. If a;=0, then s=0; •••0=^+0; .-.0=- '21p^^ ' •• 'ilp^' 3 8 r/ 9j92fl;\f 1 Hence «=27^[^1+^ -ij- The cubical parabola is therefore rectifiable. Ex. 114. Find the length of an arc of the semi-cubical parabola whose equation is (?y±W_8p! M.ns. z- 2^^ 27 • Ex. 115. Find the length of the arc of a cycloid. The differential equation of the cycloid is ydy dx= I \ yiry—y^ And z=f{dx'+df^^VTrf{2r-y)'^dy, =— 2-\/2j-(2r— 2/)+C, by Art. 184. Taking this between the limits, y=Q and y='ir, we have s=4r=:half the arc. Hence the length of the cycloid is equal to Sr, or four times the diameter of the generating circle. 278 INTEGRAL CALCULUS. Ex. 116. Determine the length of the curve whose equation is e^+1 between the limits x=l and a;=2. By differentiating and squaring, we have %^=(e2'-l)2' Separating this by the method of rational fractions, we have =log. (e'+l)+log. {e-\)-x+Q =\og.(e^—l)—x-\-C. Ifow, when £B=:1, /"^log. (e^— 1)— 1+C. Ifa;=2, /=log. (e*-l)-2+C. .•.2=log. (e^+l)— 1. But log. e=l; .-.s^log. re2+l)-log.e, =:log. (e Ex. 117. Find the length of the line whose equation is y—ox. Ans. z='\/x'^+y^. Ex. 118. Find the length of the spiral of Archimedes whose equation is r=aB. dr=ade; :.d(P=^. dz=z'\/dr^+rMB\ =xA dr^ — \l dr^-\-r^'> =-.drVa^+r^; s=- / drVa^+r^, BECXIFICATION OF PLANE CUEVES. 279 lfr=0, 2=0; a then O^glog. a+C; .-. C=-2log.a; • • 3= — -! — +K log. — ! ^i^ — ■2a ■^. a Ex. 119. Find the length of the curve whose equation is 2/^=R"— a;% from £B=0 to a!=R. The whole length of the four branches is4x-|R=6R. Ex. 120. To find the length of an arc of the logarithmic spiral. We have for the equation e=:log. r; dr ... de=—. Substitute this in dz^Vdr'^+r^dd'' and reduce, we have ds=drV2; :.z=rV2+C. If we estimate the arc from the pole where r=0, we shall have 2=»-v/2. Hence the length of an arc of a Naperian logarithmic spiral, estimated from the pole to any. point of the curve, is equal to the diagonal of a square described on the ra- dius vector. Ex. 121. Determine the length of the involute of a circle. Let C be the centre of a circle whose radius is r; APR a portion of the involute. Put the angle OCA=fl, and x and y the co-ordinates of the point P, the origin of rectangular axes being at C. Then OP=aro OA=the portion of the string un- wound, and OP=re. 280 INTEGEAl CALCULUS. Now QQ^sin. 6; .-. OD=r.sin. 0, (1) CD QQ= COS. 9; .-. CD=»-.cos. 9. (2) OB Also op^cos.e; .-. OB=>-9.cos.0; (3) PB . •qP= sin. e ; .-. BP=rd . sin. 0. (4) But a:=CE=r.cos. e+re.sin. 0, (5) yz^FE—r . sin. 0— ?'0 . cos. 0. (6) If AP=:s, then dz=Vdx^-\-dy\ Differentiating (5) and (6), we obtain clx=:—r sin. 6dd+rdd sin. 0+»'0 cos. 6dd =rd COS. Odd, (1) dy=zr COS. 6dd~rd0 cos. d+rd sin. ddd =zrBsiD. ede. (8) Substituting these values in the differential of the arc and reducing, we have dz=r9de; QUADEATUEB OF CuEVES. 210. The quadrature of a curve is the expression of its area. When this expression can be found in finite algebraic terms, the curve is quadrable, and rday be rep- resented by an equivalent square. We have shown that the differential of the area of a curve referricd to rectangular axes is ydx. Hence, to find the area of a curve when so referred, we have the following Rule. From the equation of the curve find the value of y, and multiply if by dx; or,fnd the differential ofx and mndtiply thai by y, then simplify and integrate between the proper limits. Ex.122. Let y=ax; (1) then ydxz=iaxdx, fydx = afxdx / .*. area^^aa!^. QUADEATtJEB OF CUEVES. 281 But from (1) we have xy^^ax^; :. area=:4a;y, ■which is the area of a triangle — half the product of the base and perpendicular. Ex.123. Let y^=2px; then y—V-^x, 1 ydx—{2px)^dx; area=(2j>)5 ix^dx, 1 2{2pxYx which is the area of the common parabola. Ex. 124. To find the area of a circle. We have y=i ■y/r'^—x'' ; .•. ydx zzzdx-y/r"^— aj^, fydx = fdx ■\/r'^ — x^. = lxVr'^—x^-\-lr^ sin.-' -, by Formula B. .: area= / dx -y/r^ ^tc^ = ^Trr'^ =area of a quadrant. Hence area of the circle^Trr^. It will be seen by the diagram that the area "of any segment of the circle, as ABPC, denoted by f ydx, is composed of two parts ; the first part being the area of the triangle ABP—lx'Vr^—x'h the second part the x' area of the sector APC=Jr2 sin.-' — . Ex. 125. To find the area of a circle whose radius is unity. The equation of the circle whose radius is unity is y=Vi—x'^=l—^^—^x*—^x^—j^x^—, etc.; .-. ydx=.dxy\ —x'^= dx — ^^dx — |a;*(fe — -^x^dx — .-j-f^x^ife — , etc. 282 INTEGRAL CALCULUS. .-. a,i-ea= fdxV^ —x^— x-ix^-^x^—ri^x''-T^s^9-, etc. + C. Now, when x=0, the area=0, and C=0. If the arc be equal to 30°, a;=J, and the area of the segment will be =.5 — .0208333— .0007812^.0000698— .0000085— , etc., =.4'783055. But since «= J, .'. 2/=iV3) and the area of the A=i. |-v/3=.2165063; .-.area of the sector of 30° =:.4'783055— .2165063 = .2617992, which, multiplied by 12, gives the area of the circle=3.14159+. Ex. 126. To find the area of an ellipse. The equation of the ellipse referred to its centre and axes is B B K^ ; or .-. ydx= -rdx ■\/A?—x\ area=-5- / dx'\/ h?—iL But fdxsf A?—v?=^% area of a circle whose radius is A. That is, it is the area of a circle described on the transverse axis of the ellipse, and consequently it is equal to ttA^. Hence the area of the ellipse is equal to B 7rA2X5;=7rAB. Ex. 127. To find the area of a cycloid. The differential equation of the cycloid is (Art. 130), dx^ y^y ■ .fydx^J-y^'y y2ry—y^ And, by applying Formula E twice, we shall have fydx= — ly\/'i'ry—y'^—%r-\/'2ry—'tp'-\-%fi versin.-' 1-). And making 2/=0 and yz=2r, we have the area be- tween those limits equal to AREA OF SPIEALS. 283 firr^ ; .•. whole area=37r»"^. That is, the area of the cycloid is equal to three times the area of the generating circle. 211. Area of Spirals. "We have, by Art. 148, the differential of the area of any segment of a polar curve ds=\r^de. Ex. 128. To find the area of the spiral of Archimedes. By Art. 134, the equation is r=a9 where a=7;— ; .: s=la^f6^dd=ia^d^^ Making 0=0 and 0=:27r, we have the area included within the first spire PM.A=^=one third of the area of a circle whose radius is equal to the radius vector at the end of the first revolution. The radius vector in every sub- sequent revolution will pass over the area previously de- scribed. Hence, to find the area at the end of the nth revolution, we must integrate between the limits e=(«— l)27r, and Q—%mt, n^—(n — 1)^ which gives ^ tt. Ex. 129. To find the area of the hyperbolic or recip- rocal spiral. The equation, by Art. 136, is -.. a a^ .: s=ia^fe-''de= ~i . -J. Ex. 130. To find the area of the logarithmic spiral. The equation, by Art. 135, is e=Iog. r,- dr .■.de=-, and s=lfrdr=ii'^+G. 28i INTEGEAL CALCULUS. If we make r—0, then s=0, and C=0; That is, the area of the Naperian logarithmic spiral is equal to one fourth of the square described on the radius vector. 212. Aeea op Suefaces of Revolution. The differential of the area of a surface of revolution is, by Art. 141, dS = 2wy V dx^+dy^; .: S=2Tr f2/Vdx^+dy\ If the'* curve were revolved about the axis of y, it would be necessary to change a; into y and y into x. Ex. 131. To find the convex surface of a cone. Let y=ax be the equation of the generating line; then dy=adx; .: dy'^=a^dx^, and dS=27ryVdx^+dy% = 2irax Vdx^+a^dx^, = 2'iraxdxVl+a^ ; .-. S=2Tra Vl +a ^fxdx, ^iraVl+a'^.x'^+C. (A) But a=-; x' •■•«^=^. and Vl+a2=-ya;2+2/2. Substituting these in (A), we have S='7ry Vx^+y^z=27ry x iVx^+yK But iVx^+y^=ha.W the slant height of the cone, and 27ry=circnmference of the base. Therefore the convex surface of a cone is equal to the circumference of the base multiplied by half the slant height. Got. Taking the limits" between x and x+h, we shall have th e conv ex surface of the frustum of a cone ='^aV\+a-'{{x+hY -x'-\ = \{x-\-h)+x]h.7raVl+^^ =-ir{a{x+h)-\-ax]h'\/T+^. Now a{x+h)=xhe radius of the greater base, aa; =the radius of the lesser base, and hVT+cfi=th& slant height of the frustum. AREA OF SUKFACES OF EEVOLUTION. 285 Hence, to find the convex surface of the frustum of a cone, we have the following RuxB. Multiply the sum of the radii of the upper and lower hoses by the slant height, and that product by 3.1416. Ex. 132. To find the convex surface of a cylinder. Let y=6 be the equation of the generating line. Then dy=0, and lied by its altitude. Ex. 133. To find the surface of a sphere. Let y=V»'^— a3^ be the equation of the generating circle. Then xdx x'^dx^ ay= — —, and dy-i=-^; .■.dS=2^yJdx^+'^^ V y^ = 2ir(fe a/sb^ + 2/^ ^2Trrdx, or S=27rrx+G. If a;=0, C=0, and S=0. If Xz=r, S=2irr2= surface gf g, hemisphere. Hence 47!r2=whole surface. That is, the surface of a sphere is equal to four of its great circles, or equal to the convex surface of the cir- cumscribing cylinder. Ex. 134. To find the surface of a paraboloid. The equation of the generating parabola is y'''z=2px, which, being difierentiated, gives dx=—; pi 286 INTEGEAL CALCULUS. 2t i Hence dS=—yd]/{p^+y^y ; by Art. 186. If we make y=0, S will become equal to zero, and Now, integrating between the limits y=0 and y=b, ■we shall have 2ffr 3 1 Ex. 135. To find the surface of an ellipsoid formed by revolving an ellipse about the transverse axis. The equation of the generating curve is B2 B xdx whence <7y=_^-^===; ^x^dx^ Substituting this value of dy'^ and the value of y, "R 1 ■which is -t{A?~x'^Y, in the differential of the surface, we shall have, by a little reduction, 2,rB A2-B2 1 A2-B2 , If we put — -^ — =e\ 2irBe we shall have