Ifpft BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF 34enrg M. Sage 1891 7673-2 /;z-/(x), or the limits of integration, become infinite 223 94. Applications of the Eule of Art. 92 . . . 225 EsiampUs XXlX 227 95. Formulae of Beduotion .... . . 229 96. Related Integrals . . 232 Examples XXX, XXXI 235, 239 CHAPTER VII. GEOMETRICAL APPLICATIONS. Volume of any Solid 97. Definition of an Area 98. Formula for an Area, in Cartesian Coordinates 99. On the Sign to be attributed to au Area 100. Areas referred to Polar Coordinates 101. Area swept over by a Moving Line 102. Theory of Amsler's Planimeter . Examples XXXII 103. Volumes of Solids 104. General expression for the 105. Solids of Revolution . 106. Some other Cases 107. Simpson's Eule . Examples XXXni 108. Rectification of Curved Lines . 109. Generalized FormuliB . 110. Arcs referred to Polar Coordinates 111. Areas of Surfaces of Revolution Examples XXXIV 112. Approximate Integration 113. Mean Values Examples XXXV 114. Multiple Integrals 241 242 245 247 249 250 252 255 256 258 259 260 262 264 266 268 270 274 275 279 281 282 XU CONTENTS. CHAPTER VIII. PHYSICAL APPLICATIONS. ART. PAGE 115. Mean Density 288 Examples XXXVI 290 116. Centre of Mass 291 117. Line-Distributions ... 293 118. Plane Areas 293 119. Mean Pressure. Centre of Pressure . . ... 296 Examples XXXVII 299 120. Mass-Centre of a Surface of Bevolution 301 121. Mass-Centre of a Solid .303 122. SoUd of Variable Density ... .... 306 123. Theorems of Pappus .... .... 307 124. Extensions of the Theorems 309 Examples XXXVIII 310 125. Moment of Inertia. Eadins of Gyration 313 126. Two-Dimensional Examples .... . . 314 127. Three-Dimensional Problems 316 128. Mean-Square of the Distances of a System of Particles from a Plane 319 129. Comparison of Moments of Inertia about Parallel Axes . . 321 130. AppUoation to Distributed Stresses 322 131. Homogeneous Strain in Two Dimensions .... 324 132. Homogeneous Strain in Three Dimensions .... 327 Examples XXXIX 329 CHAPTER IX. SPECIAL CURVES. 133. Algebraic Curves with an Axis of Symmetry .... 332 Examples XL 339 134. Transcendental Curves; Catenary, Tractrix .... 342 135. Lissajous' Curves 344 136. The Cycloid 347 137. Epicycloids and Hypocycloids 350 CONTENTS. xui ABT. PAGE 138. Special Oases 364 139. Superposition of Circular Motions. Epioyolios . . 359 Examples XLI 363 140. Curves referred to Polar Coordinates. The Spirals . . 366 141. The Lima9on, and Cardioid 368 142. The Curves r»=a''cosn0 . 370 143. Tangential-Polar Equations 371 Examples XLII 378 144. Associated Curves. Similarity 376 145. Inversion 378 146. Mechanical Inversion 380 147. Pedal Curves 382 148. Beciprooal Polars . 384 149. Bipolar Coordinates 386 Examples XTiTTT 390 CHAPTER X CURVATURE. 150. Measure of Curvature 394 151. Intrinsic Equation of a Curve . . ... 397 152. FormulsB for the Radius of Curvature ... . 400 153. Nevfton's Method 402 154. Osculating Circle 406 Examples XLIV 407 155. Envelopes ... 413 156. General Method of finding Envelopes . . . . 415 157. Algebraical Method 416 158. Contact-Property of Envelopes .... . . 418 159. Evolutes 421 160. Arc of an Evolute 426 161. Involutes, and Parallel Curves 427 Examples XLV 429 162. Displacement of a Figure in its own Plane. Centre of Rotation 433 163. Instantaneous Centre 434 164. Application to Rolling Curves 438 XIV CONTENTS. ABT. PAOE 165. Curvature of a Foiut-Boulette 440 166. Curvature of a Line-Boulette 443 167. Continuous Motion of a Figure in its own Plane . . . 444 168. Double Generation of EpioycUcs as Eoulettes .... 448 169. Teeth of Wheels 450 Examples XLVI 454 CHAPTER XL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 170. Formation of Differential Equations 456 Examples XLVn 459 171. Equations of the First Order and First Degree . . .462 172. Methods of Solution. One Variable absent .... 463 173. Yariables Separable . . . 464 174. Exact Equations 466 175. Homogeneous Equation 469 Examples XLVIII 470 176. Linear Equation of the First Order, with Constant Coefficients 473 177. General Linear Equation of the First Order .... 476 178. Orthogonal Trajectories 478 Examples XLIX 481 179. Equations of Degree higher than the First .... 484 180. Clairaut's form 485 Examples L 487 CHAPTER XII. DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 181. Equations of the Type cPyldx'=f{x) 490 182. Equations of the Type d'h/ldx^=f{y) 492 183. Equations involving only the First and Second Derivatives . 496 184. Equations with one Variable absent 499 185. Linear Equation of the Second Order 501 Examples LI g05 CONTENTS. XV AUT. PAGE 186. Linear Equation of the Second Order with Constant Co- efficients. Complementary Function 509 187. Determination of Particular Integrals 513 188. Properties of the Operator D 518 189. General Linear Equation with Constant Coefficients. Com- plementary Function 519 190. Particular Integrals 523 191. Homogeneous Linear Equation . ... 526 192. Simultaneous Differential Equations 529 Examples LII . . 536 CHAPTER XIII. DIFFEEBNTIATION AND INTEGBATION OF INFINITE SERIES. 193. Statement of the Question 541 194. Uniform Convergence of Power-Series 543 195. Continuity of the Sum of a Power-Series .... 547 196. Integration of a Power-Series 549 197. Derivation of the Logarithmic Series, and of Gregory's Series 552 198. Differentiation of a Power-Series 557 199. Integration of Differential Equations by Series . . . 559 200. Expansions by means of Difiereutial Equations . ■ 562 Examples LUI 566 CHAPTER XIV. TAYLOB'S THEOEEM. 201. Form of the Expansion 572 202. Particular Cases 574 203. Proof of Maclaurin's and Taylor's Theorems .... 576 204. Another Proof 582 205. Derivation of Certain Expansions 583 206. Applications of Taylor's Theorem. Order of Contact of Curves 585 207. Maxima and Minima 588 208. Infinitesimal Geometry of Plane Curves 590 Examples LIV 591 XVI CONTENTS ABT. PAGE 209. Functions of Several Independent Variables. Partial Deriva- tives of Higher Orders 595 210. Proof of the Conunutative Property 596 211. Extension of Taylor's Theorem 599 212. Maxima and Minima of a Function of Two Variables. Geo- metrical Interpretation 603 213. Applications of Partial Differentiation 606 Examples LV 610 APPENDIX. NUMEEIOAL TABLES. A. Table of Squares of Numbers from 10 to 100 ... 616 B. 1. Table of Square-Boots of Numbers from to 10, at Intervals of -1 616 B. 2. Table of Square-Boots of Numbers from 10 to 100, at Intervals of 1 617 C. Table of Beciprooals of Numbers from 1 to 10, at Intervals of 1 617 D. Table of the Circular Functions at Intervals of One-Twentieth of the Quadrant 617 B. Table of the Exponential and Hyperbolic Functions of Numbers from to 2-6, at Intervals of '1 . . . 618 F. Table of Logarithms to Base e 618 Index 619 CHA.PTER I. CONTINUITY". 1. Continuous Variation. In every problem of the Infinitesimal Calculus we have to deal with a number of magnitudes, or quantities, some of which maybe constant, whilst others are regarded as variable, and (moreover) as admitting of continuous variation. Thus in the applications to Geometry, the magnitudes in question may be lengths, angles, areas, volumes, &c. ; in Dynamics they may be masses, times, velocities, forces, &c. Algebraically, any such magnitude is represented by a letter, such as a or x, denoting the ratio which it bears to some standard or 'unit' magnitude of its own kind. This ratio may be integral, or fractional, or it may be ' incom- mensurable,' i.e. it may not admit of being exactly repre- sented by any fraction whose numerator and denominator are finite integers. Its symbol will in any case be subject to the ordinary rules of Algebra. A ' constant ' magnitude, in any given process, is one which does not change its value. A magnitude to which, in the course of any given process, different values are assigned, is said to be ' variable.' The earlier letters a,h, c, ... of the alphabet are generally used to denote constant, and the later letters ...u, v, w, x, y, z to denote variable magnitudes. Some kinds of magnitude, as for instance lengths, masses, densities, do not admit of variety of sign. Others, such as altitudes, rotations, velocities, may be either positive or negative. 2 INFINITESIMAL CALCULUS. l^ij. . When we wish to designate the 'absolute' value of a magnitude of this latter class, without reference to sign, we enclose the representative symbol between two short vertical lines, thus |a;|, |sina;|, lloga!|. It is important to notice that, if a and h have the same sign, \a + h\ = \a\ + \l\ (1), whilst, if they have opposite signs, |a + 6l<|a|+|i| (2). A geometrical representation of any class of magnitudes is obtained by taking an unlimited straight line X'X, and in it a fixed origin 0, and by measuring lengths OM- propor- tional on any convenient scale to the various magnitudes considered. In the case of sign-less magnitudes (such as masses), these lengths are to be measured on one side only of ; in cases where there is a variety of sign, OM must be drawn to the right or left of according as the magnitude to be represented is positive or negative. To each magnitude of the kind in question will then correspond a definite point Jlfiu the line ^'X Fig. 1. When we say that a magnitude admits of 'continuous' variation, we mean that the point M may occupy any position whatever in the line X'X within (it may be) a certain range. It will be observed that two things are postulated with respect to the magnitudes of the particular kind under consideration, viz. that every possible magnitude of the kind is represented by some point or other of the line X'X, and (conversely) that to every point on the line, within a certain range, there corresponds some magnitude of the kind. These conditions are fulfilled by all the kinds of magnitude with which we meet, either in Geometry, or in Mathe- matical Physics. It will be found on examination that these all involve in their specification a reference, direct or indirect, to linear magnitude. 1-ZJ CONTINUITY. 3 2. Upper or Lower Iiimit of a Sequence. Suppose we have an endless sequence of magnitudes of the same kind «i, oc^, a^s (1). each greater than the preceding, so that the differences Xl^ X\^ X^ XlJ. y X^ ^3 , . . . are all positive. Suppose, further, that the magnitudes (1) are all less than some finite quantity a. The sequence v^ill in this case have an ' upper limit,' i.e. there will exist a certain quantity fi, greater than any one of the magnitudes (1), but such that if we proceed far enough in the sequence its members will ultimately exceed any assigned magnitude which is less than /j,. In the geometrical representation the magnitudes (1) arc represented by a sequence of points M„ M„ M„ (2), each to the right of the preceding, but all lying to the left of some fixed point A. Hence every point on the line X'X, Ml Ms M3M4A A Fig. 2. without exception, belongs to ope or other of two mutually exclusive categories. Either it has points of the sequence (2) to the right of it, or it has not. Moreover, every point in the former category lies to the left of every point of the latter. Hence there must be some point M, say, such that all points on the left of M belong to the former category and all points on the right of it to the latter. Hence if we put fi = OM, fj, fulfils the definition of an ' upper limit ' above given. In a similar manner we can shew that if we have an endless sequence of magnitudes Xi, X2, Xg, (o), each less than the preceding, so that the differences Xi fl?g , x^ — x^ , x^ x^ . . • • 1—2 4 INFINITESIMAL CALCULUS. L^H- ^ are all positive, whilst the magnitudes all exceed some finite quantity 6, there will be a lower limit v, such that every mag- nitude in the sequence is greater than v, whilst the members of the sequence ultimately become less than any assigned magnitude which is greater than v. 3. Application to Infinite Series. Series with positive terms. The above has been called the fundamental theorem of the Calculus. An important illustration is furnished by the theory of infinite series whose terms are all of the same sign. In strictness, there is no such thing as the 'sum' of an infinite series of terms, since the operations indicated could never be completed, but under a certain condition the series may be taken as defining a particular magnitude. Consider a scries lti+U2 + Us+ ... +itn+ (1) whose terms are all positive, and let Sl = Wi, S2 = Mi + W2, S„ = lti+tt2+ ... +M», (2). If the sequence *1) *2) */l> (") has an upper limit S, the series (1) is said to be 'convergent ,' and the quantity S is, by convention, called its ' sum.' Ex. 1. The series 1 + i + i+l •!-••• If in Fig. 2 we make OMi=l, 0.4 = 2, and bisect M^A in M^, M^A in M^, and so on, the points M^, M^, M^, ... will represent the magnitudes Sj, Sj, s^, .... And since these points all lie to the left of A, whilst M^A = 1/2"-' and can therefore be made as small as we please by taking n large enough, it appears that the sequence has the upper limit OA, = 2. Ex. 2. The series 11 1 1.2 2.3 " ■■■ ^ n(n+1)^- If we write this in the form 2-4] CONTINUITY. 5 we see that «« = 1 , , which has evidently the upper limit 1. £!x. 3. Further illustrations are supplied by every arithmetical process in which the digits of a non-terminating decimal are obtained in succession. For example, the ordinary process of extracting the square root of 2 gives the series 1-414213... 1 -t i- -1 _?_ J_ A *"" 10"*" 10""'" 10'""" 10* "*" 10'"*" 10= """■■■ Since «„ is always less than 1'5, there is an upper limit. Again, if (1) be a convergent series of positive terms, and if < + < + <+...+«„'+ (4), be a series of positive terms which are respectively less than the corresponding terms in (1), i.e. Un III, whilst n2'< n/. The process can be continued to any extent, and if we imagine it to be followed out according to some definite law, we obtain an ascending sequence of magnitudes 111, -l-ls) iiji ••• J and a descending sequence n/, n/. n;,.... Also, every member of the former sequence is less than every member of the second. Hence the former sequence will have an upper limit 11, and the latter a lower limit 11', and we can assert that 11 :^ 13'. We can further shew that if the law of construction of the successive polygons be such that the angle subtended at the centre by any two successive^ points on the circumference 4] CONTINUITY. is ultimately less than any assignable magnitude, the limits n and n' are identical. For, let PQ be a side of the Fig. 4. inscribed polygon, PT and QT the tangents at P and Q. If PQ meets OT in ]Sf, we have PQ _P]Sr_ON TP + TQ~PT OP' Now if 2 be a symbol of summation extending round the polygons, we have n,=2(PQ), n,'=S(rp+TQ). Hence, by a known theorem, the ratio n, tjPQ) U'/ X{TP + TQ)' will be intermediate in value between the greatest and least of the ratios OF OP' But in the limit, when the angles POQ are indefinitely diminished, each of the ratios ON/OP becomes equal to unity. Hence the limit of Hg is identical with that of n/, or n = H'. Finally, whatever be the law of construction of the successive polygons, subject to the above-mentioned condition. 8 INFINITESIMAL CALCULUS. [CH. I the limit obtained is the same. For suppose, for a moment, that in one way we obtain the limit 11, and in another the limit P. Regarding 11 as the limit of an inscribed, and P as that of a circumscribed polygon, it is plain that n :^ P. In like manner, regarding P as the limit of an inscribed, and n as that of a circumscribed polygon, it appears that P :f» TI. Hence H = P. The definite limit to which, as we have here proved, the perimeter of an inscribed (or circumscribed) polygon tends, M'hen each side is ultimately less than any assignable magnitude, is adopted by definition as the 'perimeter' of the circle. The proof that the ratio (tt) which this limit bears to the diameter is the same for all circles is given in most books on Trigonometry. In a similar manner we may define the length of any arc of a circle less than the whole perimeter, and shew that the length so defined is unique. 5. Limiting Value in a Sequence. Suppose that we have an endless sequence of magnitudes Xi, X^, Xj (1), arranged in any definite order, and that a point in the sequence can always be found beyond which every member of it differs from a certain quantity /[^ by a quantity less (in absolute value) than o-, where o- may be any assigned mag- nitude, however small. The sequence is then said to have the 'limiting value' /t. We have had particular cases of this relation in the upper and lower limits discussed in Art. 2, but in the present wider definition it is not implied that the members of the sequence are arranged in order of magnitude, or that they are all greater or all less than the limiting value fi. The point in the sequence after which the difference of each member from /i is less (in absolute value) than o- will in general vary with the value of a, but it is implied in the definition that, however small a be taken, such a point exists. 4-6] CONTINUITY. 9 Sx. Let the sequence be that obtrained by putting n = 0, 1, 2, 3, ... in the formula 1 .2*1 + 1 sm -^—^ — IT. 2n+l 2 The sine is alternately equal to +1, whilst the first factor diminishes indefinitely. The limiting value is therefore zero. 6. Application to Infinite Series. If in the infinite series Ui + U^+ ... +Un + (1), whose terms are no longer restricted to be all of the same sign, we write Si = Mi, Si = 1h + th, S„ = Mi + M2 + ... +M«, (2), and if the sequence *ll *2) *>u {<>) has a limiting value S, the series is said to be 'convergent/ and S is called its ' sum.' The most important theorem in this connection is that if the series |wi|+|m2|+... +|w„| + (4), formed by taking the absolute values of the several terms of (1), be convergent, the series (1) will be convergent. If (4) be convergent, the positive terms of (1) must a fortiori form a convergent series, and so also must the negative terms. Let the sum of the positive terms be p and that of the negative terms be — q. Also, let Sm-in> the sum of the first m + n terms of (1), consist of m positive terms whose sum is Pm, and n negative terms whose sum. is — q^. We have, then, {p-q)-Sm+n = (p -q)- (Pm-qn) ='(p-Pm)-(q-qn) (5). If m + m be sufficiently great, p —pm and q — qn will both be less than a, where a is any assigned magnitude, however small ; and the difference of these positive quantities will be cb fortiori less than Oa > «3 > . . .. In the figure, let Mj M^Mg M M5M3 Ml :: Fig. 5. It is plain that the points M^, M3, M^, ... form a descending sequence, and that the points M^, M^, M^, ... form an ascending sequence. Also that every point of the former sequence lies to the right of every point of the latter. Hence each sequence has a limiting point, and since -'"»j •'"211+1 ^^ '"an+i; and therefore is ultimately less than any assignable magni- tude, these two limiting points must coincide, say in M. Then OM represents the sum of the given series (6). 6] CONTINUITY. 11 Ex. The series converges to a limit between 1 and 1 - J. This series belongs to the 'accidentally' convergent class. It is shewn in books on Algebra that the sum of n terms of the series l+i+h+i+- can be made as great as we please by taking n great enough. It cannot be too carefully remembered that the word ■ sum ' as applied to an infinite series is used in a purely conventional sense, and that we are not at liberty to assume, without examination, that we may deal with such a series as if it were an expression consisting of a finite number of terms. For example, we may not assume that the sum is unaltered by any rearrangement of the terms. In the case of an essentially convergent series this assumption can be justified, but an accidentally convergent series can be made to converge to any limit we please by a suitable adjustment of the order in which the terms succeed one another. For the proofs of these theorems we must refer to books on Algebra ; they are hardly required in the present treatise. We shall, however, occasionally require the theorem that if Ml +«a+...+l{„+ (7) and <+it2' + ...+M„'+ (8) be two convergent series whose sums are S and 8', respec- tively, the series (Ui ± ?ti') + (Ma +«/) + ... + (Un ± Un) + (9), composed of the sums, or the differences, of corresponding terms in (7) and (8), will converge to the sum S ± S'. This is easily proved. If s„, s„' denote the sums of the first n terms of (7) and (8) respectively, the sum of the first n terms of (9) will be s„ + s„'. Now {S ± S') - (*„ + s„') = iS- s„) ± (S' - s„') . . .(10). By hypothesis, if o- be any assigned magnitude, however small, we can find a value of n such that for this and for all higher values we shall have |«-s„|{x), (1), we must have y + Sy = (j> (x + Sx), (2), and therefore Sy = (x + Sx) — (p (x) (3). At present there is no implication that Sx or Sy is small ; the increments may have any values subject to the relation (2). Ex. J.i y = oi?, then if aj = 100, Sai = 1, we have 8y = (101)=- (100)'= 30301. 8. Geometrical Representation of Functions. We may construct a graphical representation of the rela- tion between two variables x, y, one of which is a function of the other, by taking rectangular coordinate axes OX, OT. If we measure OM along OX, to represent any particular value of the independent variable x, and ON along OF to represent the corresponding value of the function y, and if we complete the rectangle OMPN, the position of the point P will indicate the values of both the associated variables. 14 INFINITESIMAI, CALCULUS. [CH. 1 Since, by hypothesis, M may occupy any position on OX, between (it may be) certain fixed termini, we obtain Y K N H A M B Fig. 6. in this way an infinite assemblage of points P. A question arises as to the nature of this assemblage ; whether, or in what sense, the points constituting it can be regarded as lying on a curve. Times Fig. 7. S-9] CONTINUITY. l-'i In many cases, of course, there is no difficulty about the answer. For example, if, to represent the relation between the area of a circle and its radius, we make OM proportional to the radius, and PM proportional to the area, then PM oc OM^, and the points P lie on a parabola. The same curve will represent the relatiou between the space (s) described by a falling body and the time (<) from rest, since s oc t\ See Fig. 7. The general question must, however, be answered in the negative. The definition of a function given at the beginning of Art. 7 stipulates that for each value of a) there shall be a definite value of y ; but there is no necessary relation between the values of y corresponding to different values of oc, however close together these may be. Without some further qualification the definition refer- red to is indeed far too wide for our present purposes, the functions ordinarily contemplated in the Calculus being subject to certain very important restrictions. The first of these restrictions is that of ' continuity.' This implies that, as M ranges over any finite portion AB of the line OX, N ranges over a finite portion HK of the line OY, i.e. N occupies once at least every position between H and K. Further, that if the range _AB be continually contracted, the range HK will also contract, and can be made as small as we please by taking AB small enough. Since, as we shall see in Art. 10, the second of the above properties includes the first, it is adopted as the basis of the formal definition of a ' continuous function,' to which we now proceed. 9. Definition of a Continuous Function. Let X and y be corresponding values of the independent variable and of the function. Let Bx be any admissible increment of x, and Sy the corresponding increment of y. Then if, cr being any positive quantity different from zero, we can always find a positive quantity e, different from zero, such that for all admissible values of Sx which are less (in absolute value) than e the value of Sy will be less in absolute 16 INFINITESIMAL CALCULUS. [CH. I value than «r, the function is said to be ' continuous ' for the particular value x of the independent variable. Otherwise, if <\> (x) be the function, the definition requires that it shall be possible to find a quantity e such that \4>{x + h)-^{x)\<(T (1), for all admissible values of h such that | A | < e. The value of e will in general be limited by that of a, but it is implied that the condition can always be satisfied by some value of e, however small a may be. The restriction to 'admissible' values of Bx (or A) means that x + hx must be within the range of values of the independent variable. The above definition is sometimes (imperfectly) summed up in the statement that an infinitely small change in the independent variable produces an infinitely small change in the function. It follows from the definition that if x^, x^, x^,... be any sequence of admissible values of the independent variable having x for its limit (Art. 5), the sequence y^, y^, y^,... of the corresponding values of the function will have the limit y. Conversely, if this hold for every admissible sequence Xi, x^, X},... having x for its limit, the function will be contin- uous at X. Ex. 1. To shew that the function <^(«=) = ^' (2) is continuous, according^ to the above definition, for all values of X. We have {x + hf-a^=2x7i + h^ (3). Now, X being fixed, and o- being any assigned positive quantity, however small, we can always find a positive quantity £ such that 2£|a;| + £2=o- (4). This is in fact a quadratic equation to find e, and it is easily seen that one root is positive. And it is evident that if \h\ be less than the value of e thus found, we shall have I <^ (as + A) — <^ (a;) I < 0-. A general investigation of the continuity of rational algebraic functions will be given later (Arts. 14, 15). 9-10] CONTINUITY. 17 Ex. 2. Let ^{x) = ^{\-x) (5), a function only defined (so long as we recognize only 'real' quantities) for aj :^ 1. "We have 4,{i-h)==J{h), which can be made less than any assigned quantity o- by making h Ka^- The function is therefore continuous for x=\. 10. Property of a Continuous Function. If {x) be a function which is continuous from a; = a to x = h inclusive, and if (^(a) and <^(b) have opposite signs, there must be some value of x between a and h for which (/) {x) = 0. For definiteness we will suppose that ^ (a) is positive and (^(6) negative. Let A, Bhe the points of the line X'X for Fig. 8. which x = a, x = b, respectively; and let HA, BK represent the corresponding values of the function. In virtue of the continuity of (\> (x) there is a certain range extending to the right of A for every point of which the function differs (in absolute value) from HA by less than HA, and is therefore necessarily positive. Since this range does not extend as far as B, it will terminate at some point G between A and B. Moreover the value of the function at the point G itself must be zero. For if it were positive, then (in virtue of the 18 INFINITESIMAL CALCULUS. L"^'^- ^ continuity) there would be a certain range extending to the right of G for every point of which the function would be positiye ; and if it were negative there would be a certain range extending to the left of G for every point of which the function would be negative. Either of these suppositions is inconsistent with the above determination of G. We may express this theorem shortly by saying that a continuous function cannot change sign except by passing through the value zero. It follows that if (6) be unequal, there must be some value of x between a and b, such that ij) {x) = /3, where /S may be any quantity intermediate in value to (f>(a) and 0(6). For, let f(x) = ,l>{x)-^; since /3 is a constant, f(x) also will be continuous. By hypothesis, {a)-^ and 0(6)-/8 have opposite signs, and therefore f(a) and f(b) have opposite signs. Hence, by the above theorem, there is some value of x between a and b for which /(«) = 0, or In other words, a continuous function cannot pass from one value to another without assuming once (at least) every intermediate value. 11. Graph of a Continuous Function. It follows from what precedes that the assemblage of points which represents, in the manner explained in Art. 8, any continuous function is a ' connected ' assemblage. By this it is meant that a line cannot be drawn across the assem- blage without passing through some point of it. For, denoting the function by <^ (x), and the ordinate of any line by f{x), then if <^ (x) and f{x) are both continuous the difl'erence 4>{x)-/{x) will be continuous (Art. 13), and therefore cannot change sign without passing through the value zero. 10-11] CONTINUITY. 19 The question whether any connected assemblage of points is to be regarded as lying on a curve is to some extent a verbal one, the answer depending upon what properties are held to be connoted by the term 'curve.' We shall have occasion (in Art. 32) to return to this question ; but in the meantime it is obvious that a good representation of the general course or ' march ' of any given continuous function can be obtained by actually plotting on paper the positions of a sufficient number of points belonging to the assemblage, and drawing a line through them with a free hand. A figure constructed in this way is called a ' graph ' of the function. The figure shews the method of constructing a graph of the function y = a?; the series of corresponding values of x and y employed for the purpose being as follows : tx = 0, ±-5, +1, ±1-5, +2, ±2-5, [y = 0, -25. 1, 2-25, 4, 625. ' Fig. 9. It will be noticed that different scales have been adopted for X and y, respectively. This is often convenient; indeed in the physical applications of the method the scales of x and y have to be fixed independently. 2—2 20 INFINITESIMAI- CALCULUS. [CH. 1 The method of graphical representation is often used in practice when the mathematical form of the function is un- known ; a certain number of corresponding values of the dependent and independent variables being found by observation. An example is furnished by the annexed diagram, which represents the pressure of saturated steam as a function of the temperature (Centigrade). / / / 4) / o / c / / / 2 1 / - / ^ ^ 2 4 6 8 Ter 1( nperat )0 12 J res 14 16 18 Fig. 10. The reader may also be reminded of the meteorological charts which exhibit the height of the barometer or thermometer as a function of the time. The substratum of fact underlying a graph constructed in the above manner is of course no more than is contained in a numerical table giving a series of pairs of corresponding values of x and y ; but the graphical form appeals far more 11-12] CONTINUITY. 21 effectively to the mind, by helping us to supply, in imagina- tion, the intermediate values of the function. The graphical method will be freely used in this book (as in other elementary treatises on the subject) by way of illustration. It is necessary, however, to point out that, as applied to mathematical functions, it has certain limitations. In the first place, it is obvious that no finite number of isolated values can determine the function completely ; and, indeed, unless some judgment is exercised in the choice of the values of x for which the function shall be calculated the result may be seriously misleading. Again, the streak of ink, or graphite, by which we represent the course of the function, has (unlike the ideal mathe- matical hue) a certain breadth, and the same is true of the streak which represents the axis of x; the distance between these streaks is therefore affected by a certain amount of vagueness. For the same reason, we cannot reproduce details of more than a certain degree of minuteness; the method is therefore intrinsically inadequate to the representation of functions (such as can be proved to exist) in which new details reveal themselves ad infinitum as the scale is magnified. Functions of this latter class are not, however, encountered in the ordinary applications of the Calculus. In the representation of physical functions, as determined experimentally, the vagueness dye to the breadth of the lines is usually no more serious than that due to the imperfection of our senses, errors of observation, and the like. 12. Discontinuity. A function which for any particular value (xi) of the independent variable fails in any way to satisfy the condition stated at the beginning of Art. 9 is said to be 'discontinuous' for that value of x. Functions exist (and can be mathematically defined) which are discontinuous for every value of x within a certain range. But ordinarily, in the applications of the Calculus, we have to deal with functions which are discontinuous (if at all) only for certain isolated values of x. This latter kind of discontinuity, again, may occur in various ways. In the first place, the function may become 'infinite' for some particular value (x^) of x. The meaning 22 INFINITESIMAL CALCULUS. [CH. I of this is that by taking x sufficiently nearly equal to x^ the value of the function can be made to exceed (in absolute value) any assigned magnitude, however great. Examples of this are furnished by the functions \jx, which becomes infinite for a; = 0, and tan x, which becomes infinite for x^lir, &c. See Fig. 19, p. 34. Again, the time of oscillation of a given pendulum, regarded, as a function of the amplitude (a), becomes infinite for o = tt. A graph of this function is appended. U i' ff Amplitudes Fig. 11. Again, it may happen that as x approaches x^, the function tends to a definite limiting value \ (see Art. 26), whilst the value actually assigned to the function for x = Xi is different from \. Consider, for example, the function defined as equal to for a; = and equal to 1 for all other values of x. Moreover if Xi lie within the range of the independent variable, it may happen that the function tends to different limiting values as x approaches x^ from the right or left, respectively. In this case the value of the function for 12-13J CONTINUITY. 23 x = Xi (if assigned) cannot be equal to both these limiting values, and there is necessarily a discontinuity. An illustration from theoretical dynamics is furnished by the velocity of a particle which at a given instant receives a sudden impulse in the direction of motion. In this case the velocity at the instant of the impulse is undefined. Times Fif?. 12. Other more general varieties of isolated discontinuity are imaginable, but are not met with in the ordinary applications of the subject. A sufficient example is afforded by the function . 1 This is undefined for as = 0, but whatever value we supply to complete the definition, the point x = will be a point of dis- continuity. For the function oscillates between the values + 1 an infinite number of times within any interval to the right or left of 03 = 0, however short, since the angle 1/a; increases in- definitely. 13. Theorems relating to Continuous Functions. We may now proceed to investigate the continuity, or otherwise, of various functions which have an explicit 24 INFINITESIMAL CALCULUS. [CH. I mathematical definition, and to examine the character of their graphical representations. For this purpose the following preliminary theorems will be useful : 1°. The sum of any finite number of continuous functions is itself a continuous function. First suppose we have two functions u, v of the indepen- dent variable x. Then S(u + v) = (u + hu+v + Sv)-{u + v) = Bu + Sv. From the definition of continuity it follows that, whatever the value of a, we can find a quantity e such that for | Saj | < e we shall have | Sm | < ^o- and | Si; | < ^o-, and therefore (Art. 1) \Bu + Bv\ aj > 0, and negative outside this interval. From the second form of y it appears that f or a; = + oo we have y = — J- We further find, as corresponding values of x and y : Cx = -co, -3, -2, -1, --5, 0, -5, 1, 2, 3, +<», ly = --5,--67, -75, -1, -1-5, +00, -5, 0, --25, --33, --5. The figure shews the curve. 30 INFINITESIMAL CALCULUS. [CU. 1 Ex. 2. \+x 2 l+x' Here y = for x = and a; = 1, aud y = + oo for a; = — 1. Also y changes sign as x passes through each of these values. For numerically large values of x, whether positive or negative, the curve approximates to the straight line y^-x + 2, lying beneath this line for a; = + oo , and above it for a; = — oo . The figure shews the curve. X'k- -H } 1 1- ^— 'X Y' Fig. J 5. 15] CONTINUITY. 31 Ex. 3. y- 2a; 1-3?' Here y vanishes for x = 0, and for £c = + oo , and becomes infinite for x = ±\. Again, y is positive for l>a;>0 and negative for x>\. Also, y changes sign with x. X- Ex. 4. 2/ = 2a! As in the preceding Ex. y vanishes for a: = and a: = + oo , and changes sign with x. But the denominator does not vanish for any real value of x, so that y is always finite. S2 INFINITESIMAL CALCULCS. [CH. 1 EXAMPLES. I. 1. Draw graphs of the following functions* : (1) (2) (3) (4) (5) (6) (7) l-a;^ ar'. {x-l){x-2), x^-x + 1. x(l-xf, x^{l-xf. 1 1 1 -, X+-, x--. X X X ^"' jx- ix-\){x-2) {x-l){x-?. X ' x-% a^ tx? (9) \ — x + a? 2. 1 + a; + x" ■ x X l-x' (l~xf 3. Prove that the equation 2a? + 5!K'-5x-3 = has a root between — oo and — 1, another between — 1 and 0, and a third between 1 and 2. 4. Prove that the equation 2a? + 7a^ + 3x~5=0 has three real roots, and find roughly their situations. 5. Find roughly the situations of the roots of 2ir»-3a;'-36a! + 10 = 0. 6. Prove that every algebraic equation of odd degree has at least one real root ; and that every equation of even degree, whose first and last coefficients have opposite signs, has at least two real roots, one positive and one negative. * The curves should be drawn carefully to scale ; for this purpose paper ruled into small squares is useful. The numerical tables of squares, square- roots, and reciprocals, given in the Appendix (Tables A, B, C), will occa- sionally help to shorten the arithmetical work. 16] CONTINUITY. 33 16. Transcendental Functions. The Circular Functions. The first place among transcendental functions is claimed by the ' circular ' functions sin 0), cos X, tan x, &c., ■whose definitions and properties are given in books on Trigonometry. The function sin x is continuous for all values of x. For S (sin x) = sin (x + Sx) — sin x = 2 sin ^Bx . cos (x + ^Sx). The last factor is ; always finite, and the product of the remaining factors can be made as small as we please by taking Sx small enough. In the same way we may shew bhat cos x is continuous. This result is, however, included in the former, since cos X = sin {x + Itt). ... , sin a; Aijam, since tan x = , ^ cos a; the continuity of sin x and cos x involves (Art. 1 3) that of tan X, except for those values of x which make cos x = 0. These are given by aj = (n + ^) tt, where n is integral. In the same way we might treat the cases of sec x, cosec X, cot X. The figures on p. 84 shew the graphs of sin x and tan x. The reader should observe how immediately such relations as sin (—«)=— sin x, sin (tt — «) = sin x, sin (a; + tt) = — sin x, tan (a; + tt) = tan x can be read off from the symmetries of the curves. L. 3 84 INFINITESIMAL CALCULUS. Yi [CH. 1 Fig. 19. 16-17] CONTINUITY. 35 17. The Exponential Function. We consider next the ' exponential ' function. This may be defined in various ways ; perhaps the simplest, for our purpose, is to define it as the sum of the infinite series i+^+ri+rl:T3+ (^)- To see that this series is convergent, and has therefore a definite ' sum,' for any given value of x, we notice that the ratio of the (m + l)th term to the mth is aj/m. This ratio can be made as small as we please (in absolute value) by taking m great enough. Hence a point in the series can always be found after which the successive terms will diminish more rapidly than those of any given geornetrical progression whatever. The series is therefore con- vergent, by Art. 6. It is, moreover, 'absolutely' convergent. If we denote the sum of the series (1) by E(x), it may be shewn that E{x)xE{y) = E{x + y) (2). The proof involves the rule for multiplication of absolutely convergent series. Let and ■!;„ + Uj + iij +...+■«„+... / ' be two such series, whose sums are U and V respectively; and consider the series w^ + w^ + Wi+ ...+Wn+ (4), where w„ consists of aU the products of ' weight ' * n which can be formed by multiplying a term of one of the given series (3) by a term of the other ; viz. + u„^iVi + u^% (5). We will first shew that the series (4) will be absolutely con- vergent, and that its sum will be TJV, the product of the sums of the two former series. Let us write Vn=v„ +«! + ^2 + ... +•"„,(. (5). * The ' weight ' is the sum of the suffixes. 3—2 36 INFINITESIMAL CALCULUS. i 0) _ (a: + y)" ~ w! • by the Binomial Theorem. The several terms of the series (4) coincide therefore with those of the series for Eix + y). Hence the result (2). It follows from (2) that (10). and so on for any number of such factors. Also, since E(x) x E (- so) = E{0) = 1 (11), we have E{-x)=-^r7-\ (12). The function E {x) is continuous for all finite values of x. For, writing h for ^x, we have E(x-\-h)- E(x) = E{x).E(h) -E{x) = E{x){E{h)-\]. Now J5;(fe)-l=A(l + A + |^+...). It is easily seen that the series in brackets is absolutely convergent, and that its sum approaches the limit 1 as A is indefinitely diminished ; whilst the factor h can be made as small as we please. Hence E (x + h)-E (x) can be made as small as we please by taking h small enough. When X is positive, every term of the series (1) con- tinually increases with x, and becomes infinite for a; = + oo . The same holds d, fortiori for the sum E{x). Also, in virtue of (12), it appears that if x be positive E(-x) is positive and continually diminishes in absolute value as x increases, and vanishes for .-b = oo . Hence as x increases from — oo to + oo , the function E (x) continually increases from to + oo , and assumes once, and only once, every intermediate value. 38 INFINITESIMAL CALCULUS. [CH. I The accompanying figure shews the curve y = E{x). A column of numerical values of the function E{cb) is given in Table E at the end of the book. Fig. 20. IT-IS] CONTINUITY. 89 18. The number e. If we form the product of n factors, each equal to ^(1), we have, as in Art. 17, (10), {£^(])}» = ^(1 + 1 + ... to 11 terms) = ^(«) ...(1). It is usual to denote the quantity ^fc'(l), or ^ + ^+A + ^+ ^^^' by the symbol e. Its value to seven places of decimals is 6 = 2-7182818. With this notation, we have, if n be a positive integer, E{n)=e'' (3). Again, if m/w be an arithmetical fraction (in its lowest terms), we have \e (3}"= E(~ + j^ + ...ton terms] = E (m) = e™, and therefore El — ) = fi", (*)■ Hence, if x be any positive rational quantity, integral or fractional, we have E{x)=e'' (5). It follows, from Art. 17 (12), that so that the formula (.5) holds for all rational values of x, whether positive or negative*. * The investigations of Arts. 17, 18 are due, substantially, to Cauchy, Analyse AlgSbriq-ue (1821). 40 INFINITESIMAL CALCULUS. [CH. I The actual calculation of e is very simple. The first 13 terms of the series (2) are as follows : 1 + 1 =2 l-. = -000 198 413 2! 3! J_ 4! l^ 5! J_ 6! 2 7! : -5 1 8! : -166 666 667 1 9! : -041 666 667 1 10! = -008 333 333 1 11! = -001 388 889 1 12! •000 024 802 •000 002 756 •000 000 276 •000 000 025 •000 000 002 The sum of these numbers is 2-71 8281830. The error involved in neglecting the remaining terms is 1 13!''"14!"*"15!'^" which is less than or 13 ! \ 13 ■•■ 13« ■*" 13' + -"^ ' "' 12.121* and therefore does not affect the ninth place of decimals. Hence, allowing for the errors of the last figures in the above table, we may say with confidence that the result just found represents the value of e correctly to seven decimal places. In this book we shall have no need to recognize irrational indices, and the symbol eF, when x is irrational, is therefore to be considered as (so far) undefined. We may now define it as merely another symbol for the sum of the series E {so). The advantage of this definition is that the notation serves to remind us of the algebraical laws to which the function is subject. Thus we nave ^.ey = E (*■) xE{y) = E(a! + y) = ef'+v, whether x and y be rational or irrational. 18-19] CONTINUITY. 41 19. The Hyperbolic Functions. There are certain combinations of exponential functions whose properties have a close formal analogy with those of the ordinary trigonometrical functions. They are called the hyperbolic sine, cosine, tangent, &c.*, and are defined and denoted as follows : . , , , „ , a? al^ smh a; = J (e* - e"*) = a; + ^^ + ^^ + , cosh a; = ^(e»' + e-*) =1 + |i + ^ + (l)t. tanh X = coth X = sinha; cosh X ' cosha: sinh X ' sech X = cosech X = \ cosh X ' 1 sinh a; .(2). We notice that cosh a;, like cos a;, is an 'even' function of X ] i.e. it is unaltered by writing ~ x for a; ; whilst sinh x, like sin x, is an ' odd ' function, i.e. the function is unaltered in absolute value but reversed in sign by the same substitu- tion of — a; for x. The continuity of cosh x and sinh x follows from that of e* and e~* bj' a theorem of Art. 13. The figure on the next page shews the curves y^e", y = e-", together with the curves y = cosh X, y = sinh x, which are derived from them by taking half the sum, and half the difference, of the ordinates, respectively^. * They have in some respects the same relation to the rectangular hyperbola that the circular functions have to the circle. See Art. 98, Ex. 4. t The series are added according to the rule proved at the end of Art. 6. j The curve y = cosiix is kaown in Statics as the 'catenary,' from its being the form assumed by a chain of uniform density hanging freely under gravity. 42 INFINITESIMAL CALCULUS. LCH. I t—y 19] CONTINUITY. 43 Since sinh x and cosh x are continuous, whilst cosh x never vanishes, it follows that tauha; is continuous for all values of x. Fig. 22 shews the curve y = tanh x. This has the lines y = + 1 as asymptotes *, It is evident from the definitions (1) that cosh *• + sinh a; = e*, cosh x — sinh x = e~^ ... (3), whence, by multiplication, cosh^ X — sinh^ x = \ (4). From this we derive, dividing by cosh" a; and sinh" a;, respectively, sech" x=\— tanh" a;, 1 ,^. cosech" a; = coth" a; — 1 ) The forraulse (4) and (5) corresponcl to the trisjoiiometncal formuliB cos"a3 + sia"a;= 1 (6), sec"a: = l +taii"a;,1 cosec" X = cot" X iii"a;,| !+lJ .(7). * The numerical values (to three places) of the functions cosh x, sinh x, tanh a;, for values of x ranging from to 2-5 at intervals of 0-1, are given in the Appendix, Table E. 44 INFINITESIMAI, CALCULUS. [CH. I Again, from (1) and (3) we easily find sinh {x±y) = sinh x cosh y + cosh x sinh y, I cosh (a; + 1/) = cosh a; cosh y + sinh a; sinh y, J" whence, as particular cases, sinh 2i» = 2 sinh a; cosh a;, 1 , , cosh 2a! = cosh" a; + sinh' a;, J These are the analogues of the trigonometrical formulse sin (a; + y) = sin a; cos 2/ + cos a; sin y, I ..^. cos (a; + y) = cos a; cos y + sin a; sin y, J ^ and sin 2a! = 2 sin a: cos a;, 1 ,,, cos 2a! = cos' a; — sin' a; J respectively. 20. Inverse Functions in general. If y be a continuous function of x, then under certain conditions x will be a continuous function of y. This will be the case whenever the range of x admits of being divided into portions (not infinitely small) such that within each the function y steadily increases, or steadily decreases, as x increases. Let us suppose that as x increases from a to 6 the value of y steadily increases from a to j8. Then corresponding to any given value of y between « and y8 there will be one and only one value of x between a and h. Hence if we restrict V Fig. 23. 19-21] CONTINUITY. 45 ourselves to values of x within this interval, a> will be a single-valued function of y. Also, if we give any positive increment e to a;, within the above interval, y will have a certain finite increment a-, and for all values of By less than 0-, we shall have Sx<6. A similar argument holds if the increment of oo be negative. Hence we can find a positive quantity a such that, e being any assigned positive quantity, however small, | Sa; | < e for all values of Sy such that I Sy I < o". But this is the condition for the continuity of x regarded as a function of y (Art. 9). The same conclusion obviously holds, if y steadily diminishes in the interval from x = a to x = b. If we do not limit ourselves to a range of x within which the function steadily increases, or steadily diminishes, then to any given value of y there may correspond more than one value of x; the inverse function is then said to be 'many- valued.' Again, it may (and in general will) happen that through some ranges of y there are no corresponding values oi X, i.e. the inverse function does not exist. Bx. Let y = x^. This is a continuous function of x, and, if X be positive, continuously increases with x. Hence x, = ^y, is a continuous function of y. If x be unrestricted as to sign, we have two values of x for every positive value of y ; these are usually denoted by + Jy. If y be negative, the inverse function ijy does not exist. If 2/=/(«') • (1), the inverse functional relation is son^etimes expressed by *=/-(y) (2). We then have f[f-'{y)} =f{«=) = y (3), i.e. the functional symbols / and /""' cancel one another. This is the reason of the notation (2). 21. The Inverse Circular Functions. The ' goniometric ' or ' inverse circular ' functions sin~'«, cos~'a;, tan^^a;, &c. are many-valued. The functions sin~' x, cos~' x exist for values of x ranging from — 1 to + 1, but not for values outside these limits. 46 INFINITESIMAL CALCULUS. [CH. I The fuuction tan^^a; exists for all values of a;; it is many-valued, the values forming an arithmetical progression with the common difference tr. The graph of any inverse function is derived from that of the direct function by a mere transposition of x and y. The curves for sin""^ x and tan~' x are shewn in Art. 41. 22. The Logarithmic Function. The 'logarithmic' function is defined as the inverse of the exponential function. Thus if x = ey, we have y = \ogx (1). It appears from Art. 17 that as y ranges from — oo through to + 00 , e" steadily increases from through 1 to + oo . Hence for every positive value of x there is one and only one value of log x ; moreover this value will be positive or negative, according as a; 5 1. Also for a; = we have y= — , and for a; = + oo, y = + oo. For negative values of x the logarithmic function does not exifst. The ordinary properties of the logarithmic function follow from the above definition in the usual manner. The full line in Fig. 24 shews the graph of log a;. It is of course the same as that of e' (Fig. 20) with x and y inter- changed*. We can now define the symbol a', where a is positive, for the case of X irrational. Since we have, if x be rational, „!c = ga;loga (2), and the latter form may be adopted as the definition of a'" when X is irrational. , The logarithm of x, as above defined, is sometimes denoted by logeX to distinguish it from the common or 'Briggian' logarithm logiD X. The latter may be regarded as defined by the statement that y = logi„9;, if 10!' = a;, or e«"°K'" = a! (3). * The function log x is tabulated in the Appendix, Table P. 21-22] 47 48 INFINITESIMAL CALCULUS. [CII. I Hence j/logj 10 = logea;, or logi„ a; = /a loge a; (4), where ii = =—irA = -43429...* (5). '^ log, 10 '^ Hence the graph of the function log,o x is obtained from that of logeX by diminisliing the ordinates in the constant ratio /ot. See the dotted line in the figure. In this book we shall always use the symbol log x in the sense of loge X. 23. The Inverse Hyperbolic Functions. The inverse hyperbolic functions sinh~' X, cosh"' x, tanh~' x, &c. are defined in the manner explained in Art. 20 ; thus the meaning of y = sinh~' x is that X = sinhy (1), and so on. These functions can all be expressed in terms of the logarithmic function. Thus if X = smh. y = ^{6" - e-y) (2), wo have e'^-1xev-l = (3). Solving this quadratic in e", we find ey=x±s/{x''+\) (4). If y is to be real, en must be positive, and the upper sign must be taken. Hence sinh-'a; = log[a! + V(«^ + l)} (5). In a similar manner we should find that, if a; > 1 , cosh-' « = log {a; + V(«^-l)} (6). Either sign is here admissible; the quantities x + J{a?-\) are reciprocals, and their logarithms differ simply in sign. It appears on sketching the graph of cosh"' x that for every value of X which is > 1 there are two values of y, equal in magnitude, but opposite in sign. • Xhe mode of calculating this quantity will be indicated in Chapter XIII. 22-23] CONTINUITY. 49 Similarly, we should find, for «"< 1, and, for x^ > 1, tanh-"'a! = ^log- (7), coth-'a; = ilog-^ (8). EXAMPLES. II. 1. Draw graphs of the following functions : (1) cosec a;, cotaj, cot a; + tan a;. (2) cosech X, coth x, coth x - tanh x. (3) sin^ X, tan" x. (4) sin X + sin 2a!, sin x + cos 2a;. (5) logiosina;, logiotana;, from x = to x = ^ir. (6) sin a?, sin ^x, sin - . 2. Prove that the equation sin a; — a; cos a; = has a root hetween ir and f tt. 3. Prove by calculation from the series for e^ that l/e = -367879, cosh 1 = 1-5430806, sinh 1 = 1-1752012. 4. Prove that ^6 = 1-6487213, 1/^e = -6065307, cosh J =;;: 1-1276260, sinh J = -5210953. 5. If I a I < 1 6 1 the equation a cosh x + b sinh a; = has one, and only one, real root. 6. Shew that the function tanh — X is discontinuous for a; = 0. Draw a graph of the function. L. 50 INFINITESIMAL CALCULUS. [CH. I EXAMPLES, in. Prove the following formulae : 1. cosh 2a; = 2 cosh^ a; — 1 = 1 + 2 sinh' x. „ . , „ 2 tanh x , . 1 + tanh" x 2. sinh 2a! = , — - — rr- , cosh 2a! = .j — j — j-^— , 1 — tanh'a! 1— tanh'a: ^ , „ 2 tanh x tanh 2a! = -z — - — pi— , 1 + tanh' X 3. cosh" a! cos" x + sinh" x sin' x = ^ (cosh 2a: + cos 2a!), cosh' a! sin' x + sinh' a! cos' a; = J (cosh 2a; — cos 2a!). 4. cosh' x cos' x — sinh' x sin' a; = J ( 1 + cosh 2a! cos 2a!), cosh' X sin' x — sinh' x cos' a! = J (1 — cosh 2a; cos 2a!). 5. cosh' u +. sinh' v = sinh' u + cosh' v = cosh (u + v) cosh (m — v), cosh' u — cosh' V = sinh' m - sinh' v = sinh (?t + «;) sinh (m-d). 6. sinh"' X = cosh"' ^(a;' + 1), cosh"' ai = sinh"' s/{x' — 1). ^-, , l + x/(l-a;') ^_, , 1 + ^(1 H-ar") 7. sech 'a; = log ^^-^ -, cosech 'a;=log — ^^^-^ . ° x ° X 8. tanh-i^-j-^ = loga;. 9. If x = log tan {{tt + J i/-), then sinh a; = tan ^, cosh a! = seci/', tanha! = sini/', and tanh ^ a! = tan J ^. 10. sinh u + sinh v = 2 sinh ^ {u + v) cosh ^ (u- v), sinh u — sinh v=2 cosh ^ (m + 1;) sinh ^ (u — v), cosh M + cosh D = 2 cosh J («* + «)) cosh ^ (u- v), cosh M - cosh v = 2 sinh J {u + v) sinh J (m — ii). , , i , 1 sinh M cosh m — 1 11. tanh *u= — ; = — :—, cosh M + 1 sinh u _ //coshw — 1\ V \coshM + l/ 12. sinh .3m = 4 sinh°M + 3 sinh u, cosh 3m = 4 cosh^M - 3 cosh u. 24] CONTINUITY. 51 24. Upper or Lower Limit of an Assemblage. Before proceeding further with the theory of continuous functions it is convenient to extend the definitions of the terms ' upper ' and ' lower ' limit, and ' limiting value/ given in Arts. 2 and 5. Consider, in the first place, any assemblage of magnitudes, infinite in number, but all less than some finite magnitude yS. The assemblage may be defined in any way ; all that is necessary is that there should be some criterion by which it can be determined whether a given magnitude belongs to the assemblage or not. For instance, the assemblage may consist of the values which a given function (continuous or not) assumes as the independent variable ranges over any finite or infinite interval. In such an assemblage there may or may not be con- tained a 'greatest' magnitude, i.e. one not exceeded by any of the rest ; but there will in any case be an ' upper limit ' to the magnitudes of the assemblage, i.e. there will exist a certain magnitude /i such that no magnitude in the assem- blage exceeds /x, whilst one (at least) can be found exceeding any magnitude whatever which is less than im. And if /* be not itself one of the magnitudes of the assemblage, then an infinite number of these magnitudes can be found exceeding any magnitude which is less than /j,. The proof of these statements follows, as in Art. 2, by means of the geometrical representation. In the same way, if we have an infinite assemblage of magnitudes, all greater than some finite quantity a, there may or may not be a ' least ' magnitude in the assemblage ; but there mil in any case be a ' lower limit ' \ such that no magnitude in the assemblage falls below \, whilst one (at least) can be found below any magnitude whatever which is greater than \. And if \ be not itself one of the magni- tudes of the assemblage, an infinite number of these magni- tudes can be found less than any magnitude which is greater than \. 4—2 52 INFINITESIMAL CALCULUS. [CH. I 25. A Continuous Function has a Greatest and a Least Value. An important property of a continuous function is that in any finite range of the independent variable the function has both a greatest and a least value. More precisely, if 3/ be a function which is continuous from x = a to x = h, inclusively, and if /i be the upper limit of the values which y assumes in this range, there will be some value of x in the range for which y = fi. Similarly for the lower limit. The theorem is self-evident in the case of a function which steadily increases, or steadily decreases, throughout the range in question, greatest and least values obviously occurring at the extremities of the range. It is therefore true, further, when the function is such that the range can be divided into a finite number of intervals in each of which the function either steadily increases or steadily decreases. The functions ordinarily met with in the applications of the subject are, as a matter of fact, found to be all of this character, but the general tests by which in any given case we ascertain this are established by reasoning which assumes the truth of the theorem of the present Art. See Art. 47. It is therefore desirable as a matter of logic to have a proof which shall assume nothing concerning the function considered except that it is continuous, according to the definition of Art. 9. The following is an outline of such a demonstration. In the geometrical representation, let OA = a, OB = h. If at A the value of y is not equal to the upper limit /t, it will be less than /i ; let us denote it by y^. We can form, in an infinite number of ways, an ascending sequence of magnitudes whose upper limit is /t. For example, we may take y-i equal to the arithmetic mean of y^ and ft, y^ equal to the arith- metic mean of y^ and fi, and so on. Since, within the range AB, the value of y varies from y„ to any quantity short of /i. 25] CONTINUITY. 53 there will (Art. 10) be at least one value of sc for which y assumes the intermediate value y-i. Let x^ denote this value, or (if there be more than one) the least of such Mj M, M3 M, wi Fig. 25. values, of x. Similarly, let x^ be the least value of x for which y = y^, and so on. It is easily seen that the quantities x^, X2, x^,,.. (which are represented by the points ifj, M,, M„... in the figure*) must form an ascending sequence ; let M represent the upper limit of this sequence. Since any range, however short, extending to the left of M contains points at which * The diagram is intended to be merely illustrative, and is not essential to the proof. It is of course evident that any function which can be adequately represented by a graph is necessarily of the special character above referred to, for which the present demonstration is superfluous. In the figure, OK = /i,, OH = y„, ONi = y^, ON.^ = 2/2,...; and, in the mode of forming the sequence 2/0. Vi, 2/2. ■•• which is suggested (as a particular case) in the text, N^ bisects HK, N^ bisects N^K, N^ bisects N^K, and so on. 54 INFINITESIMAL CALCULUS. [CH. I y differs from /t by less than any assignable magnitude, it follows from the continuity of the function that the value of y at the point M itself cannot be other than ft,. To see that the above theorem is not generally true of dis- continuous functions, consider a function defined as follows. For values of x other than let the value of the function be (sin x)/x, and for x = let the function have the value 0. This function has the upper limit 1, to which it can be made to approach as closely as we please by taking | x \ small enough ; but it never actually attains this limit. As another example consider the function defined as equal to for all rational values of x, and equal to sin irx for all other values of x. (We have here an instance of a function which is discontinuous for every value of x.) 26. Iiimiting Value of a Function. Consider the whole assemblage of values which a function y (continuous or not) assumes as the independent variable x ranges over some interval extending on one side of a fixed value Xi. Let us suppose that, as x approaches the value Xi, y approaches a certain fixed magnitude X in such a way that by taking !« — a;i| sufficiently small we can ensure that for this and for all smaller values of | a; — ajj | the value of | y — \ | shall be less than a; where a may be any assigned magnitude however small. Under these conditions, \ is said to be the ' limiting value ' of y as x approaches the value x^ fi-om the side in question. The relation is often expressed thus : lima,=a.,2/ = A,, but in strictness the side from which x approaches the value Xi should be specified. If we compare with the above the definition of Art. 9 we see that in the case of a continuoviS function we have lim«=».^(a!) = <|>(a!,) .(1), or the 'limiting value' of the function coincides with the value of the function itself, and that if x^ lie within the range of the independent variable this holds from whichever side X approaches Xi. If, on the other hand, ^i coincides with 25-27] CONTINUITY. 55 either terminus of the range, a; must be supposed to approach Xy from within the range. Conversely, a function is not continuous unless the con- dition (1) be satisfied. Let us next take the case of a function the range of whose independent variable is unUmited on the side of x positive. If as x is continually increased, y tends to a fixed value \ in such a way that by taking x sufficiently great we can ensure that for this and for all greater values of x we shall have | y - \ | less than cr, where a may be any assigned positive quantity, however small, then \ is called the limiting value of y for a; = M , and we write limj;=„ y = Jt. There is a similar definition of lim^=_oo y, when it exists, in the case of an independent variable which is unlimited on the side of x negative. 27. General Theorems relating to Limiting Values. 1°. The limiting value of the sum of any finite number of functions is equal to the sum of the limiting values of the several functions, provided these limiting values be all finite. 2°. The limiting value of the product of any finite number of functions is equal to the product of the limiting values of the several functions, provided these limiting values be all finite. 3°. The limiting value of the quotient of two functions is equal to the quotient of the 'limiting values of the separate functions, provided these limiting values be finite, and that the limiting value of the divisor is not zero. The proof is by the same method as in Art. 13, the theorems of which are in fact particular cases of the above. Thus, let M, V be two functions of x, and let us suppose that as X approaches the value x^, these tend to the limiting values M,, «,, respectively. If, then, we write 56 INFINITESIMAL CALCULUS. [CH. I a and /8 will be functions of x whose limiting values are zero. Now (u + v) — (iti + Vi) = a + P, uv - iItV^ = aVi + ySwi + aj8, u th _ "■'"i ~ P''h V Vi~ Vi (vi + P) ' And, as in Art. 13, it appears that by making x sufficiently nearly equal to o^ we can, under the conditions stated, make the right-hand sides less in absolute value than any assigned magni- tude however small. 28. Illustrations. We have seen in Art. 26 that the limiting value of a continuous function for any value x^ of the independent variable x, for which the function exists, is simply the value of the function itself for a; = a^. It may, however, happen that for certain isolated or extreme values of the variable the function does not exist, or is undefined, whilst it is defined for values of x differing infinitely little from these. It is in such cases that the conception of a ' limiting value ' becomes of special importance. For example, consider the period'of oscillation of a given pendulum, regarded as a function of the amplitude a. This function has a definite value for all values of a between and TT, but it does not exist, in any strict sense, for the extreme values and tt. There is, however, a definite limiting value to which the period tends as a approaches the value zero. This limiting value is known in Dynamics as the ' time of oscillation in an infinitely small arc' Some further illustrations are appended Hx. 1. Take the function 1-n/(1-«^) x'' The algebraical operations here prescribed can all be performed for any value of x between + 1, except the value 0, which gives to the fraction the form 0/0. Now the definition of a quotient 27-28] CONTINUITY. 57 ajh is that it is a quantity which, multiplied by b, gives the result a. Since cmy finite quantity, when multiplied by 0, gives the result 0, it is evident that the quotient 0/0 may have any value whatever. It is therefore said to be ' indeterminate.' We "may, however, multiplying numerator and denominator of the given fraction by 1 + ^(1 - k"), put the function in the equivalent form a" and for all values of x between + 1, otlier than 0, this is equal to 1 1 + 7(1-4- Since this function is continuous, and exists for x=0, its limiting value for sc = is ^. Ex. 2. Consider the function J{\+x)- ^x. As X is continually increased this tends to assume the in- determinate form 00 — 00 . But, writing the expression in the equivalent form 1 J{\+x)+Jx' we see that its limiting value for k = oo is 0. Ex. 3. To find limj-^^Ke-". This assumes the indeterminate form oo x 0. But since ./«»'= l/(i , X or ...). we see that the limiting value for a; = oo is 0. If we write z for e'", and therefore - log z for x, we infer that lim^^o zlogz = 0. In the same way we can prove that lim^_„a;"e-* = 0, where m is any rational quantity. 58 INFINITESIMAL CALCULUS. [CH. I 29. Some Special Iiimiting Values. The following examples are of special importance in the Differential Calculus. 1". To prove that lima,=a = maF^^ (1), for all rational values of m. If m be a positive integer, we have r«TO /jWl 06"- — a" limj; = a — - — = ^VCS\x-a («"^' + a«"^ + . ■ . + «"'-=« + a™"') = ??m™~S since, the number (m) of terms being finite, the limiting value of the sum is equal to the sum of the limiting values of the several terms (Art. 27). If m be a rational fraction, =p/g', say, we put x = y^, a = Jfi, and therefore gfi _ gm ymq _ Jfnq yp _ Jj» X — a " y2 - 6? ~ y^ — b^ ' This fraction is equal to y^-bi' y-b The limiting value of the numerator is ph^\ and that of the denominator is qbi~^, by the preceding case. Hence the required limit is asbefo.. ' ' If m be negative, = — m, say, we have ^m_„m ar^-a-^ 1 a;" -a" x — a a;"a" x — a 29] CONTINUITY. 59 If n be rational, the limiting value of this is m, ■ wa'^S = - na-^^\ = ma™~', by the preceding cases. 2°. To prove that ,. sin ^ ,. tan^ , , lim9=o-^=l, lime-o ^Y~^ ^ ^' If we recall the definition of the ' length ' of a circular arc, given in Art. 4, these statements are seen to be little more than truisms. If, in Fig. 4, the angle POQ be 1/nth of four right angles, then n . PQ will be the perimeter of an inscribed regular polygon of n sides, and n (TP + TQ) will be the perimeter of the corresponding circumscribed poly- gon. Now, if = Z POA = irjn, we shall have chord PQ _ PN' _ sm 9 arc PQ ~aTcPA~ 6 ' TP + TQ PT . tan 6 ^^^ arcPQ ~arcP^~ ' Hence the fractions sin 1 tan -r ^""^ -0- denote the ratios which the perimeters of the above-mentioned polygons respectively bear to the perimeter of the circle. Hence, as n is continually increased, each fraction tends to the limiting value unity (Art. 4). In the above argument, it is assumed that is a stib- multiple of it. But, whatever the value of the angle POQ in the figure, we have chord PQ < arc PQ, and TP + TQ> arc PQ ; i.e. (sin 0)10 is less than 1, and (tan0)/0> 1. Hence these fractions must have respectively an upper and a lower limit ; and it follows from the preceding that neither of these limits can be other than unity. 60 INFINITESIMAL CALCULUS. [CH. I The following numerical table illustrates the way in which the above functions approach their common limiting value as 6 is continually diminished. 71 ejrr (am 0)10 (tan 0)10 4 •25 •90032 1-27324 5 •20 •93549 1-15633 10 •10 •98363 103425 20 •05 •99589 1^00831 40 ■025 ■99897 1-00206 00 1-00000 1-00000 The third and fourth columns give the ratios which the peri- meters of the inscribed and circumscribed regular polygons of n sides respectively bear to the perimeter of the circle. 3°. To prove that limA_o — j— = 1- .(3). We have e*-l , h (n-^+^+. 3 ' 3.4 The series in brackets is convergent, and its sum has the limit 1 when h = 0. Hence by taking h small enough, the difference between (e* — l)//i and 1 can be made as small as we please. If we put A = log[l+-), the theorem (3) takes the form lim„= z n ■=1. or whence ..g(i+£) lim„_„ (l H — j =^ .... .(4). and therefore 29-30] CONTINUITY. 61 30. Infinitesimals. A variable quantity which in any process tends to the limiting value zero is said ultimately to vanish, or to be 'infinitely small.' Two infinitely small quantities are said to be ultimately equal when the limiting value of the ratio of one to the other is unity. Thus, in Fig. 4, p. 7, when the angle POQ is indefinitely diminished, NA and AT are ultimately equal. For, by similar triangles, OP or OJV~ OP' OP-ON _ OT-OP ON ~ OP ' NA_ ON *"■ AT' OP' and the limiting value of the ratio ONjOP is unity. It is sometimes convenient to distinguish between diffe- rent orders of infinitely small quantities. Thus if u, v are two quantities which tend simultaneously to the limit zero, and if the limit of the ratio v/u be finite and not zero, then V is said to be an infinitely small quantity of the same order as u. But, if the limit of the ratio v/u be zero, then v is said to be an infinitely small quantity of a higher order than u. More particularly, if the limit of v/u™' be finite and not zero, V, is said to be an infinitesimal of the mth order, the standard being u. Thus, in the figure referred to, NT is an infinitesimal of the second order, if the standard be PN. For PN' = ON.NT, NT 1 1 whence p]p= ON' " 02 '" ^''^ ^™'*- In calculations involving quantities which are ultimately made to vanish, only infinitesimals of the lowest order present need as a rule be retained; since the neglect ab initio of any finite number of infinitesimals of higher order will make no difference in the accuracy of the final result. We shall have frequent exemplifications of this principle. 62 INFINITESIMAL CALCULUS. [CH. I A quantity which in any process finally exceeds any assignable magnitude is said to be ' infinitely great.' And if one such quantity u be taken as a standard, any other v is said to be infinitely great of the mth order, when the limit of vju™ is finite and not zero. EXAMPLES. IV. 1. Shew geometrically that the sequence p + q p + q \p + qj ' in has the upper limit 1 + - . q 2. Find the upper and lower limits of the magnitudes ri'+l where 9i = l, 2, 3 3. If a and 5 be positive, and a>h, the function ae" + be-" ^ + «-" has the upper limit a and the lower limit h. 4. Find the limiting values, for a; = 0, of sin ax, . sinh ax and . sinh a; 1 — cos X . lim_ , (sec x — tan x) = 0. 8. Prove that lim^„ >/(L±i^zV(lzL^)^ 1_ X 9. Prove that X 5. Trace the curves sin X y- X 6. Prove that linix-o 7. Prove that 80] CONTINUITY. (J3 10. Prove that lim ^H£^ = 0, and hence shew that lim \/n=\. n=oo V 11. Find the limiting values, for aj = 0, of sin"' a; , tan~'a! and . X X 12. A straight line AB moves so that the sum of its intercepts OA, OB on two fixed straight lines OX, OY is constant. If P be the ultimate intersection of two consecutive positions of AB, and Q the point where AB is met by the bisector of the angle XOY, then AP=QB. 13. Through a point ^ on a circle a chord AP is drawn, and on the tangent at .4 a point T is taken such that AT = AP. If TP produced meet the diameter through A in Q, the limiting value oi AQ when P moves up to A is double the diameter of the circle. 14. A straight line AB moves so as to include with two fixed straight lines OX, OZ a triangle jIO-S of constant area. Prove that the limiting position of the intersection of two consecutive positions of AB i& the middle point of AB. 15. A straight line AB oi constant length moves with its extremities on two fixed straight lines OX, OY which are at right angles to one another. Prove that if P be the ultimate intersection of two consecutive positions of AB, and N\he foot of the perpendicular from on. AB, then AP = NB. 16. Tangents are drawn to a circular arc at its middle point and at its extremities; prove that the area of the triangle contained by the three tangents is ultimately one-half that of the triangle whose vertices are the three points of contact. 17. If PGP' be any fixed diameter of an ellipse, and Q V any ordinate to this diameter; and if the tangent at Q meet CP produced in T, the limiting value of the ratio TP :PV, when PV is infinitely small, is unity. CHAPTER II. DEEIVED FUNCTIONS. 31. Definition and Notation. Let y be a function which is continuous over a certain range of the independent variable x; let hx be any incre- ment of X such that x-'rhx lies within the above range, and let hy be the consequent increment of y. Then, x being regarded as fixed, the ratio I w will be a function of hx. If as 8a; (and consequently also iy) approaches the value zero, this ratio tends to a definite and unique limiting value, the value thus arrived at is called the 'derived function,' or the 'derivative,' or the 'differential coefficient,' of y with respect to x, and is denoted by the symbol i ■ • ■«■ More concisely, the derived function (when it exists) is the limiting value of the ratio of the increment of the function to the increment of the independent variable, when both increments are indefinitely diminished. It is to be carefully noticed that in the above definition we speak of the limiting value of a certain ratio, and not of the ratio of the limiting values of hy, 8x. The latter ratio is indeterminate, being of the form 0/0. The symbol dyjdx is to be regarded as indecomposable, it is not a fraction, but the limiting value of a fraction. The fractional appearance is preserved merely in order to remind us of the manner, in which the limiting value was approached. 31] DERIVED FUNCTIONS. 65 When we say that the ratio Sy/Sx tends to a unique limiting value, it is implied that (when a; lies within the range of the independent variable) this value is the same whether Sx approach the value from the positive or from the negative side. It may happen that there is one limiting value when Bx approaches from the positive, and another when Sx approaches from the negative side. In this case we may say that there is a ' right-derivative/ and a ' left- derivative,' but no proper 'derivative' in the sense of the above definition. The question whether the ratio SyjSx really has a deter- minate limiting value depends on the nature of the original function y. Functions for which the limit is determinate and unique (save for isolated values of x) are said to be ' differentiable.' All other functions are excluded ab initio from the scope of the Differential Calculus. A differentiable function is necessarily continuous, but the converse statement is now known not to be correct. Functions which are continuous without being differentiable are, however, of very rare occurrence in Mathematics, and will not be met with in this book. There are various other notations for the derived function, in place of dy/dx. The derived function is often indicated by attaching an accent to the symbol denoting the original function. Thus if y = 't>i^) (3), the derived function may be denoted by y" or by ^' (*•). Since Sy^^(x+S^-^xJ ox bx we have, writing h for 8x, J.' / \ T 6 (x + h) - (b (a) (/) («) = hmft^o ^^^^^ ^ — ^^-^ (4). The operation of finding the differential coefficient of a given function is called ' differentiating.' If x be the inde- pendent variable, we may look upon d/dx as a symbol denoting this operation. It is often convenient to replace 66 INFINITESIMAI, CAI.CULUS. [CH. II bhis by the single letter D ; thus we may write, indifferently, dx' dx^' y' for the differential coeflScient of y with respect to x. 32. Geometrical meaning of the Derived Func- tion. In the annexed figure, let OM=x, ON=x + hx, PM = y, QN=y + hy, Fig. 26. and draw PR parallel to OX. Let QP produced cut the axis of a; in S. Then Sx-pR~m~^^''^^^ (^> As 8a; is indefinitely diminished, Q approaches P, and it follows that if the derived function exist the line PQ tends to a definite limiting position PT, such that tanPrX = ^. dx .(2). It appears then that the assemblage of points (Art. 11) which represents any differeutiable function has at each 31-32] DERIVED FUNCTIONS. 67 of its points a definite direction, or a definite ' tangent-line.' And the derived function is the tangent of the angle which this line makes with the axis of x. The question as to whether a continuous function can be represented by a curve depends, as already stated (Art. 11), on the meaning which we attribute to the term ' curve.' In its ordinary acceptation, the word implies not merely the idea of a connected assemblage of points, but also the existence of a definite tangent-line at every point, and (further) that the direction of this tangent-liae varies continuously as we pass along the assemblage. That is, it is implied that the ordinate y is a differentiable function of the abscissa x, and that the derived function dyjdx is itself a continuous function of x. These conditions will be found to be satisfied, save occasionally at isolated points, by all the functions met with in the ordinary applications of the Calculus. And whenever we speak of a ' curve,' we shall, for the present, not attach to the term any connotation beyond what is contained in the above statements. It is convenient to have a name for the property of a curve which is measured by the derived function. We shall use the term ' gradient ' in this sense, viz. if from any point P on the curve, we draw the tangent-line, to the right, the gradient is the tangent of the angle which this line makes with the positive direction of the axis of x. Fig. 27. If this angle be negative, the gradient is negative. If the tangent-line be parallel to the axis of x, the gradient is zero. If 5—2 68 INFINITESIMAL CALCULUS. [CH. II it be perpendicular to the axis of x, the gradient is infinite. When for a particular value of x we have a right-derivative and a left-derivative, difierent from one another, then on the corre- sponding curve there are two branches making an angle with one another. The value of dyldx is then discontinuous. The figure illustratfes some of these cases. 33. Physical Illustrations. The importance of the derived function in the various applications of the subject rests on the fact that it gives us a measure of the rate of increase of the original function, per unit increase of the independent variable. To illustrate this, we may consider, first, the rectilinear motion of a point. The distance « of the point from some fixed origin in the line of motion will be a function of the time t reckoned from some fixed epoch. The relation between these variables is often exhibited graphically by a ' curve of positions,' in which the abscissae are proportional to t and the ordinates to s ; see Fig. 7, p. 14. If in the interval S< the space Ss is described, the ratio Ss/8i is called the 'mean velocity' during the interval S< ; i.e. a point moving with a constant velocity equal to this would accomplish the same space 8s in the same time Si!. In the limit, when ht (and consequently also S«) is indefinitely diminished, the limiting value to which this mean velocity tends is adopted, by definition, as the measure of the ' velocity at the instant t.' In the notation of the calculus, therefore, this velocity v is given by the formula ds "-dt (^)- In the graphical representation aforesaid, v is the gradient of the curve of positions. Again, the velocity v is itself a function of t. The curve representing this relation is called the 'curve of velocities.' If 8w be the increase of velocity in the interval ht, then hvjht is called the ' mean rate of increase of velocity,' or the ' mean acceleration' in this interval. The limiting value to which the mean acceleration tends when ht is indefinitely diminished is called the ' acceleration at the instant t.' If this acceleration be denoted by a, we have " = 5^ (2)- 32-34] DERIVED FUNCTIONS. C9 In the graphical representation, a is the gradient of the curve of velocities. In the case of a rigid body revolving about a fixed axis, if be the angle through which the body has revolved from some standard position, the ' mean angular velocity' in any interval U is denoted by 8^/8^, and the ' angular velocity at the instant t,' by s w Again, if to denote this angular velocity, the ' mean angular acceleration in the interval ht' is denoted by 8- 4°. If y = tana; (7), we have Sy _ tan {x + h) — tan x _ sin {x + h) cos x — cos {x + h) sin x hx h- h cos X cos {x + h) _ sin h 1 h ' cosaicos («+ '/)' Hence, in the limit, J = ^=setf^ (8). ax cos* a; This shews that the gradient of the curve y = tan x, between the points of discontinuity, is alwaj^s positive; see Fig. 19, p. 34. 5° If y = e'' (9), Sy e*+''-e* e*-l ^ we have ^ = r — = — t — e^. hx h h 35-36] DERIVED FUNCTIONS. 73 It was shewn in Art. 29, 3° that eft — 1 limft=o —J- = 1- Hence ^=e^ (10). More generally, if y = ^^ (H)) we have ^ = hm,.„ ^ = lim,,=„ -j^ . k^^ = k^ (12). In particular, if y = e~^ (13), we have ~ =-e-* (14). Again, if a be any positive quantity, we have by the definition of Art. 22 (2), «• a Hence if 2/ = a* (15), we have, by (12), ^ = log a . e^i^s* = log a . a^ (16). 36. Rules for differentiating combinations of simple types. Differentiation of a Sum. 1°. Let 2/ = M+C7 (1), where m is a known function of x, and (7 is a constant. We have y + Sy = M + 8m + (7, and therefore hy = Su, Sy _ Su Bx Sx' or, in the limit, g = ^ (2). This fact, that an additive constant disappears on differ- entiation, obvious as it is, is very important. The geometrical 74 INFINITESIMAL CALCULUS. [CH. H meaning is that shifting a curve bodily parallel to the axis of y does not alter the gradient. 2°- Let y=u + v (3), where u, v are given functions of x. As in Art. 13, we find By=Su + Sv, , ,, , Sy Su Sv and therefore r^ = f" + t" ■ ox ox ox Hence, since the limiting value of a sum is the sum of the limiting values of the several terms, dy^du_^d^ ^ ^^^ .^. dx dx dx ^ Again, if y = u + v + w (5), , dy d , . dw we have -^ = t- (tt + «) + -r- dx dx dx _du dv dw . dx dx dx ' by a double application of the preceding result. In this way we can prove, step by step, that the derived function of the sum of any finite number of given functions is the sum of the derivatives of the separate functions. JUx. The derived function of is mA„x'^-^ + (m - 1) A-^x^-''+ ... + A^_j^. 37. Differentiation of a Product. 1°. If y = Gu (1), where (7 is a constant, and u a function of x, we have y + Sy = (7 (m + hu), and therefore By = Ghu. Hence Tr- = G =r- , ox ox and, proceeding to the limit, ^ = 0*^ (2) dx dx • -v /■ 36-37] DERIVED FUNCTIONS. 75 Hence a constant factor remains attached after the differ- entiation. The geometrical meaning of this result is that if all the ordinates of a curve be altered in a given ratio, the gradient is altered in the same ratio. Cf Fig. 24, p. 47. 2°. Let y = uv (3), where u, v are both functions of x. As in Art. 13, we find Sy ={u + Su) (v + Sv) — uv = vSu + uSv + SuBv, and therefore rr- = v-s- + (u + Bu) ^r- . ox ox ox Hence, proceeding to the limit, and making use of the principle that the limit of a product is the product of the limits, we have dy _ du dv ,., dx~ dx dx If we divide both sides of this equation by y, = uv, we obtain the form 1 dy _1 du I dv y dx u dx V dx' This result is easily extended ; thus if y = uvw, we have, writing y = zw, where z = uv, 1 dy _1 dz 1 dw y dx z dx w dx _1 du 1 dv \ dw ,,. udx V dx w dx by a double application of the preceding result. And so on for any finite number of factors. If we multiply both sides by y, = uvw ..., the generalized form of the last result becomes dy du dv dw ,„, -^ =vw...-r +uw...^r + '^i't> ■■■ -i- + (0), dx dx dx dx or, in words : The derived function of a product is found by differenti- ating with respect to x, so far as it is involved in each factor 76 INFINITESIMAL CALCULUS. [CH. II separately, the other factors being treated as constants, and adding the results. Ex.1. If y=u.u.u to m factors = M™ (7), we have 1 dy I du I du ^ ^ m du - -S-= - ----{ =- + ... to m terms = — 5- , y dx u dx u ax u ax whence S = '"^"'''£ ^^> A general proof of tliis result, free from the restriction that m is a positive integer, is given in Art. 39. Ex.2. If y = sina;cosa; (9), d/u d , . . . d , , we have -f-= cos x -^ (sin x) + sin a; -=- (cos x) dx dx^ ' dx^ ' = cos X . cos X — sin x . sin x = cos' a; — sin^ a; = cos 2a; (10)- Ex.3. If y = a-'smx (11), we have 7i~'''^T' ^^"^ ^^ + sin a; -5- (x') ^oa'cos x+ 2a3sina! (12). 38. Differentiation of a Quotient. Let 2/ = | (1), where u, v are given functions of x. As in Art. 13, we find 5, _u + Su u _ vSu — vSv ^ v + Bv V v(v + Sv)' Su Sv , By Bx Bx whence s- = . , ., , . ox v{v + ov) du dv Hence, in the limit, 3^= (2). In words: To find the derived function of a quotient, from the product of the denominator into the derived function of 37-38] DERIVED FUNCTIONS. 77 the numerator subtract the product of the numerator into the derived function of the denominator, and divide the result by the square of the denominator. ^=1'S5 (^)' we have t- = 1 + 2x, -,- = — 1 + 2x, ax ax whence ^ = (1 + 2^) -«^ + «=') + (1 - 2^) (1 + «^ + «^') dm (1 - cc + x'f = -1(1^^ {\-x + xy ^ >■ The particular case 3/ = ^ (5) is worthy of separate notice. We then have g ^_1 1 g^ ^ v + hv v~ v(v + E:v)' Bv Sy _ Sx Sx v(v + Sv) ' <^ ^ _ 1 ^ (6) dx v' dx ^ '' This might of course have been deduced by putting u = l, du/dx = in the general formula (2). -^»=-2. If 2/ = J, (7), where m is a positive integer, we have du 1 „_, du _^ ,du i=-^---"'SS=— "'^ (8); see Art. 39. The formula of this Art. might also be deduced from the result of Art. 37, 2°. If y = ujv, we have u = vy, and there- fore 1 du _1 dv 1 dy udx V dx y dx ^ ^' 78 INFINITESIMAL CALCULUS. [CH. II 1 dy _1 du 1 dv ,, „v . y dx udx V dx this is equivalent to (2) above. The following examples are important : sin a? ,, ,, 1". If v = tana;= (11), •^ cos X d , . . . d . . cos a; -1- (sin a;) — sm x -j- (cos x) dy dx^ ' dx^ we find J- = „„„, dx cos" X cos'' a; + sin'' a! ,,„ = =sec''a! (12). cos'' a; ^ ^ This agrees with Art. 35, 4°. Similarly, if y = cotx (13), we find -^ = — cosec" X (14). dx 2°. If « = seca; = (15), " COSiB , dy 1 d . , sin a; we have j^ = — -j- (cos x) = — r— (lb). dx cos" X dx^ ' cos" x Similarly, if y = coseca; (17), e. A ^y cos a; , we find -Y- = ^-r (1°)- dx &iv?x If, as explained in Art. 31, we employ the symbol D to denote the operation of differentiating with respect to x, the results of Arts. 36 — 38 may be summed up as follows : D{u + v) = Du+Bv (19), I>{uv) = vI)u + uBv (20), vBu — uBv 7i ^^^^ n9=' V' 38] * DERIVED FUNCTrONS. 79 EXAMPLES. VL Verify the following differentiations : 1. y = x{\-x), I)y=\-1x. 2. y=.x{\-xf, % = (I-x)(l-34 3. y = ^{\~xf, Dy = x^--->-(\-xf-'-{m.-lm+n)x\. 4. 2/ = (a;-l)(a;-2)(a;-3), Dy = Zo^-\1x+\\. 5. y = x(\-x)'.{\+x)\ i)2/ = (l-a;)(l+a;)^(l + 3a;)(l-24 6. 2/ = (l+a:2)(l-2a;2), Dy = - Ix (\ ^ i,x\ 7. .=(.H.iy. i),=2(..i)(i-i^). a; (1 1 -xf (1 + xy 4a; (1 -xy 8- 2^=1 — . % = in 1+a^ 10- 2/=T3^.. % = 12. y = a; sin x. By = sin a; + a; cos x. 13. y = gs'cosx, Z>2/ = 2a! cos a; - a;^ sin a;. 14. y = sin' a;, cos x, Dy = 2 sin a; - 3 sin' x. , _ sin X r^ X cos x — sin a; 15. y = ^^, Dy = ^ . , „ a; „ sin a; — a; cos x 16. v = —. , Dv = . " sma;' ^ sm^a; , _ tan X „ a; — sin x cos a; "■ 2' = ^- ^y= ..eos'a. • 18. y = tan^ a;, .^2/ = 2 tan a; sec^ a;. 19. y = sec" x, Dy = 2 tan x sec" a;. „„ sin X „ cos' X - sin' a; 20. y = -= , Dy = 1 + tan X " (sin x + cos a;)" 80 INFINITESIMAL CALCULUS. [CH. II „, 1 + sin cc _ 2 cos X 21- y^T — TT^rz' ^y 1— sina;' (1— sina;) 1 — cos X _ 2 sin x 1+cosa;' (l+cosa;)^' 23. y = xe°, Dy = {x+\)e'. 24. y = aw"" Dy = ( 1 - x) e'". 25. 2/ = a!'"e", Dy = (x + m) af'-^e'. 26. y = e°' sin a;, Z)y = e"" (sin a; + cos x). 27. y = e"' cos X, By = e° (cos x — sin a;). 28. 2/ = ^. %=7 = + 1' ''~{^+\f 39. Differentiation of a Function of a Function. If y = F(u) (1), where u=f{x) (2), the symbols F, f denoting given functions, then l=|-S=^'<«);/» W For, if Zx, By, Su be simultaneous increments, we have Sy _ By Bu Bx Su,' Bx' identically ; and therefore, since the limit of a product is the product of the limits, dy dy du dx du ' dx ' A useful application of the formula (3) occurs in the theory of rectilinear motion. Thus if, as in Art. 33, we denote by v and a the velocity and the acceleration, respectively, of a moving point, we have ds dv , , . " = 16' ''=di W- Hence if v be regarded as a function of the space described (s), we have dv ds du ds dt a = — -^ = v^ (.5). 39] DERIVED FUNCTIONS. 81 Similarly, in the case of a rigid body rotating about an axis, the angular acceleration, when the angular velocity is regarded as a function of 0, will be given by dia dO dill ,„. Tedt'°''''d-6 (^)- The following deductions from (3) are important : P. If y = F{x->ra) (7), then, putting u = x+ a, dujdx = 1, the formula (2) gives | = ^'(- + '^) (8)- The geometrical meaning of this is that shifting a curve bodily parallel to the axis of x does not alter the gradient. 2° If y = F{lcx) (9), we have, putting u = kx, du/dx = k, g = 'fcJ^'(^*) (10). 3°. If 2/ = «™ (11), •where m is any rational quantity, we have F (u) = M™, F' (u) = mu^-\ and therefore- -^ = mu''^^ ~ (12). In particular, in the cases m = ^, vi = — ^, we find (13), d . _ 1 du dx 2f/u dx ' d 1 _ 1 du dx^/u 2m* dx , respectively. We add a few examples on the above rules. £!x. 1. If 2/ = sin'"a; (14), that is, y = 14™, where u = sin x, we have By = mu™-^ ■-r- = msin^-'x cos x (15). L. 82 INFINITESIMAL CALCULUS. [CH. II Ex.% If y=J{a''-tc') (16), we have Dy = D {a' - x^ = H"" " '"T* • -0 (a' - x') (17). V(«'-a^)' X ^-3- ^' y=:jw^) ('')• If we put u = x, v= ^(a' — a?), — X we have Du = 1, and Dv = .. ^ — -j, , as in the preceding Ex. The rule for differentiating a fraction then gives a? = "'' , (19). EXAMPLES. Vn. 1. 2/=(a! + a)'"(a! + 6)", i)y = (a; + a)™"^ (jc + 6)"-' {(m + n)x + mb + na}. rytU — i. ^"(TTl^' '^^""(l+i«)''+i" 3. 2^ = a; 7(1+4 -^^ = 2 ^(1 +«=) ' 4. ,=V{(. + i)(.+2)}, i^y= ,^^(,yi")^,,2)} - 1 — 3a; 5. y = {l + x)^{l-x), ^y = 2^{i-xy 6. ,, = (l-a;)V(l+«=»), ^y = - ^^J(ity - 8£ 89] DERIVED FUNCTIONS. 83 (l-a,-)^(l-a;=)' X 14. ?/ = sin 2 (x — a), Dy = 2 cos 2 (a; - a). 15. y = sin^2a;, iJy = 2 sin 4ji;. 16. y = ^(1 + sin a;), Dy = ^J{l-smx), 1-7 i 2 n 2 sin a; 17. y = tan-" x, Dy = cos" a; 2 sin X cos'o! 3 sin^a; 18. y = sec^ x, Dy = 19. y = tan' x, Dy = " ' " COS' a; 20. y =i= sin' x + cos' x, Dy = 3 sin a: cos x (sin x — cos a;). 21. y = smx — ^sin"x, Dy = cos^x. 22. 2/ = tan a; + |^tan' x, Dy = sec* a;. „„ sin" X 2 sin a; . , 23. y = — J- , Dy = 5 — {x cos x — sm x). X X 24. y = ^, Dy = 2xe^. 25. , = ^, ^2/ = «"(2-^)- 26. y = e"" ^ Dy = cos a; e™"'. 27. y = e^''^, Dy = sin 2x6'^'''. 28. 2/ = 6°^ sin /JaJ) -Oy = e™ (a sin /Sa; + /3 cos ^.■r). 29. 2/ = e" cos ;3a;, i)y = ««« (a cos ^x — ^ sin /3a;). 6—2 84 INFINITESIMAL CALCULUS. [CH. II 30. y = sin mx sia nx, Dy = n sin mx cos nx + m, cos wm; sin nx. 31. y = J(asiii'x+Bcos'x), Dy = \U-S) „ . T° ^"L ;r^. * ^^ ^ /' i' ^V ^''^(asin^ai + yScos'ia;) 40. Differentiation of the Hyperbolic Functions. 1°. If 2/ = sinha; (1), we have ^ = i) { ^^ ~^~" \ = ^ (£1^^ - Be''') = \{^ -{■e~'') = cosh.x (2). Similarly, if y = cosh x (3), we find -5^ = sinha; (4). 2°. If 2/=tanha' (5), we have dy _ j^ sinh x _ cosh x D sinh x — sinh x D cosh x dx cosh X cosh'' a; cosh'' a; — sinh^ a; ,„ = ^^¥— =^^''^^ (^)' by Art. 19 (4). Similarly, if 2/ = cotha; (7), we find T^ = — cosech^ a; (8). 3°. If y=:sechx .(9), we have, by Art. 38 (6), — D cosh X cosh'' X _ . sinh a; cosh" a; ^ ^' Similarly, if y=cosecha! (11), n J dy cosh a; .,„. we find -^= -.-^nr- (12). l = ^(elL)- dx sinh^ X 40-41] DERIVED FUNCTIONS. 85 EXAMPLES. VIII. Verify the following diflferentiations : 1. y = sinh" x, Dy = sinh 2a;. 2. y = cosh^ X, Dy = sinh 2a!. o 4. 1,2 n 2 sinh a; 3. w = tanh''aj, Dy= rs — . " ' ^ cosh' X 4. 2/ = sinh a; + ^ sinh' a;, i)^/ =-= cosh' x. 5. 2/ = tanh x — |^tanh' x, Dy = sech* x. 6. 2/ = cosh X cos X iJy = 2 sinh a; cos x. + sinh a; sin x, 7. y = cosh a; sin x Dy = 2 cosh x cos a;. + sinh X cos x, cosh X - cos a; _ 2 sin x sinh a; 8. « = ^r-T ^ , Dy = 7-;—; : r; . ° smh X + sm a; (sinh a; + sin a;)" 41. Differentiation of Inverse Functions. If 2/ be a continuous function of .t, then tinder a certain condition (see Art. 20), which is fulfilled in the case of most ordinary mathematical functions, x will be a continuous function of y. If Sa;, %y be corresponding increments of x and y, we have Sy Sa;_ Bx' Sy ' identically. Hence, since the limit of the product is equal to the product of the limits, dydx^^ dx'dy Hence, it being presupposed that yisa. differentiable function of X, it follows that x is in general a differentiable function of y, and that the two we find -p = tana! (12). Ex. 2. If j/ = logtan|a! (13), we have -r- = I T- • -O tan Ix - =— . sec^ Aa; . A ax tanja; '' tan^a; ^ ^ = ^ (14). sin a; ^ ' Similarly, if y = log tan (^tt + ^a;) (15), we should find ^= (16). ax cos a; ^ ' ^a;. 3. Let y = Jlogi±| (17), = |log(l+a;)-Jlog(l-a;). _ ,1^1 -a;"^*l-a! l-ar" l^a;. 4. Let y = ^og{x+ J{a? +\)} (19). ^-- I=*i4^-*i4-. = r^ (!«)■ 42-43] DERIVED FUNCTIONS. 91 Wehave f^= 1^^. D {x + J{a? ±1)} = x+J{a?+\) V + TP+l)/ (20). 1 43. Logarithmic Differentiation. In the case of a function consisting of a number of factors it is sometimes convenient to take the logarithm before differentiating. Thus if 11 = (i), ■wehave log2/ = logt«i+logM2+logM3 + ... - log Di - log v^ -\0gV3- .. .(2), and therefore, by Art. 42, 3°, 1 dy _ 1 dui 1 du^ 1 dug y dx III dx ^2 dx u^ dx 1 dvi 1 dVi 1 dvs ,_. Vi dx Vi dx «3 dx This is a generalization of the results of Arts. 37, 38. we have log y = 4 log (a + a;) + J log (6 + a;) - J log (a - a;) - J log (b - x). TT !<% 1 f 1 1 1 11 Hence -:r- = i 1 + i — + + i — f y dx '' {a + x b + x a—x b — x) _ a h («5 + 6) {ah — xF) ~ ^^^ ^ W^^ " {a'-a?) {V-x") ' dy _ (a + 6) (oib — a!*) Sx " \a - x)i (b - x)i {a + x)i {b + x)i 95 92 INFINITESIMAL CALCULUS. [CH.-II EXAMPLES. IX. Verify the following differentiations : 1. y = x log X, Dy = 1 + log x. 2. y = af" log X, Dy = a;"-^ (1 + m log x). 3. y = log sin x, Dy = cot x. 4. y = log cos a;, Dy = — tun x. 5. y = log tan x, Dy=2 cosec 2a!. 6. 2/ = log sinh x, Dy = coth x. 7. y = log cosh X, Dy = tanh x. 8. y = log tanh x, Dy = 2 cosech 2a!. X 1 9- y=^°gT:rz' ^y- l+a:' ^ a!(l+a;)' 10- 2'=i°sttz' ^y=- l+a:' ^ l-ai=" a; 1 V(a;^+1)' ^~a!(a;«+l)- TO , \-^- ^X 1 \-^x' " {l-x)Jx- 13. 2/ = log (7(^ + 1) ^^^_ 1 + V(a;-1)}, ^ 2V(a:=-l)- V(a;^+l)-a;' ^^ " a: " V(ar> + 1) " 15. 2, = log(a!-l)-^^^,, % = (^3yy3. l+a; + a:= 2(l-a;«) 17. y = sin-i(l-a;), Dy= ^ ^{2x-xy 18. 2/ = a; sin~^ x, Dy = sin~' x + X V(i -«=')■ 44] DERIVED FUNCTIONS. 93 1 19. 2/ = cot ^x, iiy = — 20. y = sec"' x. By = l+a;^' 1 x^{^-\y 1 21. 2, = cosec-.. ^2' = -,;/(^_l^- 22. 2/ = sin-' a; Dy = Q. + sin-' V(l - s^), 23. y = *3.ii-' a; + tan-' - , Dy = Q. 24. y = sin-'{2a,V(l-a=% 2)y=-7^jf^^. 25. 2, = tan-'j^, ^2' = ra- 26. y=t^u-^{J{^+\)-x}, Dy = -^^~^y 27. y = sin"' (cos x), Dy = —\. A -3? „ 2 28. y = cos-'^^„ % = T^. 2^' 2'=x/(t^^)' ^^= (i+=«ya-^)^ - on V(l+a^)-a: „ - "^ V(l+a;^) + a;' " V(l + »=')( V(l+ar')+a;P • 31 ,,-V(l+^)jV(ir-) z>,/- i i 32. ,=41±51±41Z^) .,= ^ V(l+a^)-V(l-aiT ar'V(l-a^) a^' 44. Differentiation of the Inverse Hyperbolic Functions. P. If y = smh.-'x (1), we have a; = sinh y, -g- = cosh y = tj{a? + 1), 94 INFINITESIMAL CALCULUS. [CH. II and therefore | = v(rT^ ^^^- There is no ambiguity of sign, for cosh y is essentially positive. 2°. If y = cosh->a; (3), we have ai = cosh y, -j- = sinh y — ± >J{x^ — 1). ^^^■^"^ l=±V(^ ^*^- For any given value of x, greater than unity, there are two possible values of y, one positive, the other negative, and for these dyjdx has opposite signs. [Cf. Fig. 21, Art. 19, inter- changing X and y.] 3°. If y = i3iah-'x (5), we have so = tanh y, 3- = sech" y = l—a?, and therefore -^ = ^ -„ (6). ax \— x^ This agrees with Art. 42, Ex. 3. It is to be noticed that y is real only when a; < 1. See Fig. 22. Similarly, if y = coth^'a; (7), -fi'^'i S=-^i («>' X being necessarily > 1, if y is real. EXAMPLES. Z. Verify the following differentiations : 1. 2, = sech-a,, i)2, = __L_. 2. 2, = cosech-a,, By=-^^-j^^ 3. y = sin~^ (tanh x), By = sech x. 4. y = tan~^ (sinh x), By = sech x. 5. y = tan~' (tanh ^x), By — \ sech as. 6. y — tanh"' (tan \x), By = ^ sec x. 44-4!5] DERIVED FUNCTIONS. 95 EXAMPLES. XI. 1. What is the geometrical meaning of the theorem ^,l,{kx) = k'{kx)1 2. If 2/ = tan-'-, ,, , _ vDu-uDv prove that Vy = - — = = — . 3. Assuming that — j = 1 + w + a;^ + ... +a;'', deduce, by differentiation, the sum of the series l + 2a!+3:B2+ ...+na;»-', and test the result by putting a; = 1. Hence shew that, if |a;|{a;,y,...} (1). In particular, in the aforesaid case of a surface, if we denote the altitude PN by z, we have z = (F,y) (2). Let us now suppose that all the independent variables save one («) are kept constant. Then the function u may or 45-46] DERIVED FUNCTIONS. 97 may not be a differentiable function of x\ if it is differenti- able, its derived function with respect to a; is called the 'partial differential coefficient' or 'partial derivative' of it with respect to x, and is conveniently denoted by dujdx. Thus du ,. ix+hx,y,...)-^{x,y,...) g^-Iima.=o g^ ...(3). In like manner clu,. (x,y + By,...)-(x,y,...) g^-lims^=o ^ -W- In the case of the surface (2) it is plain that the partial derivatives dz dz dx' dy are the gradients of the sections {HK, LM, in the figure) of the surface by planes parallel to the planes ZOX, ZOY respectively. Ex.\. If « = ai'»y'' (5), we have — = ma:'"-'^", — = nx^y'^-'- (6). Ex. 2. Assuming that in a gas the pressure (jp), volume («), and temperature (&) are connected by the relation ^ = -^ ('). V V wehave ^=-^' sd = ^ (^>- 46. Implicit Functions. An equation of the type ^(«'.2/) = (1) in general determines y as a function of x ; for if we assign any arbitrary value to x, the resulting equation in y has in general one or more definite roots. These roots may be real or imaginary, but we shall only contemplate cases where, for values of x within a certain range, one value (at least) of y is L. 7 98 INFINITESIMAL CALCULUS. [CH. ll real. The term ' implicit ' is applied to functions determined in this mannein, by way of contrast with cases where y is given 'explicitly' in the form y=/(*) (2). If we regard z = 4>{x,y) (3) as the equation of a surface, then (1) is the equation of the section of this surface by the plane !S = 0. If the plane xy be regarded as horizontal, the sections a = C, where G may have different constant values, are the 'contour-lines.' If we require to differentiate an implicit function, we may seek, first, to solve the equation (1) with respect to y, so as to bring it into the form (2). It is useful, however, to have a rule to meet cases when this process would be inconvenient or impracticable. It will be sufiScient, for the present, to consider the case where («, y) is a rational integral function of x and y, i.e. it is the sum of a series of terms of the type J.m^„a;'"^/", where m, n may have the values 0, 1, 2, 3,.... Since, by hypothesis, (f) (x, y) is constantly null, its derived function with respect to a; will be zero. Now by Arts. 35, 37, 39, we have Hence, if <^(a;,2/) = S^m.Ti^^y" (4), we have 2^m,„mfl;"'-'^" + '%Am,nnx™'y^^j- = (5). In the notation of Art. 45, this may be written ?^ + ?^^ = (6) dx dy dx dy dx ,^, dx=~q • ^^>- dy It will be shewn in the next Chapter that the results (6) and (7) are not limited to the above special form of (x, y) ; but the present case is sufficient for most geometrical applications. 46] DERIVED FUNCTIONS. 99 EXAMPLES. XII. 1. Sketch the contour-lines of the surface and describe the general form of the surface. 2. Also of the surface az = xy. 3. If ^=f(x + y), ., . Sz Sz prove that 5- = 5" , dx dy and give the geometrical interpretation of this result. 4. If r:=J{x^ + f), ,, , 8r dr prove that r-— = x, r — = ?/. ^ Sa; ' By -^ 5. If « = atan"'-, X dz ay dz ax prove that ^---« 5. ^ -Zs a- ax or + y' By ar + y^ 6. If z=f(x' + f), ., . dz dz prove that .r- ■ ::^ = x : v. ^ dx dy " If z -A^- 4.1. i. dz dz . prove that x — + ij — = 0. ^ dx "dy 8. If aiie'+2hxy + by' + 2gx+2fy + c = 0, dy ax + hy + g prove that dx hx + by +f' 7—2 CHAPTER III. APPLICATIONS OF THE DERIVED FUNCTION. 47. Inferences from the sign of the Derived Function. li y = (as), and if Sx, hy be simultaneous increments of X and y, the limiting value of the ratio hyjhx when hx is indefinitely diminished is, by definition, 0' («). Hence, before the limit, we may write | = f(^) + <^ (1), where a- is an ultimately vanishing quantity. A numerical example of the manner in which the ratio 8j//8a! approximates to its limiting value may be of interest. We take the case of y^logy^x, for the neighbourhood of x = 1. The limiting value ia here ^ = ^=•43429.... ax X The numbers in the second column are taken from the printed tables. Sx Sy SyjSx ■1000 •041393 •41393 •0500 •021189 ■42379 •0100 •0043214 ■43214 0050 •0021661 ■43321 •0010 •00043408 ■43408 ■0005 •00021709 ■43419 •0001 •000043427 ■43427 47] APPLICATIONS OF THE DERIVED FUNCTION. 101 Let US first suppose that ' («) > 0. Since the limiting value of a- is zero, we can by taking Sx small enough ensure that 0. That is, by (1), 8y will have the same sign as Sec for all admissible values of Sx which are less in absolute value than a certain magnitude e. In the same way, if By will have the opposite sign to Sx for all admissible values of Sai which are less in absolute value than a certain quantity e. If the independent variable be represented geometrically as in Fig. 1, Art. 1, and if a; = OM, where if is a point within the range considered, we may say that if ^' (is) be positive there is a certain interval to the right of M for every point of which the value of the function '{x) be negative, the words 'greater' and 'less' must be interchanged in this statement. When M is at the beginning or end of the range of x, the intervals referred to lie of course to the right or left of M, respectively. It follows that if <^' (x) be positive over any finite range, the value of ^ (x) will steadily increase with x throughout the range ; i. e. if x^ and x^ be any two values of x belonging to the range, such that x^ > x^, then For (j){x}, being by hypothesis differentiable, and therefore continuous, must have (Art. 25) a greatest and a least value in the interval from x^ to x^ (inclusive). And the preceding argument shews that the greatest value cannot occur at the beginning of the interval, or in the interior ; it must there- fore occur at the end. Similarly the least value of ^ (x) must occur at the beginning of the interval. In the same way it appears that if (f>' (x) be negative 102 INFINITESIMAL CALCULUS. [CH. Ill over any finite range, then (j> (x) will steadily decrease as x increases, throughout this range; i.e. if Xi and a?2 be any two values of tv belonging to the range, such that x^ > x^, then ^ (a^a) < (a;i). The geometrical meaning of these results is obvious. When the gradient of a curve is positive the ordinates increase with x ; when the gradient is negative the ordinates decrease as x increases. The graphs of various functions given in Chapter I. will serve as illustrations. The converse statements that if <^ (x) steadily increases with X throughout any range, 4>'{x) cannot be negative for any value of x belonging to this range, and that, if ^(a;) steadily decreases as x increases, <^'{x) cannot be positive, follow immediately from the definition of <^' (x). Again, even if '(«) vanish at a finite number of isolated points, provided it be elsewhere uniformly positive, <^ («) will steadily increase. Suppose, for example, that <^'(«i) = 0, and that with this exception ^'{x) is positive in the interval from a; = «! to a; = x^, where x^ > x^. The least value of (x) cannot then occur within this interval, or at the upper extremity {x = x^. It must therefore occur at the lower extremity {x = Xi). Hence (a^a) > (aJi). The same conclusion is arrived at if '(x) = throughout the interval; in the latter case (t>'{xi) cannot be either positive or negative (Art. 47) and must therefore vanish, since it is by h3rpothesis finite. The geometrical statement of this theorem is that if a curve meets the axis of x at two points, and if the gradient is every- where finite, there must be at least one intervening point at which the tangent is parallel to the axis of x. See, for example, the graph of sin x on p. 34 ; also Fig. 13, p. 27. 104 INFINITKSIMAL CALCULUS. [CH. Ill Ex. 1. If ^ (ic) = (a; - a) (x - b), ■we have 4>' ("') = 2x-{a + b). Hence ' (x) must vanish for some intermediate value of x. This is in agreement with Art. 47, where it was shewn that the equation x = tan x has a root between ir and |7r. It is to be carefully noticed that, in the above demonstra- tion, the conditions that {a;) and <^'(^) should each have a definite (and therefore finite) value throughout the interval from x = a to x=b are essential. The annexed figures exhibit various cases when the conclusion does not hold, owing to the violation of one or other of these conditions. A slightly more general form of the theorem of this Art. is that if '(«) will vanish for some intermediate value of co. This follows by the same argument, applied now to the function ^ (x) — /8. 49. Application to the Theory of Equations. If '(x) = (1) 48-49] APPLICATIONS OF THE DERIVED FUNCTION. 105 will lie between any two real roots of «^(^) = (2). This result, which is known as 'RoUe's Theorem,' is important in the Theory of Equations. It is an immediate consequence that at most one real root of (2) lies between any two consecutive roots of (1). That is, the roots of (1) separate those of (2). JSx. 1. If <^ (x) = 4a!» - 2la? + ISa; + 20, we have <^' (x) = 120? - 42a! + 18 = 6 (2a; - 1) (a; - 3). Hence the real roots of <^ (x) = 0, if any, will lie in the intervals between — oo and J, ^ and 3, 3 and + oo , respectively. Now, for as = — 00 , J, 3, + 00 , the signs of (x) are — , +, — , +, respectively, so that ij> (x) must in fact vanish once (by Art. 10) in each of the above intervals. Hence there are three real roots. The figure shews the graph of (a!). Fig. 33. If by continuous modification of the form of ' (x) = which lies between must coalesce with them. Hence a double root of <^(a!) = is also a root of {x) = {x-arx{oo) (3). where % (x) is a rational integral function. Hence f {x) = {x- ay-^ [rx (x) + (x-a) x' (ix:)} (4) ; i.e. (x — ay~^ will be a common factor of <^(«) and ^'{x). And it- is easily seen that (x — a)'^^ will not be a common factor unless ^ {x) is divisible by (x — ay. Hence the multiple roots of ^ (x), if any, are to be detected by finding the common factors of («) and <}>' (x) by the usual alge- braical process. Ex.2. If (x) = oci^-9x' + ix + l2, we have ^' (x) = 403* — 1 8a; + 4. The usual method leads to the conclusion that x—2 is a common factor of <^ (a;) and <^' (x) ; whence we infer that (x — 2)' is a factor of {x) = {x-2f{x + l){x + 3). Ex. 3. To find the condition that the cubic ar' + jai + r = (5) should have a double root. The double root, if it exists, must satisfy 3ar'+g=0 or x = ±J{-lq) (6). Substituting in (5), we find r = ±^sl{-W} or T' = --4W (7), which is the required condition. 50. Maxima and Minima. A ' maximum' value of a continuous function is one which is greater, and a ' minimum' value is one which is less than the values in the immediate neighbourhood, on either side. More precisely, the function ^(a;) is a maximum for 49-50] APPLICATIONS OF THE DERIVED FUNCTION. 107 a! = 0Ci,ii two positive quantities, e and e', can be found such that (j) {(Ci) is greater than the value which {x) has a determinate and finite derivative at all points of the range considered. The argument of Art. 47 then shews that if <^{Xi) be a maximum or minimum, ^'(«i) cannot dififer from zero. For if it be either positive or negative, there will be points in the immediate neighbourhood of x^ for which <^ {x) will be greater, and others for which it will be less, than ^{xi). Hence, in the case supposed, a first condition for a maximum or minimum value of ^(«) is that ^'{x) should vanish. This condition is necessary, but it is not sufficient. To investigate the matter further, we will suppose that on each side of the point x^ there is a certain interval throughout which <})'(x) is altogether positive or altogether negative* Now if (}>' (x) be positive for all values of x between x^ — e and Xi, ^ (x) will (Art. 47) steadily increase throughout the interval thus defined ; and if <}>' (x) be negative for all values of X between Xi and x^ + e', <^ (x) will steadily decrease throughout the corresponding interval. Hence if both these conditions hold, (xi) cannot be the greatest value which the function assumes within the interval extending from Xi — e to Xi + e. We may express this shortly by saying that the necessary and sufficient condition in order that {a!i) may be a minimum value of '{x) should change sign from — to + as a; increases through the value Xi. In geometrical language, when the ordinate of a curve is a maximum the gradient must change from positive to negative; when the ordinate is a minimum the gradient must change from negative to positive. This is abundantly illustrated in our diagrams; see, for example, Figs. 13, 17, 18, 33. jEx. 1. The distance (s), from an arbitrary origin, of a point moving in a straight line is a maximum when the velocity (dsjdt) changes from positive to negative, and is a minimum when the velocity changes from negative to positive. Thus, in the case of a particle moving upwards under gravity, we have 1 o ds s = ut-lgt\ -j^ = u-gt. Hence dsjdt changes from positive to negative as t increases through the value ujg. The altitude (s) is therefore then a maximum. Ex. 2. To find the rectangle of greatest area having a given perimeter. Denoting the perimeter by 2fls, the lengths of two adjacent sides may be taken to be x and a — x; hence we have to find the maximum value of the function x{a-x) (1). The derivative of this is a — 2x, which changes sign from + to — as X increases through the value \a. The rectangle of greatest area is therefore a square. The graph of the function (1) has been given in Fig. 13, p. 27. Ex. 3. To find the maxima and minima of the function ^{x) ^ ia? - 2\3? + \9,x + 10 (2). We have ^' (oc) = 1 2 (a; - J) (a; - 3) (3). This can only change sign when x passes througli the values ^ and 3. Now when a; is a little less than \, the signs of the second and third factors are -, — ; whilst when a; is a little greater than J they are +, -, Hence as x increases through the value \, 0' (a;) changes sign from + to — . In a similar manner we find that as x increases through the value 3, ' {x) changes 50] APPLICATIONS OF THE DERIVED FUNCTION. 109 sign from - to +. Hence <^ (x) is a maximum when x-^, and a minimum when x=3. If we substitute in (2) we find that the maximum value is 24i, and the minimum value - 7. See Fig. 33, p. 105. '2'r Ex.L If '^(«') = r^ W' we find ^'(«,) = M1^)... (5). This can only change sign for a;=±l. As a; increases (alge- braically) through the value — 1, 1 — aj'' changes sign from - to +. As flj increases through +1, 1 - a;^ changes sign from + to — . Hence for a; = — 1 we have a minimum value - 1 of (a;), and for as = 1 a maximum value 1. See Fig. 17, p. 31. If Ex.6. If '^H = i3^» (6). we have ,^' (^) = ^11^ (7). Here ^' (x) is always positive, and the function ^ (x) has no finite maxima or minima. See Fig. 16, p. 31. Ex. 6. To find the right circular cylinder of least surface for a given volume. If X denote the radius and y the altitude, the surface is 27ra!' + 'iirxy, and if the given volume be iira?, we have a?y = 2a». Hence, eliminating y, the expression to be made a minimum is 0?+ — , X the derived function of which is This changes sign as x increases through the value a, and the change is from — to + . Hence x = a makes the surface a mini- mum ; and since y then = 2a, the height of the cylinder is equal to its diameter. The reader may verify that with these proportions the surface is l"1447...of that of a sphere of equal volume. 110 INFINITESIMAL CALCULUS. [CH. Ill Whenever the derived function {x)bo!? (8). This makes ' (x) = 3a?, which vanish s, but docs not change sign, as X increases through the value 0. Hence <^ (x), though ' stationary,' is not a maximum or minimum for x = 0. Fig. 34 shews the graph of the function af. In most cases of interest, the derived function ^'(x) is 50-52] APPLICATIONS OF THE DERIVED FUNCTION. Ill continuous as well as determinate (and Unite). It can then only change sign by passing through the value zero ; and it is further evident from Art. 10 that the changes (if there are more than one) will take place from + to — , and from — to +, alternately. The maxima and minima will therefore occur alternately. See Fig. 18, p. 34. 51. Exceptional Cases. It will, however, occasionally happen that ^' («), though generally continuous, becomes discontinuous for some isolated value of X ; and if the discontinuity be accompanied by a change of sign as x increases through the value in question, we shall have a maximum or minimum, by the same argument as in Art. 50. Ex. If we have -a^x^ •(1). .(2). 4.{x)-. As X increases through the value 0, this changes from - co to + 00 . Hence <^ (a;) is a minimum for x = 0. See Fig. 35. Y Again, in Fig. 32 there occurs a point where <^' (x) is dis- continuous, passing abruptly from a finite positive to a finite negative value. The ordinate is then a maximum. 52. Algebraical Methods. It is to be noticed that many important problems of maxima and minima can be solved by elementary algebraical methods, without recourse to the Calculus. This is especially the case with questions involving quadratic expressions. These are all easily treated by the method of 'completing the square.' 112 INFINITESIMAL CALCULUS. [CH. Ill Ex. 1. Thus, in the problem of Ex. 2, Art. 50, we have x(a — x) = \a' — {x — |a)'. Since the last term cannot fall below zero, this expression has its greatest value {^a^) when x = ^a. Ex. 2. The expression 2a^ — 3a: + 2, may be put in the form 2(ar'-|a;+l) = 2(a;-f)» + |. Hence the expression has the minimum value |-, corresponding to a.- = |. Again, the solution of many important problems comes at once from identities such as ayy = \{(,x + yy-(x-yy] (1), (a!-\-yf = (ai-yf + ^ (2), fl^ + 2/^ = i{(a; + 2/)» + (a;-y)»} (3). Thus: The product (xy) of two positive magnitudes, whose sum (« + y) is given, is greatest when they are equal ; The sum of two positive magnitudes whose product is given is least when they are equal; The sum of the squares of two magnitudes whose sum is given is least when they are equal. Ex. 3. To find the greatest rectangle which can be inscribed in a given circle. If 2cB, 2y be the sides, we have to make xy a maximum subject to the condition that a? + y^= a\ where a is the radius of the circle. Now 'ixy = a? + y^-{x-yf^a^~{x-yf (4), which is obviously greatest when x = y. Hence the greatest inscribed rectangle is a square. Ex. 4. To find the minimum value of acotO + hta,uO (5), for values of 6 between and ^tt. The product of a cot 9 and 6 tan is constant, hence their sum is least when they are equal, i.e. when tane = (o/6)i (6). The minimum value of the sum is therefore 2ai6i. 52] APPLICATIONS OF THE DERIVED FUNCTION. 113 Ex. 5. To find the greatest cylinder which can be inscribed in a frustum of a paraboloid of revolution cut off by a plane perpendicular to the axis. Supposing the paraboloid to bo generated by the revolution of the curve y' = '>^ax (7), about the axis of x, then if h be the length of the axis, and x the abscissa of the end of the cylinder nearest the origin, the volume of the cylinder is .y-Q,-x) = ^f{h~^^^ (8). Now the sum of the quantities y^ and ^ah — y^ is constant ; their product is therefore greatest when they are equal, i.e. when y'=2aA, or x = \h (9). The height of the cylinder is therefore one-half that of the frustum. EXAMPLES. XIII. 1. Verify the theorem of Art. 48 in the following cases ; (1) i,(x) = {x-ar{x-h)\ (2) ^{x)=\ogp^, ^ ' ^^ ' "(a + 6) a;' (3) 0(a,) = (^-^y-^). 2. Prove that the curves 2/ = £K<-6a:' + 9a!= + 4a!-12, and J/ = a;* - a;' - Sa;" + OK - 2, touch the axis of x, and find where they cut it. Trace the curves. 3. Prove that when x increases through a root of ^ (x) = 0, <^ ix) and <^' (tc) will have opposite signs just before, and the same sign just after the passage. Does this hold in the case of a double root? 4. If, for a > a; > 0, («!) = a, L. 8 114 INFINITESIMAL CALCULUS. [CH. Ill a' and, for x>a, (a;) = a 5 , whilst for a; = a, <^ {x) = 0, prove tliat (x) and 1^' (x) are continuous from a; — to a; = 00 . Trace the curve y = (x). 5. Prove that the expression (a:- 1)6"+!, is positive for all positive values of oe. 6. Prove that in the rectilinear motion of a point, the velocity is a maximum or_,a minimum when the acceleration changes sign. Illustrate this from the simple-harmonic motion s= a cos nt. 7. Find the maxima or minima of the function x*-8si?+ 22a^ - 24a; + 12. 8. Prove that the function 2a'-3a!»-36a; + 10 is a maximum when x=—2, aiid a minimum when x = 3, 9. The function 4k= - 18a? + 27a! - 7 has no maxima or minima. 10. Find the stationary points of the function a;' — 5a!* + Sa;* + 1, and examine for which of them the function is a maximum or minimum. 11. Prove that the function 10a!«- 12a!' + 15a!* - 20a;' + 20 has a minimum value when a;=l, and no other maxima or minima. 12. Prove that the function a!^(oa!-a!') is a maximum when x = fa. 13. The function -. r— ; r (a + x){b + X) is a maximum when a!= J{ab), and a minimum when x = — ij(ab). 52] APPLICATIONS OF THE DERIVED FUNCTION. 115 14. Prove that the function {x + lf has a maximum value -^j, and a minimum value 0. 15. Prove that the expression 1+x + x^ 1 - x + x' has a maximum value 3, and a minimum value ^. 16. The function !(gr'+l) x^-nP+l has a maximum value 2, and a minimum value — 2. 17. The function '"J'^^'Z^l has two maxima, each = J, and two minima, each = — J . 18. Prove that cos 6 + sin 6 is a maximum when 6 = ^ir. 1 9. Prove that sin {6 - a) cos (6 - /8) is a maximum or a minimum when according as n is even or odd. 20. Find the maximum ordinate of the curve y = xe'". Trace the curve. 21. The curve y = xlogx has a minimum ordinate — '3618..., Trace the curve. 22. Prove that the ratio of the logarithm of a number {x) to ;he number itself is greatest when x = e. 23. Prove that the expression a cos 6 + b sin 6 has the maximum and minimum values + Jla' + b''). 8—2 116 INFINITESIMAL CALCULUS. [CH. Ill 24. Prove that it a>b the expression a cosh x + b sinh x has the minimum value J{a' — b^), but that if a < 6 it has neither a maximum nor a minimum. 25. Prove that the function cosh X + cos X has a minimum value when x = 0, but no other maxima or minima. 26. Prove that the function cosh X cos X has a maximum value when ce = 0, a minimum value when x-j^n (nearly), and a series of alternate maxima and minima corre- sponding to x = mr + \ir, approximately, where n=\, 2, 3.... 27. Prove that the function mi(x — Xjf + m.2{x-X2y+ ... +m„(a!-a;„)' is a minimum when miSCi + mjCCj + ... + m„x„ TOi + TOa + . . . + ?»„ 28. The velocity of waves of length \ on deep water is proportional to \/(^9' where o is a certain linear magnitude; prove that the velocity is a minimum when k = a. 29. The inclination of a pendulum to the vertical, when the resistance of the air is taken into account, is given by the formula = ae-** cos (nt + e) ; prove that the greatest elongations occur at equal intervals ir/n of time, and that they form a series diminishing in geometrical progression. 30. The, force exerted by a circular electric current of radius a on a small magnet whose axis coincides with the axis of the circle, varies as X where x is the distance of the magnet from the plane of the circle. Hence prove that the force is a maximum when x = |a. 62] APPLICATIONS OF THE DERIVED FUNCTION. 117 31. Find the straight line of quickest descent from a, given point to a given vertical straight line, assuming that the time of sliding a distance s from rest at an inclination 6 to the hori- zontal is \g sm OJ [The minimum time is 2 (ajg)^, corresponding to 6 = \ir, where a is the horizontal distance of the point from the given straight line.] EXAMPLES. XIV. 1. The rectangle of least perimeter for a given area is a square. 2. The rectangle of given perimeter which has the shortest diagonal is a square. 3. The greatest rectangle which can be inscribed in a given triangle has one-half the area of the triangle. 4. A rectangle is inscribed in a right-angled triangle, so as to have one angle coincident with the right angle ; prove that its area is a maximum when the opposite corner bisects the hypo- thenuse. Shew also that under the same circumstances the perimeter of the rectangle has neither a maximum nor a minimum value. 5. Find the rectangle of greatest or least perimeter which can be inscribed in a given circle. 6. If through a given point A within a circle a chord PAQ be drawn, the sum of the squares of the segments PA, AQ is least when the chord is perpendicular to the diameter through A, and greatest when the chord coincides with the diameter. 7. Given a fixed straight line, and two fixed points A, B outside it, it is required to find a point P in the straight line such that AP'^ + PB^ shall be a minimum. 8. Find the square of least area which can be inscribed in a given square ; and the square of greatest area which can be circumscribed to a given square. 118 INFINITESIMAL CALCULUS. [CH. Ill 9. A straight line drawn through a point (a, h) meets the (rectangular) coordinate axes in P and Q, respectively; prove that the minimum value of OP + OQ is a + 2J{ab)+b. 10. A straight line is drawn through a fixed point (a, h) ; prove that the minimum length intercepted between the co- ordinate axes (supposed rectangular) is 11. A rectangular sheet of metal has four equal square portions removed at the corners, and the sides are then turned up so as to form an open rectangular box. Shew that when the volume contained in the box is a maximum, the depth will be iia + b-^ia'-ab + b')}, where a, b are the sides of the original rectangle. 12. At what distance from the wall of a house must a man whose eye is 5 feet from the ground station himself in order that a window 5 feet high, whose sill is 20 feet from the ground, may subtend the greatest vertical angle 1 13. It is required to cut from a cylindrical tree-trunk a beam of rectangular section of maximum flexural rigidity ; prove that the breadth of the section must be ^ the diameter, and its depth -866 of the diameter. (Assume that the flexural rigidity varies as the breadth and as the cube of the depth.) 14. A straight road runs along the edge of a common, and a person on the common at a distance of one mile from the nearest point (A) of the road wishes to go to a distant place on the road in the least time possible. If his rates of walking on the com- mon and on the road be 4 and 5 miles an hour, respectively, shew that he must strike the road at a point distant 1-| miles from A. 15. Find at what height on the wall of a room a source of light must be placed in order to produce the greatest brightness at a point on the floor at a given distance a from the wall. (Assume that the brightness of a surface varies inversely as the square of the distance from the source, and directly as the cosine of the angle which the rays make with the normal to the surface.) 52] APPLICATIONS OF THE DERIVED FUNCTION. 119 16. Two particles P, Q describe fixed straight lines inter- secting in 0, with constant velocities u, v. Prove that if A, B be simultaneous positions of the particles, and if OA = a, OB = b, L A OB = 111, the distance PQ will be least after a time au +bv — (av + hu) cos w II? — 2md cos ta-i v' ' and that the least distance will be {av ~ hu) sin w ^{u^ - luv cos (0 + v^) 17. Prove that the greatest rectangle which can be inscribed in a segment of a parabola bounded by a chord perpendicular to the axis has a length equal to -I that of the segment. 18. The greatest rectangle which can be inscribed in a given ellipse has its diagonals along the equi-conjugate diameters. 19. If the length of a tangent to an ellipse intercepted 'between the axes be a minimum, the tangent is divided at the point of contact into two portions equal to the semi-axes of the ellipse, respectively. 20. If a tangent to an ellipse includes with the principal axes (produced) a triangle of minimum area, it is parallel to one of the equi-conjugate diameters. 21. A circular sector has a given perimeter ; prove that the area is a maximum when the angle of the sector is 2 radians, and that the area is then equal to the square on the radius. 22. If a triangle have a given base, and if the sum of the other two sides be given, the area is greatest wlien these two sides are equal. 23. A quadrilateral has its four sides of given lengths, in a given order; prove that its area is greatest when it can be inscribed in a circle. 24. If the power required to propel a steamer through the water vary as the cube of the speed, the most economical rate of steaming against a current will be at a speed equal to IJ times that of the current. 120 INFINITESIMAL CALCULUS. [OH. Ill EXAMPLES. XV. [The following results may be assumed : (1) The curved surface of a right circular cylinder of height h and radius a is iirah; (2) The volume of the same cylinder is •ira^h ; (3) The curved surface of a right circular cone of height /*, base-radius a, and slant side I is nral; (4) The volume of the same cone is \Tra^h ; (5) The surface of a sphere of radius a is 47ra° ; (6) The volume of the same sphere is |^ira°.] 1. The cylinder of greatest volume which can be inscribed in a given sphere has a volume equal to '5773 of that of the sphere. 2. The cylinder of greatest superficial area which can be inscribed in a given sphere has a surface equal to "8090 of that of the sphere. 3. The cylinder of greatest volume for a given superficial area has its height equal to the diameter of the base, and its volume is '8165 of that of a sphere having the given superficial area. 4. Find the cylinder of greatest volume for a given surface ; and find the ratio of its volume to that of the sphere of equal surface. [-81 65.] 5. Find the proportions of a thin open cylindrical vessel in order that the amount of material required may be the least possible for a given volume. [The height must equal the radius of the base.] 6. A cylinder is inscribed in a right circular cone; prove that its volume is a maximum when its altitude is ^ that of the cone, and that its volume is then ^ that of the cone. 7. If a cylinder be inscribed in a right circular cone the curved surface is a maximum when the altitude of the cylinder is J that of the cone. Shew also that the total surface of the cylinder cannot have a maximum value if the semi-angle of the cone exceeds 26° 34' [=tan-'i]. 53] APPLICATIONS OF THE DERIVED FUNCTION. 121 8. The cone of greatest volume which can be inscribed in a given sphere has an altitude equal to § the diameter of the sphere. Prove, also, that the curved surface of the cone is a maximum for the same value of the altitude. 9. If a right circular cone be circumscribed to a given sphere, its volume will be a minimum when the altitude is double the diameter of the sphere. Shew also that the semi-vertical angle will be 19° 28' [= sin"' |]. 10. The right circular cone of greatest surface for a given volume has an altitude equal to ^2 times the diameter of the base. 11. From a given circular sheet of metal it is required to cut out a sector so that the remainder can be formed into a conical vessel of maximum capacity ; prove that the angle of the sector removed must be about 66°. 53. Geometrical Applications of the Derived Function. Cartesian Coordinates. We have seen (Art. 32) that if i|r denotes the angle which the tangent, drawn to the right, at any point of the curve y = H«=) ■ (1). makes with the positive direction of the axis of x, then dy _ dx = tan -^Jr . .(2). With the help of this formula, several magnitudes connected with a curve may be expressed in terms of x, y, and dyjdx. Y Fig. 36. 122 INFINITESIMAL CALCULUS. [CH. Ill If the tangent and the normal at the point P meet the axis of a; in r and G, respectively, and if M be the foot of the ordinate, then TM is called the 'subtangent' and MG the 'subnormal.' Hence we find subtangent = TM = MP cot ^Jr = y ' dx (3), subnormal = MG = MP tan ^ = y ^ (4), tangent = TP = MP cosec •^ normal = PG = MP sec ■>jr = y {t), y = xit) (1) giving the coordinates in terms of a subsidiary variable t. For example, in Dynamics, the coordinates of a moving particle may be given as functions of the time. If we take any convenient series of values of t, we can calculate the corresponding values of *• and y, and so plot out as many points as we please on the curve. If Sx, Sy, Bt be simultaneous increments of x, y, t, we have Sy Sy Sx 53-55] APPLICATIONS OF THE DERIVED FUNCTION. 125 and therefore in the limit, when St is indefinitely diminished, "°*=I=I4: «• Ex. 1. In the ellipse a; = acos<^, ^ = 6sinc^ .'(3), we have tan it = ~ I -=- = — cot a Ex. 2. In the case of a. projectile moving under gravity, we have x = a + uj, y = b + v„t-yf (5), , , , dy Idx v — qt whence tan \li=~ l^r= — ■ ^ dt \ dt M„ 55. Polar Coordinates. Let P, P' be two neighbouring points on a curve, and let r, 6 be the polar coordinates of P, and r + Sr, ^ + 8^ those of P. If we join PP", and draw FN perpendicular to OP", we have PN= OP sin PON= r sin h0, P'N= OP'-0]Sf = r + Sr-rcos 8(9 = Sr + r-(l - cos Bd). Fig. 38. When SO is indefinitely diminished, the ratio of sin 8^ to 86 tends to the limiting value unity, and 1 — cos BO, = 2 sin"" ^B0, 126 INFINITESIMAL CALCULUS. [CH. Ill is a small quantity of the second order. Hence we may tan PP'0 = p^=-g^ + a (1), where o- is a quantity whose limiting value is zero. Hence ultimately, when P' coincides with P, we have, if - Ex. \. In the circle r = 2a&va.d (4) we have log r = log 2a + log sin 6, and therefore — =„ = cot 6, r dd whence cot<^ = cot5, or = (5). o Fig. 39. * The argument, which is an application of a principle stated in Art. 30, may be amplified as follows. We have, exactly, and the limiting value of this is evidently rdBjdr. 55] APPLICATIONS OF THE DERIVED FUNCTION. 127 Ex. 2. When the radius vector of a curve is a maximum or a minimum, it is in general normal to the curve. For if drldO = 0, we have cot = 0, or <^ = Jir. EXAMPLES. XVI. 1 Prove that the condition that the tangent to a curve should pass through the origin is X dx ' Prove that a pair of straight lines can he drawn through the origin, each of which touches all the curves obtained by giving c different values in the equation y = c cosh - . 2. Prove that the perpendicular drawn from the foot of the ordinate to the tangent of a curve is y- Hi))'- Hence shew that in the catenary y = c cosh xjc this perpen- dicular is constant. 3. Prove that the perpendicular from the origin on the tangent is (»-|)M-(l)}* Verify that in the circle this perpendicular is constant, and that in the rectangular hyperbola xy =li? it is equal to JW^)' 4. In the exponential curve (Fig. 20, p. 38) the subtangent is constant, and the subnormal is y^/a. 128 INFINITESIMAL CALCULUS. [CH. Ill 5. In the catenary y = c cosh xjc, the subtangent is c coth xjc, the subnormal is \c sinh ixjo, and the normal is y''lo. 6. The subtangent of the curve yn^^n-ig. IS nx. 7. Prove that the curve (I) -"i +(^^ =2 touches the straight line -+ j- = 2 at the point (a, b), whatever the value of H, 8. In the curve of sines y = b sin — , " a' the subtangent is a tan x/a, the subnormal is ^V/a . sin 2a;/a, and the normal is 6 sin - . ^ / 1 1 + 1 cos' - ) . 9. Prove that the curves y = e~'^ainl3x, y = e~'^ touch at the points for which ^x = 2Mir + ^jt, where n is integral. Sketch the curves. 10. Prove that in the parabola a sin'^' the focus being pole, = Tr -^d, and hence shew that the tangent makes equal angles with the focal distance and with the axis. 11. Two adjacent points P, P' on a curve being taken, straight lines PR, PR are drawn at right angles to the radii ; prove that the limiting value of FR, when P' coincides with P, is drjdd. 56] APPLICATIONS OF THE DERIVED FUNCTION. 129 12. If (j!) be the angle which the tangent to a curve makes with the radius vector drawn from the origin, prove that dy tan = dx dy' ""^ydx 13. If a curve be constructed with the velocity {v) of a moving point as ordinate, and the space described (s) as abscissa, the acceleration will be represented by the subnormal. 14. If a curve be constructed with the kinetic energy {\miF) of a particle as ordinate, and the space s as abscissa, the force will be represented by the gradient of the curve. 56. Mean-Value Theorem. Consequences. The following very important theorem is an extension of that given in Art. 48. If a function <^(a;) be continuous, and have a deter- minate derivative, throughout the interval from x = a to x = b, then i%:^M=f(.0 (1) where x^ is some value of x between a and b. Consider the function ■a ^(,)-^(a)-^-^^l^^ix-a) (2). This is, under the conditions stated, continuous from x = a to x = b, and it obviously vanishes for each of these values of X. Hence its derived function ^^(,)_*(^^) ^3) must vanish for some value (xi, say,) of x between a and 6. This proves the statement (1). The meaning of this result, and the nature of the proof, should be carefully studied. The geometrical interpretation L, 9 130 INFINITESIMAL CALCULUS. [CH. Ill is as follows. In the annexed figure, we have and tan<2Pi^ = |f = ii^^) (4). Y ^^^ f r — — ^ o ' \ D Fig. 40. The theorem therefore asserts tliat (under the restrictions stated) there is some point between P and Q where the tangent to the curve y='j> (x) is parallel to the chord PQ. The equation of the chord PQ is ^ , , (b)-,f>{a), , y = 4,{a) + r^l^_^^ > {x-a). •(5), as is easily verified, and the expression (2) therefore measures the difierence between the ordinate of the curve and that of the chord. This difierence vanishes at P and Q, so that there must be one point at least between P and Q at which it is a maximum or a minimum. Ex. 1. If we have {x) = a?, ^(b)-4.{a) 6^^— -*'■"' which is equal to the value of ^' {x) for x — \{a + h). This is equivalent to the statement that any chord of a parabola is parallel to the tangent at the extremity of that diameter which bisects the chord. 56] APPLICATIONS OF THE DERIVED FUNCTION. 131 The fraction i^)' ("-) (6), b — a that is, the ratio of the increment of the function to that of the independent variable, measures what may be called the ' mean rate of increase ' of the function in the interval b — a. Hence the theorem expresses that, under the conditions stated, the mean rate of increase in any interval is equal to the actual rate of increase at some point within the interval. For instance, the mean velocity of a moving point in any interval of time is equal to the actual velocity at some instant within the interval. Some other modes of stating the result (1) are to be noticed. The fact that Xi lies between a and b may be expressed by putting Xi = a + 6{b-a) (7), where stands for ' some quantity between and 1.' The precise value of will in general depend on the values of a and b. If we further write a + h ior b, we get the very useful form t(S±ItL^M^^'^a + 0h) (8) or {a) + h'{a + 0h) (9). Again, if we write x for a, and Sx for h, we have S (x) must be constant throughout that range. For if ^ (x) vary, let a and 6 be two values of x for which it has unequal values. The fraction b-a ^^^^' will then be different from zero, and there will therefore be some intermediate value of x for which ^' (x) will diifer from zero, contrary to the hypothesis. 9—2 132 INFINITESIMAL CALCULUS. [CH. Ill Moreover, if two functions («;) and ■^Qe) have equal derivatives for all values of x within a certain range, they can only differ by a constant. For, by hypothesis, .l>'(x)-ylr'(x) = (13), ^{'/'(«')-t(^)}=0 (14). Hence ^ (a;) — i|r (a;) = const (15), by the preceding case. £!x. 2. If the normals to a curve all pass through a fixed point, the curve must be a circle. For, by hypothesis, if the fixed point be taken as pole, we have, in the notation of Art. 55, '(x)dx (4). The vanishing quantities dx, dy are called ' differentials.' * The student need not take exception to the above mode of expression, which is purely conventional. Its use is * It is on account of the position which it occupies in this formula that ' (x) is called the 'differential coefficient.' 56-58] APPLICATIONS OF THE DERIVED FUNCTION. 133 simply to express the fact that in calculations involving the quantities Sx and Sy, which are afterwards made to approach the limit zero, we may at any stage replace Sy by <^' (x) Bx, whenever it is plain that the omission of quantities of the second order will make no difference to the accuracy of the final result. 58. Calculation of Small Corrections. The equation Sy = '(x)Bx (1) may, moreover, be employed as an approximate formula to find the effect on the value of a function of a small change in the independent variable, since (as we have seen) the out- standing error will be merely a small fraction of '(x)Sx provided Sx be sufficiently small. An important practical application is to find the error, or the uncertainty, in a numerical result deduced from given data, owing to given errors or uncertainties in the data. Ex. 1. To calculate the difference for one minute in a table of log sines. If ^ = logio sin X, we have dyjdx = ju, cot x, and 8y = /* cot aSa;, approximately, provided Saj be expressed in circular measure. Putting Sx = circular measure of 1' = = -0002909, iOoOU we find 8y = -0001 263 x cot x. Tlie numerical factor agrees with the difference for 1', in the neighbourhood of 45°, given in the tables. Ex. 2. Two sides a, 6 of a triangle and the included angle G are measured ; to find the error in the computed length of the tliird side c due to a small error in the angle. Wehave c" = a^ + 6^ - 2as6 cos C (2), and therefore, supposing (7 and c alone to vary, cSc = ah sin (78(7, whence 8c = — sin (78(7 = a sin 58(7 (3). 134 INFINITESIMAL CALCULUS. [CH. Ill result may also be obtained geometrically ; thus, if in 3 L liCB' = SC, and 5iV be drawn perpendicular to AB", This r€ the figure we have, ultimately, Sc = B'N=BB' cos BB'N= aW . sin CB'A = aSO . sin B, neglecting small quantities of the second order. Again, to find the error in c due to a small error in the measured length of a, we have, on the hypotliesis that a and c alone vary, c8c ={a—b cos C) Sa = c cos BSa, or Sc = cosBoa (4), a result which, like the former, admits of easy geometrical proof. The above method is defective in one respect, in that there is no indication of the magnitude of the error involved in the approximation. This is supplied, however, by the theorem of Art. 56. It was there shewn that Sy = (})' (x + 6Bw) Sx (5), where is some quantity between and 1. Hence if A and B be the greatest and least vabies which the derived function assumes in the interval from a; to a; + Sx, the error committed in (1) cannot be greater than \{A—B) Sx \. 58-59] APPLICATIONS OF THE DERIVED FUNCTION. 135 59. Maxima and Minima of Functions of several Variables. We close this chapter with a few indications concerning the extension of some of the preceding results to functions of two or more independent variables. In the first place let us seek for the maxima and minima of a function u = 4>{x,y) (1). A first condition is that we must have simultaneously t-'>- I-"--; • (^). where the diiferential coefficients are ' partial,' as in Art. 45. For if u be greater (or less) than any other value of the function obtained by varying x, y within certain limits, u will d fortiori be a maximum (or minimum) when y is kept constant and x alone is varied. This requires in general (Art. 50) that d^jdx = 0. Similarly, u must be a maximum (or minimum) when x is kept constant and y alone varies ; this requires that di^jdy = 0. As before, these conditions, though necessary, are not sufficient. The further examination of the question, in its general form, is deferred till Chapter XIV. ; but it often happens that the existence of maxima and minima can be inferred, and the discrimination between them can be effected, by independent considerations. The conditions (2) then supply all that is analytically necessary. Ex. To find the rectangular parallelepiped of least surface for a given volume. Let X, y, z be the edges, and a' the given volume. Since xyz = d? (3), the function to be made a minimum is u = xy -vyz-\-zx = xy ■¥ — ■\- — (4). The conditions dujdx = 0, du/dy = give ar'y = a', xi/ = a% the only real solution of which is a; = 2^= a, whence, also, z = a. 136 INFINITESIMAL CALCULUS. [CH. Ill It appears from (4) that, x and y being essentially positive in this problem, there is a lower limit to the surface of the paral- lelepiped. And the above investigation shews that this limit is not attained unless the iigure be a cube. As in Art. 52, the solutions of various problems can be deduced from known algebraical identities, such as a?Jry'' + z'=\[{x + y + zy^-{y- zf + (z- xf +{<^-yy] (5), yz + zx + xy = x' -\- y^ + z^ — ^ [{y — zf -\-{z — xf + {<«-yf] (6). Thus : If a straight line be divided into three segments, the sum of the squares on these is least when the segments are equal ; The surface of a parallelepiped inscribed in a given sphere (a? + y^ + z^ = a') is greatest when the figure is a cube. 60. Total Variation of a Function of several Variables. Let u = <^{x,y) (1), be a continuous function of x and y, and further let us suppose that the partial derivatives du du dx' By ^^' are also continuous functions of x and y. Let 8u be the increment of u due to increments Sx and Sy of the independent variables ; i.e. Bu = ^(x + Sx,y + By) - Similarly, if y alone were varied, the increment of u would be Qhy (6), where the limiting value of Q, when Sy is indefinitely diminished, is «■=! (')• Let us now suppose that the actual variation from x, y to x-\-hx,y + ^y is made in two successive steps, in the first of which X alone, and in the second of which y alone is varied. The total increment of u will then be hu = Phx + Q'hy (6)*, where Q differs from Q owing to the fact that the starting point of the second variation is now {x + hx, y) instead of (a?, y). To find the form which (6) assumes when hx and hy tend simultaneously to the value 0, preserving any assigned ratio to one another, we put hx = aU, 8y = ^St (7), when a, /S are finite, and St is made ultimately to vanish. We have, then ^ = Pa + Q'/3 (8). In virtue of the assumed continuity of the derivatives (2), the limiting value of the right-hand side, when St = 0, is I\a+Q,0 or g^« + g^^ (9). Hence, the smaller Sx, Sy are taken, provided they have some * If we write ^='l>x (^. 2/). S^"^*" ^"' ^^' „ (k (x + Sx, y) - (h (x, y) , „ , . we have P= 21^--!^ — gg -fe (* + «i S^. y). { x + dx,y + 5y)-{x + Sx,y) _ „,«,„■, Q = j^ = (py{x + Sx, y+e^Sy), where B^ , 8^ lie between and 1. 138 INFINITESIMAL CALCULUS. [CH. Ill definite ratio to one another, the more nearly does it become true that ^«4>^-^|^^ ;••('«)• in the sense that the ratio of the two sides is ultimately one of equality. This result is often expressed in the form ^""^i^^+Ty^y (">• The symbols dx, dy, du are then called ' diiferentials,' and du is called the ' total differential ' of u. The equation (10) shews that in the neighbourhood of a maximum or minimum, the variation of u is of the second order of small quantities, since we then have £=». 1=0 p* by Art. 59. Thus, at a point of maximum or minimum altitude on a surface the tangent plane is in general horizontal. As already indicated, the converse is not necessarily true. See Art. 51. The preceding theorem can be readily extended to the case of any number of independent variables x, y, z.... We have ultimately. 61. Application to Small Corrections. The theorem of the preceding Art. can be applied after the manner of Art. 58 to the calculation of small corrections. Ex. 1. In the case of Art. 58, Ex. 2, the total error in c, due to errors ^a, 86, 8(7 in the observed values of the two sides and the included angle, is to be found from »<^)='£-'-.y?-^'»a (,, which gives c^ -{a — h cos (7) So + (6 - a cos C) Sb + ah sin (7SC, or Sc = cos -SSos + cos j186 + a sin 58(7 (2). 60-62] APPLICATIONS OF THE DERIVED FUNCTION. 139 Ex. 2. If A be the area of a triangle, as determined from a measurement of two sides a, b, and the included angle G, we have A = ^ffl6sinC (3), whence log A = log ^ + log a + log b + log sin (4). Hence, differentiating, 8A Sa Sb ^„,„ _. -ir= — + ^+ cot CSC (5). A a b ^ ' This gives the ' proportional error,' i.e. the ratio of tlie error (8A) to the whole quantity (A) whose value is sought. An important point brought out by the investigation of Art. 60 is that the small variations of a quantity due to independent causes are superposed. This follows from the linearity of the expression for 8u in terms of Sx, By, Sz,.... Thus, in determining the weight of a body by the balance, the corrections for the buoyancy of the air, and for the inequality of the arms of the balance, may be calculated separately, and the (algebraic) sum of the results taken. The error involved in this process will be of the second order. 62. Differentiation of a Function of Functions, and of Implicit Functions. Another important application of the formula (11) of Art. 60 is to the differentiation of a function of functions, and of implicit functions. 1". Thus if u = (j}{a;,y) (1), where x, y are given functions of a variable t, we have, ultimately, hi _ d4> Bx d(f> By -,, li^d^Bi^dyBt ^^^' du _ dcj) dx d4> dy , . °'' Tt~didi'^d^tt ^ ''• This may be applied to reproduce various results obtained in Chap. II. To conform to previous notation we may write y = ^{u, v), 140 INFINITESIMAL CALCULUS. [CH. Ill where u, v are given functions of x ; the formula (3) then takes the shape dy d4> du 3(^ dv , ., dx du dx dv dx Thus, if {u, v) = uv (5), we have d^jdu = v, dfjtjdv = u, _ , „ d (uv) du dv ... and therefore -^ — =v-=- + u-j- (o), CliX CbX OnC in agreement with Art. 37. Again, if (u,v) = u'" (7), we have dy) = o (9), then dififerentiating this equation with respect to x, we have d4> dx d(j> dy _ ^ dx dx dy dx ' |^ + |^? = (10). ox oy dx This is an extension of a result given in Art. 46. EXAMPLES. XVII. 1. Prove that in a table of logarithmic tangents to base 10 the difference for one minute in the neighbourhood of 60° will be -00029, approximately. 2. The height A of a tower is deduced from an observation of the angular elevation (a) at a distance a from the foot ; prove that the error due to an error 8a in the observed elevation is 8A = a sec" aSa. If a =100 feet, a = 30°, and the error in the angle be 1', prove that hh= •47 inch. 62] APPLICATIONS OF THE DERIVED FUNCTION. 141 3. In a tangent galvanometer the tangent of the deflection of the needle is proportional to the current ; prove that the propor- tional error in the inferred value of the current, due to a given error of reading, is least when the deflection is 45°. 4. The distances (cc, x') of a point on the axis of a lens, and of its image, from the lens, are connected by the relation 1 \_\ prove that the longitudinal magnification of a small object is {x'fxf. 5. Verify the theorem of Art. 56 in the case of <^ (x) = a?. 6. Prove that if cf (*') ^^ continuous and differentiable, except for x = Xj^, when it becomes infinite, then <^'(a3]) is also infinite. 7. The error in the area (S) of an ellipse due to small errors in the lengths of the semi axes a, h is given by 8^^ 8a 86 S a'^ b' 8. If the three sides a, 6, c of a triangle are measured, the error in the angle A, due to given small errors in the sides, is „ . sin ^ 8a . ^ 86 , ^ Sc 8^ = -. — TT-. — 7^ cot C -Y- — cot 5— . sin Ji sm (j a b c 9. If the area (A) of a triangle be computed from measure- ments of one side (a) and the adjacent angles (B, C), shew that the proportional error in the area, due to small errors in the measurements, is given by 8A „8a c 8B b SO A a a sin B a sin G ' Also, verify this result geometrically. 10. If a triangle ABC be slightly varied, but so as to remain inscribed in the same circle, prove that Sa 86 So - A + S + r* = 0- cos A cos B cos C 142 ■^ INFrNITERIMAI. CALCULUS. 11. If the density (s) of a body be inferred from its weights ( W, W) in air and in water respectively, the proportional error due to errors SW, 8W' in these weighings is 8s W &W SW + ■ s W-WWW-W' 12. A crank OP revolves about with angular velocity to, and a connecting rod PQ is hinged to it at P, whilst Q is con- strained to move in a fixed groove OX. Prove that the velocity of Q is (1). OP, where S is the point in which the line QP (produced if necessary) meets a perpendicular to OX drawn through 0. 13. An open rectangular tank is to contain a given volume of water, find what must be its proportions in order that the cost of lining it with lead may be a minimum. [The length and breadth must each be double the depth.] 14. Given the sum of three concurrent edges of a rectangular parallelepiped, find its form in order that the surface may be a maximum. 15. Prove that the parallelepiped of greatest volume which can be inscribed in a given sphere is a cube. 16. Prove that the rectangular parallelepiped of greatest volume for a given surface is a cube. 17. If a triangle of maximum area be inscribed in any closed oval curve the tangents at the vertices are respectively parallel to the opposite sides. 18. If a triangle of minimum area be circumscribed to a closed oval curve, the sides are bisected at the points of contact. 19. The triangle of maximum area inscribed in a given circle is equilateral ; and the triangle of minimum area circum- scribed to the circle is also equilateral. 20. A polygon of maximum area, and of a given number (n) of sides, inscribed in a given circle is regular; and a polygon of minimum area, of n sides, circumscribed to the circle is also regular. APPLICATIONS OF THE DERIVED FUNCTION. 143 21. Assuming that the rectangle of greatest area for a given perimeter is a square, explain how it follows immediately that the rectangle of least perimeter for a' given area is a square. What inferences can be drawn in lilce manner from the results of Examples 14 and 16, above? 22. The polygon of n sides, -which has maximum area for a given perimeter, or minimum perimeter for a given area, is regular. (Assume the result of Example 23, p. 119.) Hence shew that the figure of maximum area for a given perimeter, or of minimum perimeter for a given area, is a circle. 23. By the regulations of the parcel post, a parcel must not exceed six feet in length and girth combined ; prove that the most voluminous parcel which can be sent is a cylinder 2 feet long and 4 feet in girth, and that its volume is 2 546 cubic feet. CHAPTER IV. DKEIVATIVES OP HIGHER ORDERS. 63. Definition, and Notations. If y be a function of x, the derived function dyjdx will in general be itself a differentiable function of x. The result of differentiating dyjdx is called the 'second differential co- efficient,' or 'second derivative.' If this, again, admits of differentiation, the result is called the 'third differential coefficient,' or 'third derivative'; and so on. If we look upon djdx as a symbol of operation, the first, second, third, . . . wth derivatives may be denoted by d I dy fdy fdY Tx-y' [Tx)-y' [d-xj-y'- UJ-^- respectively. The more usual forms are dy d^y d'y d^y dx' ds^' do^' '" dc^' which may be regarded as contractions of the preceding, although (historically) they arose in a different manner. Again, writing D for djdx, as in Art. 31, we have the forms Dy, D'y, D^y, ... B-y. If y = (l> (x), the successive derivatives are also denoted by '{x\ "{x), '"(x),... .^<»'(*). Occasionally it is convenient to adopt the briefer notation y', y", y"', ... 2/<»). 63] DERIVATIVES OF HIGHER ORDERS. 145 There are a few cases in which simple expressions for the nth derivative of a function can be found. The more im- portant of these are given in the following examples. Ex. 1. If y=A„+A^x+ A^+ 1^1? + ...+A^x'^, (1), we have Dy= A^ +2A^+ iA3a?+ ...+mA^x^-^, D"'y= 2 . 1^2 + 3 . 2^3 + .. . + m (»?i - 1) A^,af ■(2), (3). D'^y = m{m-\) {m-2) ... 2.1A^, and therefore B'^+^y = 0, D'"*^y = 0, etc Hence the mth derivative of a rational integral function of the with degree is a constant, and all the higher derivatives vanish. £!x.± If y = ^ we have Dy = Ae*^, B^y -- A^e*^, . . . , and, generally, J)^y = A"e** Hence, putting k = log a, we have iya?'=(\ogaf'a?' Ex. 3. If y = sinl3x we have By = j8 cos ^x, B'y = — /3° sin ^x, B^y = - fi^ cos j3x, B*y= ^sm/3x, and so on. Otherwise, we have By = 13 sin {^x + ^ir), and therefore B'y = ^ sin (j3x + Jx + \ir), and, generally, B"y = /3" sin (px + ^mr) Ex. 4. If y = cospx we have By = — y8 sin px, B''y = — ;8^ cos Pxj Bh'j^ P^smjSx, B^= ^^cos^a;, •W, .(5). .(6). •(7), •(8). and so on. Or, whence and, generally, L. ...(9). .(10), •(11). By = p cos (fix + Jtt), B'y = 13" cos (fix + iw + ^7r), B^y = ^ cos {px + ^mr) (12). 10 146 INFINITESIMAL CALCULUS. [CH. IV Hx.5. If y^e'^cos^x (13), .(14). we find Dy = e"^ (a cos fix — ^ sin ^x), jy^y = ^ {(a= - ;8^) cos ^x - 2a;3 sin ySa;} Similarly, if y = e!^&\VL^x (15), we have Dy = e"^ (a sin fix + 13 cos fix) ) D^y = e<^{{o!'- fi') s.in fix + 2afi cos fix}] ^ General formulae may be obtained, in these cases, by putting a = rcos^, fi = rsme (17). This makes B . e™ cos fix = e"^ (a cos fix — fi sin fix) = re"^ cos {fix + 0), and by repeated application of this result we find D"'. e'^ cos fix = r''e^ cos {fix + n0) (18). Similarly, i)" . e"" sin fix = r''e^ sin {fix + nO) (19). ^£c. 6. If y = loga; (20), we have Dy = x-\ D'y = -x-\ i5'2/ = - l.-S.ic-^, .••, and, generally, i)"2/ = -l.-2.-3...-(M-l)a;-« =(-r-^^ (21). 64. Successive Derivatives of a Product. Leib- nitz' Theorem. If u, V be functions of x, we have by Art. 38 (20), I){uv) = Bu.v + u.Dv (I). If we differentiate this again, we have B' (uv) = D{Du.v) + B{u. Dv). Now, by the rule referred to, we have I>{Du.v) = B'u.v + I)u.Dv, I) {u . Dv) = Bu . Dv + u .B^v, whence B''(uv) = B^u.v + 2Bu.Bv + u. 0=y (2). 63-64] DERIVATIVES OB' HIGHER ORDERS. 147 The general formula for the nth derivative of a product is D» (uv) = B^u . V + nD'^'u . Bv + "'Y~ ' B'^'^u .BH + ... + nBu . B^H + u.B'v (3), the coefficients being the same as in the Binomial Theorem. This formula is due to Leibnitz. To see the truth of (3), consider the process of formation of the first few derivatives of uv. Using the acceat notation, we have D (uv) = u'v + uv (4). Differentiating this again, D^ (uv) = u"v + u'v' + u'v' + uv" = u"v + 2u'o' + uv" (5). The next differentiation gives B\uv) = u"'v + 2u"v' + u'v" + u"v' + 2u'v" + uv'" (6), where in the first line we have differentiated the first variable factor in each term of (5), and in the second line the second variable factor. The result is n\uv) = u"'v + 3u"v' + 3u'v" + uv'" (7) It appears that the numerical coefficient of the rth term in (7) is the sum of the coefficients of the rth and (r — l)th terms in (5); and it is evident from the nature of the successive steps that this law will obtain for all the subsequent derivatives. Now this is precisely the law of formation of the coefficients in the expansions of the successive powers of a + b; and since the coefficients of D(uv) are the same as those of the first power of a + b, it follows that the coefficients in the expanded form of D'^{uv) will be the same as those of {a + 6)". Ex.\. If y = xu (8), we have D^y = xD^u + nDx . D'^'hi = xB'^u + nD^-^u (9), since D'x = 0. 10—2 148 INFINITESIMAL CALCULUS. [CH. IV Thus if y = xsiQpx (10), we have B^y = xTfi sin ^x + ID sin ^x = -/?*»; sin /3a; + 2/3 cos ;S^ (11). Again, if y = x\ogx (12), we have B^y = a;Z>" log x + w/)"-' log x ~\ J 3.11-1 '^ \ I g.n-1 » by Art., 63 (20). Hence i>"y=(-r^-^ (13). Bx.2. If y = ^u (14), we have B"v = e"^ . B''u + n . Be"' . B^-'u + '^\~ ' BH'" . B^-'^u + . . . " 1.2 = e'^(i)''tt + rao2)''-'M + ^?^^^a''Z>"-='M + ...) (15). 1 • J Thus, if y^eT'wa.^x (16), wc have B'y-e'-^{D' sin px + 2aB sin fix + a" sin px) = ^{{a'-l3'')smPx+2al3cospx} (17), in agreement with Art. 63 (16). 65. Dynamical Illustrations. The second derivative is especially predominant in the dynamical applications of the Calculus. Thus, in the case of rectilinear motion, if s be the distance from a fixed origin, we have seen (Art. 33) that the velocity (v) and the acceleration (a) are given by the formulae ds dv ,,, '' = dt' " = Z« (!)• Hence, in the present notation, we have d /ds\ d?s '^~dt\dt) ~ df ^■'i' i.e. the second derivative of s (with respect to the time) measures the acceleration. 64-65] DERIVATIVES OF HIGHER ORDERS. 149 So also the angular acceleration of a body about a fixed axis is given, in the notation of Art. 33, by di=d? ^"*^- Ex. 1. If s be a quadratic function of t, say s = At' + £t+G (4), ds we have -v- = 2At + B, at 5='^ (5). i.e. the acceleration is constant. Ex. 2. In ' simple-harmonic ' motion we have « = acos (nt+ e) (6), whence -r = — na sin (nt + e), at ^ ' --=-^ — — v?a cos{nt+ e) = — n''s (7), i.e. the acceleration is directed always towards a fixed point (the origin of s) and varies as the distance from that point. Ex.3. If s = A cosh nt + B sinhnt (8), ds we have -j- = nA sinh nt + nB cosh nt, at dh -j-^ =n'A cosh n* + ji^5 sinh n< = re^s (9), at i.e. the acceleration is from, a fixed point, and varies as the distance. EXAMPLES. XVm. Verify the following differentiations : 1. y = al'{l-xf, D-'y = 2-Ux+l2a?. 2. 2/ = l/iar' {I - ^x), D^y = ix.{l- x). 3. y = -^ijx{l-x)(p+lx-a?), B'y = ^fxx {x - 1). 4. y = ^^{3P-ilx + ix'), I)'y = ip.{x-iiy. 150 INFINITESIMAL CALCULUS. [CH. IV 1 r., 24 (1 - lOar^ + 5a^) 8. y=(l-a;)-"', i)»y = m(TO+l)...{OT+w-l)(l-a;)-'»-''. 1 + a; _ „ 2 . n ! l-a;' ■^ (l-aif^" 10. y = sin'^ a;, D^y - 2 cos 2*. 11. y = cos'' !B, D^y = 2"-' cos (2a! + Jtitt). 12. 2/ = sec X, B^y = 2 sec' x — sec aj. 13. y = x^ sin a;, i)''y = 4a: cos x- (a? — 2) sin x. 14. y = sin' x cos a;, /)'y = 6-60 sin^ as + 64 sin^ x. 15. y = sin a; sinh x, D^y = 2 cos x cosh a;. 16. 3/ = cos X cosh a;, B^y = - 2 sin a; sinh x. 17. 2/ = sin a; cosh a;, D^y = 2 cos a; sinh x. 18. y = cos X sinh a;, D^y = - 2 sin a; cosh x. 19. 2/ = sin-^a;, i)«2/ = __^^. 20. The first five derivatives of tan x are l + «^ 2«(l + «^), 2 (1 + 3*'') (1 + <"), 8< (2 + 3<=) (1 + /=), 8(2 + 15i''+15<^)(l+<'), where < = tan x. D''y = (x + n)e'. 24. 2/ = ar'e", -D"y = {a;^ + 2wa; + n{n- 1)} e^. 21. 2/ = a!'lo3^, 22. 2/ = a5nog*', 23. 2/ = a;e% G6] DERIVATIVES OF HIGHER ORDERS. 151 25. By applying Leibnitz' Theorem to the differentiation of the identity prove that r t(t—V) r(r — l)(r — 2) where m^ = m (m — 1) (m - 2) . . . (m - r + 1). 26. The equation -j-^ + /; -j- + w^'s = is satisfied by « = ^e-^** cos (o-i + c), for all values of A and e, provided 27. The equation -^ + 2n -^ + n^'s = is satisfied by s = (.4 + Bt) e""'. 28. If a; = <^(«), 3/ = x(<). , , ^ cPy 'dt'd&'"dt'd& prove that -p^ = , . 66*. Geometrical Interpretations of the Second Derivative. In Art. 56 an important property of the derived function was obtained by a process which consisted virtually in a comparison of the curve y = ^(p) (1) with a straight line 2/ = -4 +£« (2), the constants A, B being determined so as to make (1) and (2) intersect for two given values of x. We proceed, in a somewhat similar manner, to compare the curve (1) with a parabola y = A+Bai + Gx'' (.3), * Arts. 66, 67 can be postponed. 162 INFINITESIMAL CALCULUS. [CH. IV where the constants A, B, G are determined so as to make (1) and (3) intersect for three given values of x. 1°. We will first suppose these values of w to be equi- distant ; let them be a — h, a, a + h. The equations to determine the constants are then A+B(a-h) + G{a-hy = tj){a-h),\ A+Ba +Ca^ =i>(a), ■ (4). A+B(a + h) + Gia + hy = i>(a + h) J Let us now write F{x) = (t>{a;)-{A+Bx + Gx') (5), i.e. F(x) denotes the difference of the ordinates of the curves (1) and (3). By hypothesis, F{x) vanishes for cc = a — h, and for x = a; hence, by Art. 48, the derived function F' {x) will vanish for some intermediate value of x, that is F'{a-e,h) = (6), where 1 >6i>0. Again, since F{x) vanishes for x = a, and for x = a + h, we shall have F'(a + ej,) = (7), where 1 > ^2 > 0. By a further application of the same argument, siace the function F' (x) vanishes for « = a — ff^h and for a; = a + $ih, its derived function F"(x) will vanish for some intermediate value of a;; we have therefore F"(a + eh) = (8), where 6 is some quantity lying between —0i and 0^, and d fortiori between + 1. Since, by (5), F"(x) = ri^)-^G (9), it follows that, for some value of 6 between + 1, ^"{a + eh) = 2G (10). Now from (4) we find ^(a + h) - 2^{a) + ^{a-h) = 2Glf (11), and therefore .|,(a + A)-2.^^(a) + .^(«-;.)^^„^^^g^^ ^^^^^ 66] DERIVATIVES OF HIGHER ORDERS. 153 Hence In the same way we could prove that Hrr,^,ii^±M::^^^±M±^) = ^"(„) (14). If the difference (a) be denoted by Sy, the expression {<^ (a + 2/t) - <^ (fls + A)} - {<^ (a + /t) - .^ (a)} may be denoted by S {By) or S^y. Hence the formula (14) is equivalent to i;„ % _ "(a) (16), ultimately. It appears that the chord is above or below the arc according as "{a) is positive or negative. 2°. We will next suppose that two of the three points at which the curves (1) and (3) intersect are coincident. More precisely, we suppose that for x = a the curves not only intersect but touch, and that they intersect again for x = a+h. The conditions that, for x=a,y and dyjdx should have the same values in the two curves, are B+20a='(a)\ ^ '' while the third condition gives A+B(a + h) + C(a + hy = (t3{a + h) (18). With the same definition oi F{x) as before, we have F(a) = 0, F(a + h)=0 ....(19), and therefore F'{a+e,h) = (20), where 1 > ^i > 0. Again, since F' (x) vanishes for x = a, and for a; = a + dji, we have F"{a+6h)=0 (21), where 6i>d>0. , Now from (17) and (18) we find ^{a + h)-4>(a)- h^'(a)=:Gli^ (22). Hence, by (9) and (21), (f>(a + h)=4>(a) + h' (a) + Ih'^" (a + Oh). . .(23). This very important result will be recognised, later, as a particular case of Lagrange's form of Taylor's Theorem (see Chap. xiv). It includes as much of this theorem as is ordinarily required in the dynamical and physical applications of the subject. 66] DERIVATIVES OF HIGHER ORDERS. 155 From (23) we deduce lim,^, ^^"^ + ^> - ^,(^> - ^'^^ (^> = W' (a) ...(24). In Fig. 43, let OA=a, AH = h, and let AP, HQ be the corresponding ordinates of the curve (1). If QH meet the tangent at P in V, we have QH=cj)(a + h), VH=^ (a) + hcj}' (a). Pig. 43. Hence (24) asserts that QV=iAHK"(a) (25), ultimately. Hence, ultimately, the deviation of a curve from a tangent, in the neighbourhood of the point of contact, is in general a small quantity of the second order. If " (a) is positive or negative. The formulae (16) and (25) have an interesting appli- cation in the theory of Curvature. See Chap. x. 156 INFINITESIMAL CALCULUS. [CH. IV 67. Theory of Proportional Parts. Let us make the curves y = '^W (1). and y=A+Bx + Ca? (2), intersect for x = a, x = a + zh, x = a + h, where 1 >z>0. We find (l-z)(t>{a) + z^{a + h)-{a + 2h) = z(l-e)h'G...(3); and consequently, by the method of the preceding Art., (1- z)(l}(a) + z(j)(a +h)- ^(a + zh) = y (1- z)h''^" (a + eh) (4), where 1 > ^ > 0. This result, which includes the theorems of Art. 66 as particular cases, is here introduced for the sake of its bearing on the theory of ' proportional parts.' Suppose that (^ («) is a function which has been tabulated for a series of values of x at equal intervals h. Let a be one of these values, and suppose that <^ (x) is required for some value of X between this and the next tabular value a + h; say for a + zh, where 1 > 2^ > 0. In the method of ' proportional parts,' the interpolation is made as if the function increased uniformly from x= a to x= a + h, i.e. we assume 4>ia + zh)-(l> (a) _ z . <}>{a+h)-(p{a) ~1 *■ '*' or (f){a+zh) = (l-z}(a) + z^(a+h) (6). The formula (4) gives the error involved in this process, which is equivalent to assuming that the arc of the curve (1) between x = a and x = a + h may be replaced without sensible error by its chord. The maximum value of ^: (1 — z) is J, by Art. 50, Ex. 2. Hence if R denote the greatest value which " (x) assumes in the interval from x== a to x = a + h, the formula (4) shews that the error >kh''-R (n 67-68] DEKIVATIVES OF HIGHER ORDERS. 157 Ex. 1. In a seven-figure logarithmic table, the logarithms of all numbers from 10000 to 100000 are given at' intervals of unity. Now if ^(x)=l0gi„ 03 (8), we have •^"W = -S ^^)- Hence, putting A=l, in (7), we find that in the interpolation between log],,?! and logio(re+l) the error involved in the method of proportional parts is not greater than •05429-=-™" (10). Thus for m= 10000, where it is greatest, the error does not exceed -000000000543, and is therefore quite insensible from the stand-point of a seven- figure table. It appears from (4) that the method may be expected to fail whenever j/' {x) is large. The differences are then said to be ' irregular.' -ffic. 2. If <^(a;) = logi(,sina! (11), we have <^" (sc) = — /«. cosec" a; (12). Hence, putting h = j^ = -000291, wefind |AV (a;) = - -00000000460 cosec" a: (13). Since cosec'' 18°= 1047, it appears that in a table of log sines at intervals of 1' the error of interpolation may amount to half a unit in the seventh place when the angle falls below 18°. 68. Conca-vity and Convexity. Points of In- flexion. Just as ^'(«) measures (Art. 33) the rate of increase of (^ («), so ^" {x) measures the rate of increase of 0' (x). Hence if <^" {x) be positive the gradient of the curve 2/=<^(^) (1) increases with x ; whilst if " («) be negative the gradient decreases as x increases. If (\)" (x) = 0, the rate of change of the gradient is momentarily zero, and we have a ' stationary tangent.' The simplest case of this is at a ' point of inflexion,' i.e. a point Bt which the curve crosses its tangent ; see p. 159. 158 INFINITESIMAL CALCULUS. [CH. IV A curve is said to be concave upwards at a point P when in the immediate neighbourhood of P it lies wholly above the tangent at P. Similarly, it is said to be convex upwards when in the immediate neighbourhood of P it lies wholly below the tangent at P. If the curve, to the right of P, lie above the tangent at P, it is easily seen from Art. 56 that within any range (how- ever short) extending to the right of P there will be points at which ^'(x) is greater than at P. Hence, by Art. 47, the value of ^"(«) at P cannot be negative. The same conclusion holds if the curve, to the left of P, lie above the tangent at P. Fig. 44. Similarly, if the curve, either to the right or left of P, lie below the tangent at P, the value of ^" (x) at P cannot be positive. It follows that the curve is concave upwards when " (x) cannot be either positive or negative, and therefore (since it is assumed to be finite) must vanish. This condition, though essential, is not suffi- cient. It is further necessary that ^" («) should change sign 68] DEKIVATIVES OF HIGHEB ORDERS. 159 as X increases through the value in question. Suppose, for instance, that to the left of P the curve lies below the tangent at P) and that to the right of P it lies above it. It appears then from Art. 56 that there will be points of the curve both to the right and to the left, in the immediate neighbourhood of P, at which the gradient is greater than at P, i.e. the gradient is a minimum at P, and <^" («) must , therefore change (Art. 50) from negative to positive. Fig. 45. If the crossing is in the opposite direction, the gradient is a maximum at P, and f^" (x) changes from positive to negative. Ex. 1. If we have y" = lox. ■(2), This changes from — to + as a; increases through 0. Hence we have a point of inflexion ; see Fig. 34, p. 110. Ex. 2. This makes y= 2x y 1+3? „ _ 4a; (a;' - 3) Q.+x'f • .(3). 160 INFINITESIMAL CALCULUS. [CH. IV Hence there are three points of inflexion, viz. when x = and when x = d= J3. See Fig. 17, p. 31. Ux. 3. In the curve of sines 2/ = &sin- (4), 1 II b . X 11 we have y — — ; sin - = — -^ . a' a a' Hence y" changes sign, and there is a point of inflexion, whenever the curve crosses the axis of x. See Fig. 18, p. 34. Ex. 4. In the curve 2/ = a3* (5), we have y" — \'2i3?. This vanishes, but does not change sign, when a; = 0. Hence we have a stationary tangent, but not a point of inflexion in the strict sense. It is in fact obvious, since x* is essentially positive, that the curve lies wholly on one side of the tangent at the origin. 69. Application to Maxima and Minima. The criterion of Art. 50 for distinguishing maxima and minima values of a function {x) can also be expressed in general in terms of the second derivative <^" («). Since ^" («) is the derivative of ^' (x), it appears that if, as X increases through a root of (x). See Fig. 17, p. 31. It may happen, however, that a value of x which makes ^'(«) = Oalso makes "(«) = 0. It is easily shewn that in this case ' (cc) = will occur between any two roots .of ^ {x) = 0. Similarly, at least one root of (f)" (cc) = will occur between any two roots of ^' (*•) = 0, and so on. Moreover, since an r-fold root of ^ («) = is an {r — 1)- fold root of ^' (x) = 0, it will be an (r — 2)-fold root of ^" (a;) = 0, . . . , and finally a simple root of ^ "■"" («) = 0. Hence the necessary and sufficient conditions for an r-fold root of {x), ^'{x), "{x), ..., .^(-"(a;) (1) should simultaneously vanish. Ex. If {x) = 2a^ + 5x^ + 4:a^+2x^ + 2x+l, we have <)i'(a;) = 10a3*+ 20k'+ 12a;^+ 4a!+ 2, " (x) = 4 (10a.'' + 15a!^ + 6a; + 1). These all vanish for a; = - 1, which is therefore a triple root of 1^ (a;) = 0. We find, in fact, that ^{x) = {x+lf{2a?-x+l). L. 11 162 INFINITESIMAL CALCULUS. [CH. IV EXAMPLES. XIX. Prove that, in a table of natural sines at intervals of 1', the error of proportional parts never exceeds •0000000106. 2. Shew that in a table of natural tangents the method of proportional parts fails for angles near 90°. Also prove that the limit of error for angles near 45°, when the tangents are given at intervals of 1', is ■0000000423. 3. Shew that in a table of log tangents the method of proportional parts fails both for angles near 0° and for angles near 90°. Shew also that the maximum error involved in the method is least for angles near 45°. 4. Prove that the curve y = \ogx is everywhere convex upwards. 5. Prove that the curve y = x log X is everywhere concave upwards. Trace the curve. 6. Find the maximum ordinate, and the point of inflexion, of the curve y = xe'". Trace the curve. [The maximum ordinate corresponds to x=l ; the inflexion to a: =2.] 7. Shew that the curve y = e~" has inflexions at the points for which a; = * —r^ ; and trace it. 8. Find the maximum and minimum ordinates, and the inflexions, of the curve -it" y = a;e . Trace the curve. [The maximum and minimum ordinates are given by as = =t ^1 ; the inflexions by a; = 0, ± ^f.] DERIVATIVES OF HIGHER ORDERS. 163 9. A certain function <^ (x) is constant and forOb. Prove that <^ (x) and ^' (as) are continuous, but that " {x) is discontinuous. Trace the curve y = 4> {x). 10. Shew that y = x'{i-x) has an inflexion at the point (1, 2). Trace the curve. 11. Shew that y = x^{l-x-') t 1 5 \ has inflexions at the points v^—fni oa ) • Trace the curve. 12. Find the points of inflexion of the curve 13. Shew that y = -, r-„ (x — ay has a point of inflexion at (— 2a, — |a). Trace the curve. 14. Find the pointsH)f inflexion of the curve and trace the curve. [a; = 0, ± aJZ.'] 15. Shew that the curve \-x y~l+x^' has three points of inflexion, and that they lie in a straight line. Trace the curve. 16. Prove that the equation a!»-10a:2+15a;-6 = has a triple root. 11 -2 164 INFINITESIMAL CALCULUS. [CH. IV 17. Prove that the equation of - 5af + 5ai^ + da? - Ua? - 4a; + 8 = has a triple root ; and find all the roots. 18. If PiT, P'N' be two neighbouring ordinates of a curve y = 'l> (x), and if QH, any intermediate ordinate, meet the chord FF' in V, prove that QV=iFH.HIir'.^"{c), ultimately, where c is the abscissa of some point between If and N'. 19. Shew that in the formula tj) (a + h) = ({> (a) + hcj) (a + 6h) of Art. 56, the limiting value of 0, when h is infinitely small, is in general J. What is the geometrical meaning of this result ? 20. Shew that the variation in the value of a function, in the neighbourhood of a maximum or minimum, is in general of the second order of small quantities. 21. Explain why the rate of a compensated chronometer, at any particular temperature, differs from the rate at the temperature of exact compensation by an amount proportional to the square of the difference of temperature. 22. Shew that, in a mathematical table calculated for equal intervals of the variable, the maximum error of interpolation by , proportional parts, in any part of the table, is one- eighth of the 'second difference' (i.e. of the difierence of the differences of successive entries). CHAPTEE V. INTEGRATION. 71. Nature of the Problem. In the preceding chapters we have been occupied with the rate of variation of functions given a priori. The Integral Calculus, to which we now turn, is concerned with the inverse problem; viz. the rate of variation of a function being given, and the value of the function for some particular value of the independent variable being assigned, it is required to find the value of the function for any other assigned value of the independent variable. In symbols, it is required to solve the equation l=^(^) OX where <^ {x) is a given function of x, subject to the condition that for some specified value (a, say) of oo, y shall have a given value (b). For example, the law of velocity of a moving point being given, and the position of the point at the time {xl 166 INFINITESIMAL CALCULCS. [CH. V the equation (1) becomes |-S^« •■■■•; W Hence if, as is the case in most practical applications of the subject, y be restricted to be continuous, we have, by Art. 56, y=.ir{x)+G (4), where is a constant. The precise value of G is indeter- minate, so far as the equation (1) is concerned ; G is therefore called an ' arbitrary constant.' Its use is that it enables us to satisfy the remaining condition of the problem as above stated. Thus iiy = biovx = a, we must have h = ^^{a)+G, whence y — h = ■<^ (ai) — ■^ (a) (5). JCx. Given that the velocity of a moving point is m + gt, we have it=''^3'-T^'^^^3n (6), whence 8~ut + \yt^+G (7). Determining G so that s = s^ for t = t^, we have s~s, = u(t-t,) + y{f'-t,^) (8). If, as in Art. 31, we use the symbol D for the operator djdx, the equation (1) may be written %=<^(^) (9), and its solution may, consistently with the principles of algebraic notation, be written y = iy'<^{x) (10), the definition of the ' inverse ' operator D~^ being that D[D-^4,{x)]==4>{x) (11). The function I)-^(f)(x) (12), when it exists, is called the 'indefinite integral' of (j) (x) with re.spect to x. It is more usually denoted by S^{a>)dx (13). 71-72] INTEGRATION. 167 The origin of this notation will be explained in the next chapter ; in the meantime (13) is to be regarded as merely another way of writing (12). The distinction between ' direct ' and ' inverse ' operations is one that occurs in many branches of Mathematics. A direct operation is one which can always be performed on any given function, according to definite rules, with an unambiguous result. An inverse operation is of the nature of a question : what function, operated on in a certain way, will produce an assigned result? To this question there may or may not be an answer, or there may be more than one answer (cf Art. 20). In the case of the operator D"^, we have seen that if there is one answer, there are an infinite number, owing to the indeterminateness of the additive constant G. Whether there is, in every' case, an answer is a matter yet to be investigated ; but we may state, although this is rather more than we shall have occasion formally to prove, that every continuous function has an indefinite integral. In the rest of this chapter we shall be occupied with the problem of actually discovering indefinite integrals of various classes of mathematical functions. 72. Standard Forms. There are no infallible rules by which we can ascertain the indefinite integral I)~^^(so) or ^^{x)dx of any given continuous function (^ {x). As above stated, integration is an inverse process, in which we can only be guided by our recollections of the results of previous direct processes. The integral, moreover, although in a certain sense it always exists, may not admit of being expressed (ia a finite form) in terms of the functions, whether algebraic or trans- cendental, which are ordinarily employed in mathematics. The following are instances : and the list might easily be extended indefinitely. 168 INFINITESIMAL CALCULnS. [CH. V The first step towards making a more or less systematic record of achieved integrations is to write down a list of differentiations of various simple functions; each of these will, on inversion, furnish us with a result in indefinite integration. The arbitrary additive constant which always attaches to an indefinite integral need not be explicitly introduced, but its existence will occasionally be forced on the attention of the student by the fact of integrals of the same expression, arrived at in different ways, dififering by a constant. The student should make himself thoroughly familiar with the following results, which are fundamental: ^ - — ' fa;''dx = .~\ai''+\ (A) [except for m = — 1], T-.x^ = ruff' ax ^.e** = A;e**, ( ^'>dx = ^e^'>, (G) -V- . sin a; = cos x, I cos xdx = sin x, (D) J- . cos a; = — sin «, I sin xdx = — cos x, (E) -J- . tan X = sec'' x, j sec^ xdx = tan x, (F) -J- . cot x= — cosec' X, / cosec' xdx = — cot x, (G) d^ ■ _^x 1 [ dx . _j re ^ ,„ da;-^™ a~^/(fl^-xy j V(a' - a''') ~ ^'° a' ^^' d . ,x a r dx 1 ^ , « / r^ / T-.tan-'-= - =- tan-' - , (I) ^ dx a d' + m^ J a^ + x^ a a ^ ' T- . sinh X = c3feh x, I cosh xdx = sinh x, (J) T- . cosh X = sinh x, I sinh xdx = cosh x, {K) * As to the question of sigD^ see Art. 41. 72-73] INTEGRATION. 169 T- . tanh X = sech" «, / sech" xdx = tanh x, (L) J r -J- . coth x = — cosech' x, I cosech" xdx = — coth x, (M) d ■ ._!'"_ 1 r dx . ,_iX = log ^^^- ',* (AO d , , X 1 f dx , , X . cosh""' = log ^ + ^('^~^°> .* (0) d . , ,x a f dx l,,,a! T- . tanh-' - = — -„ , -z z = - tanh-' - dx a a'—x' J a?—x^ a a d ,, ,x a f dx l,,,a; . coth-' - = - r . -r = - - coth-' - dx' a x' — a' h [«=>«'] =i^log^-^. (Q) 73. Simple Extensions. To extend the above results, we first notice that the addition of a constant to x makes no essential difference in the form of the result (cf Art. 39, 1°). Thus, obviously, I {x+aydx= -(a; + a)™+' (1), J ' m + 1^ ' ^ ' { dx , , J^T^=''^g^(^+«) (2), r dx ^ r dx _ . _^ x — a J ^/(2ax -x^)~J Via"" -{x- af] ~ ^^^ ~ir ^'^^' and so on. Some further illustrations occur in Arts. 74, 75. * As to the sign, see Art. 44. i/ 170 INFINITESIMAL CALCUf.US. [CH. V Again, if x be multiplied by a factor k, the integral has the same form as before, except that it is divided by this factor (see Art. 39, 2°). Thus I sinkxd!Xi = — vcosia; (4), h =-\os(ax + b) (5), and so on. Agaiu, we have the theorems fGudx=GJudx (6), f(u + v + w+ ...)dx=JudsB+Jvd!v+Jwdx + (7); since, if we perform the operation d/dx on both sides we get in each case an identity, by Arts. 36, 37. Thus the indefinite integral of a rational integral function A^^ + A,x"^^ + ...+A„,^,x + Am (8), is — ^ A,x"^' + -A^x"' + ... + ^A^^x-' + AmX (9). m + 1 m Again, suppose we have a rational fraction of the form q^ (10). x + a ^ ' By division this can be reduced to the sum of a rational integral function and a fraction -^ (11). The former part can be integrated as above, and the integral of (11) is ^log(a;+a) (12). Ex. 1. /(a;-l)l«Za!=3-^(a:-l)t+i = |(a;-l)l Ex.2. j-^^ = ^log(2x-l). Ex. 3. /sin' xdx= J/(l — cos 2a!) dx = ^x — \ sin 2a;. Ex. 4. Jtan^ xdx = J(sec^ x—l)dx = tan x — x. 73-74] INTEGRATION. 171 Ex.6. /2^efe = /{K+i-H-|+8(2^)}'^- = \a? + ^«^ + ^x + ^ log (2a! - 1). EXAMPLES. XX. Find the indefinite integrals of the following expressions * : 1 1 1. r^, 7^^^,. 2. 2a!- 1' (2a! -If 3. {x~l)\ ^^TTj. 4. Jx, -L, i±^. \ /'//.._ 1 \J Jx Jx 1 +x 1 +x 5. -77^^ — r, „o^ , . 0. 1 1 1-a!' (i-xy (x-iy, 1 (a!-l)^- 1 1 ^(2 + a!) ' J{3-2.:) l-2a! 3 + a! ' 2 + a! 3-x' 1 +a! l-a:' \-x 1 +x' !-»!=' \ + 0(? X IV / iv 3 + a! 3-a; \ as/' \ a;/ a!* a? 9. i^, ^-j::. 10. 1 +03' 1 -a!' 11. = 5, = 5. 12. cos'' a:, cof^a:. 1 + a:' 1 — x^ 13. (cos X — sin aj)^. 14. cosh^ a:, sinh" a!. 1-a:™ 1 + »;'""+' 15. tanh'a!, coth^a;. 16. 1 - X ' 1 +a! 74. Rational Fractions with a Quadratic Denomi- nator. We next shew how to integrate any fixpression of the form /(-> (1), af+pai + q where F(pB) is rational and integral. If necessary, we first divide the numerator by the denominator until the remainder is of the form ax + b. We thus get the function (1) expressed as the sum of a rational integral function and a fraction "^ + ^ (2) si?+px + q ^ ' * The student should test the accuracy of his results by differentiation. 172 INFINITESIMAL CALCULUS. [CH. V The former part can be integrated as in Art. 73 ; it remains only to consider the form (2). We take first the case 1 /3) oo'+px + q The form of the result will depend on whether p' = iq. l{p^ < 4iq, we have x^ + px + q=^{x + ^py + {q-ip') = {se-ay + ^, where a, /3 are real. Now /< dx 1. _,« — a ,.. by an obvious extension of Art. 72 (T). Ifp' = 4!q, we have «" +px + q = {x + ^py, ^"•^ i(*+ip?^"^+¥ ^^^' If p'' > 4-q, we have a choice of methods. In the first place, writing ai?+px + q = (x + ^py-(lp''-q) = (x-ay-^\ where a, y8 are real, we have by Art. 72 (Q) /, dx 1 , , , a; — a 1 , X— a — B .>,. , =2;s^°g^3^ (n- The more usual method of treating this case depends on the fact that when p^ > 4q the quadratic expression can be resolved into real factors; thus x^+px + q = {x- a') (x - ^'), where a' = a + /3, )8' = a — /3. With a proper choice of the constants A, B we may then put (x-a')(x-^) x-a''^x-^ ^ '' * It is assumed that x>a+p. The modifications necessary in other cases may be easily supplied. 74] INTEGRATION. 173 viz. this will be an identity, provided l=A{x-^') + B(x-oi') (9), i.e. provided A+B = 0, A^' + Ba' = -1 (10), ^=7^" ^ = -7^' ^"^- Hence f daj _ 1 j /" dx _ f dx ) ]{x -a'){x- /3') ~ a' - ^'\]^^^' j^^j = Z3^'l°g^3^ (^2), which agrees with the last form in (7). When we have once learned that the two sides of (8) can be made identical, the proper values of A, B are most easily found as follows. We first multiply both sides of the identity by a; — a', and afterwards put x = a'. This gives the value of A. Again, multiplying both sides by a; — /8', and afterwards putting X = /S', we find B* i* dx f dx 2 , , a; - i „ 1 /• ' provided ax + b = A(x — ^') + B(x — a'), i.e. provided A+B = a, Al3' + Ba' = -b (18), . aa'+b „ a0'-\-b .^^. ^=73^" ^^W^' (i9>- k 74] INTEGRATION. 175 It is unnecessary, however, to go through this work in every case, as the values o{ A, B can be found more simply by the artifice explained on p. 173. The integration of (17) then gives I (^_;,^(t^_^^) dx = A log (^-a') +B log {x-0'). . .(20). 2x — 3 Ex. 5. To integrate -. _. , ' — rv . ° {x-%){x + V) A B A Bsuming that this = ^ h =- , tJC — £l u3 T J. we find -4 = J, B=^. The required integral is therefore J log (k - 2) + 1 log (a; +1). EXAMPLES. XXI. C\ +x 1. [^ „ dx = tan-' X + log ^(1 + 3?) 2. |j^^=K-log^(l+a;^). t dx _ . 2a; + ] "• jl+3a! + 2a^~ ^^"^TT iV + a; + 1 , , /a , V 2 2a; — 1 8. -= =-c?a3 = a3 + log(a;'-a!+l) + -75 tan-i — ^^- : Jx'-x + l ^ J^ ij^ 9. /(f^''^=^"^('"-^)-^ 1 76 INFINITESIMAL CALCULUS. [CH. V "■ l {x -1)"(i _ 3) '^ - =" - 3 log («= - 2) + 8 log (x - 3). 12. j'^^^dx = x + {y5 + l}log(2x-^5-\) -(|V5-l)log(2« + V5-l). ^^- /t:5" c?a; = »: + K + la^ + ... + gT;,^"^""'- 14. } ' dx ==x-ia? + W-...+ (-)™-i ^r-!-^ ar"»-'. aa; + 6 75. Form / . „ , d =^ • A somewhat similar treatment can be applied to functions of the above type. 1°. If A be positive, the form is equivalent to "" + ^ (1). '^{x'' + px + q) Consider, in the first place, the form ^/{x'+px + q) ^ -*• By completing the square, the expression under the root- sign may be put in one or other of the shapes (x - ay ± ^. Now, by Art. 72, {N), (0), dx These functions have the alternative forms, ^^^ / vf(^-«Vm ="^«h-'V ^*^- , x-a + i^{{x-aftP'} log ^ -• 75] INTEGRATION. 177 log^-'^+V(g'+^^+g) (5;; cf. Art. 23. In the more general case (1), we assume ax + b = \ (x + ^p)+fj, (6), which is satisfied by \ = a, fi = b - ^pa (7). Hence r ax + b J _■. [ a' + jp 7 r dx J 'sJix''-\- px+q) *~ J^/ix'+px+q) * ^ J \/{x''+ px+q) (8). The former of these two integrals is obviously equal to \/(x'' +px + q), and the latter has been dealt with above. 2°. We will next suppose that, in the form placed at the head of this Art., the coefficient A is negative. Without loss of generality we may put it = — 1. Consider, first, the function 1 .9) \/{q+px-x^) ^ '' Unless the quadratic expression be essentially negative, in which case the function would be imaginary for all real values of X, it can be put in the shape ^-{x- ttf. Now f-TToT-^ ^,=sin-^" (10), In the more general case of the function ax + b sj{q ^-px-x^) ^ '' we assume ax + b =X{\p — x) + ij. (12), or \ = -a, fjL = b + ^pa (13). !•■ 12 178 TNFINITESIMAI. CALCULUS. [CH. V Hence / • ax + b J f jp - a; , ( dx J ^(q+px-x') '*"' - ^J V(g +px -a^r'"^ ^ j s/{q+px-a?) (14). The former of these two integrals is equal to ^J{q + px~ a?), and the latter has been treated above. Ex. 1. f 1+'" ^-,- /■J?±il±L f?a; c^ - /V(l -«:-«=') '^*'"'*i^/{|- (a! + in = -V(l-«=-«=^) + isin-'^ ^x 2 / iB+i J _ /"_^±i)_±i_ J 7 V(a'' + a'+l) ;V(a^ + a;+l) = V{a^+a!+l) + Jsinh-^^ = 7(a!' + a; + 1) + 1 sinh-' ' . '-'■ u(:^>-iw^^ V(l-a^) C xda. _ [ dx r xdx -jj(i-x')-j:jo^ 75-76] INTEGBATION. 179 EXAMPLES. XXIL 5. j^j-p-m = <=o«h'' (s^' - !)• J J{x{x-1)} ^ ' „ [ dx 1 . , _, 3aj + 1 O. / —m ;; 1;-^ = — r„ Sinh ^ 8 r dx J ja + 2x+: r tfo; 1 . _j 3a; - 1 JV(l + 2a!-3a;^) = 73'''' ^2~ ' f ^/" , = cos-'(l-^). j ^(2asa3 - x^) \ a] ^{^ax — a?) 0. I /( jdx= J{x{a-x)} + ^asm~' 2*— a 76. Change of Variable. There are two artifices of special use in integration ; viz. the choice of a new independent variable, and the method of integration ' by parts.' To change the variable in the integral M =/<^ (x) dx • •(!) from X to t, where a; is a given function of t, we have, by Ai't. 39, du dudx , , .dx ,„. -dt^diTt = '^^''^di (2)' 12—2 180 INFINITESIMAL CALCULUS. [CH. V and therefore, by the definition of the inverse symbol /, u Hence j(oo)da> = j(a>)^^dt (3)* Conversely, whenever a proposed integral is recognized to be of the form j^(w)^^«' ^*^' we may replace it by J(f)(u)du (5), which is often easier to find. The following are important cases : 1°. J{x + a)dx = f(j>(u)du (6), where u = a! + a. 2°. J(ka!)da; = ~J(l>(u)du (7), where u — hx. These results have already been employed in Art. 73. 3°. S^{pif)xdx-=^j^iu)du (8). where u = aP. The following are examples of (8). „ ^ [ dx [ xdx } xii+x')" j ^^jr+^) = log X J(l+a?)- dx * Hence the rule : After the sign J replace dx by -=- dt. 76] INTEGRATION. 181 xdx ■i/(.-^-.40''" 1 1 "-1 ar^-l = ilog '-'■ /ra=^/i^=i-- _ . t xdx ^ [ du . . , u = i sin ' - . ' a? The student will, after a little practice, find it easy to make such simple substitutions as the above mentally. 4°. Occasionally the integration of an algebraical func- tion is facilitated by the substitution x^.ljt, d^/dt = -llt\* Thus = sinh~^ at a = — sinh~' - a X - - 1 '" {()\ Similarly, | — rr- -r = — cosh"' - = a^"g a + V(a--^'') ^^^)' * The substitution is equivalent to writing dt . dx - - for — t X 182 INFINITESIMAL CALCULUS. [CH. V and I — -fT-^ — rr = --sin-'- (11). More generally, the integral ]{x + a)^/{Ax-' + Bx + C) ^^^^ is reduced by the substitution x + a=ljt to one or other of the forms discussed in Art. 75. Again, the substitution x = 1/t gives f dx __ [ dt _ r tdt J(a' + aPf~ Jt\a^ + t-'f~ i(l + aT)3 1 1 , 1 X (13). aVCa" + «''') ](a?- a?f " ~ o^ VC*' - a") ^"^''^' ^^^^°™ /(l^l^t (1^) can, by 'completing the square,' be brought under one or other of the preceding cases. 77. Integration of Trigonometrical Functions. 1°. ha,nxdx=\—^ — dx J Jcosx fd (cos a;) , = — I — ^ = — log cos X J cos X ° = logsec« (1). Similarly /cot a; da; = log sin a; (2). 76-77] INTEGRATION. 183 Again, by the same artifice, fsina; , _ fd (cos os) J cos° aa J cos'' a; = = secfl! (3). cos a; ^ In a similar manner I ■ „ dao = — cosec x (4). Jsm^a; ^ ' Cf. Art. 38, 2°. dm 2°. f dx _ f jsina;~ji 2 sin ^OB cos ^x _ ^ /sec^ ^ajrfa; _ fd (tan ^a;) '^ J tan Aa; j tan ^x J tan Ja; = log tan ^a; (5). From this we deduce dx f dm ^C Jcosx jsir ■ = log tan (Jtt + ^x) (6). /sin(^7r + a;) The formulae (1) to (6) rank almost as standard results, and should be remembered. 30 ( ^ -f ^ ■ Ja + 6cosa; j(a + 6)cos*^a; + (a — 6)sin''ia; _ f sec' ^xdx ,h-. ~' J(a + b)+{a-b)ta,u^x ^ ^' If we put tan ^x = u, this takes the shape du 'Iw- •, + b) + (a-b)u' ^^^' and so comes under one or other of the standard forms (/), (P),(Q)ofArt. 72. Similarly, with the same substitution, f ^ _,« [ ^'^ (9). Ja + 6sina! ]a + 26m + au^ 184 INFINITESIMAL CALCULUS. [CH. V Ja=cos''a! + 6''siii=a; Ja= + 6-tan=a! ^ '' If we put tan x = u, f du 1 [ d (bu) 1 , ,bu = -7 taii""^(-tana;J (11). The analogous results involving hyperbolic functions may be noted. We easily find / tanh iB (fa; = log cosh K, / coth a; eic = log sinh a; . . . (12), I — r-r— dx = - secha;, / . , . dx = — cosech a:... (13), ] cosy X J smh' £C :logtanh|a; (14), h ' sinh X dx - /■ e'dx l^x='j^r''^--''' (^^)- Similarly the forms C dx r dx J a + b cosh x' ) a + l ■ b sinh X can be integrated by the substitution tanh \x = u. 78. Trigonometrical Substitutions. The integration of an algebraic function involving the square root of a quadratic expression is often facilitated by the substitution of a trigonometrical or a hyperbolic function for the independent variable. Thus : the occurrence of >J{a? — x^) suggests the substitu- tion a; = ci sin 0, or x = a tanh u ; that of \/{oi? — a^) suggests x = a sec 6, or x = a cosh u ; that of tj{x^ + a") suggests x = a tan ^, or a; = a sinh u. 77-78] INTEGRATION. 185 Ex.\. Tofind fj{a,'-x')dx (1). Putting x = a sin 6, dx = a cos 6 dO, we find fiji"''' —x')dx = os^/cos" 6 dd = K/(l+cos2e)c^0 = Ja^ (0 + J sin 26) = !«'' sin-i - + ^ V(a^ -x'} (2). Cli Ex.2. Tofind ^M±p^ (3). Putting x = a sinh u, dx = a cosh u du, we obtain the form J'coth'' u du, which = _[( 1 + cosech* u)du = u- coth u = sinh-g-'>/('^+"') (4). ^jq. 3. To find [,-= ^ dx J{l-x)J{l-x')- If we put X = cos 0, dx = — sin 6 d9, the integral becomes jl-cosfl ^Jsm^^fl -^ EXAMPLES. XXIII. „ Cx^dx ,, l+sc' ia?dx ,^ _, , 3. I ° dx = ^ (log a;)". 4. /sin a; cos xdx = ^ sin'' a: 186 INFINITESIMAL CALCULUS. [CH. V f COS a: , , ,, . . 6. I =-^ — -. — da; = log (1 + sm x), j 1 + sm a; °^ " /■ sin a; , 1 , / , \ I 5 dx = — T- log {a + cos a;). J a + b cos a; 6 ^ ' 7. /sin a; cos' a; dx = — \ cos* a;. „ /■ sin X cos a; . 1 , , . , . » v 8. / 1 , . „ dx=^ ^r-jr r log (as cos- a; + 6 sin' a;). ] a cos'' X + sin- x 2 (6 — a) ^ ' 9. /tan' xdx = ^ tan" a; + log cos x. 0. I — J— «&! = Jsec'a;. 11. /sec* a; ^0; = tan a; + i tan' a;. / COS £C 2. /(sec a; + tan x)dx = log = — sin a; /(sec a; — tan x)dx = log (1 + sin a;). { dx , ( dx I T = cosec X — cot a;, I = — = - cota; — cosec x. J l+cosx / 1 - cos a; . C dx f dx ^ .4. 1 = : — = tan X — sec x, 1 = ; — = tan x + sec x. j 1 + sm a; / 1 — sin a: - f dx O- |-^-5 5— = tan a; - cot a;. J SI " I sm'' X CDS'* X „ f dx , , 6. I -; 5— = sec X + log tan ix. j Sin a; cos" x o i 7. I -; ^ = A sec" a; + log tan x. j sin X cos" a; ■^ ° ®- j l+tana !^^'""^^^"^^''"^'"'^^''^'"^' 9. It j-= -T^tan"' ( -^tana;). ;l+cos'a! ^2 \^2 y 20. /a;V((*= + ar')da: = J(a» + ar')*. /• c^a: 1 a;" ja!(ai"+l)~w ^^a!'' + l* 22. Evaluate J^{x' + a^dx and j^(a?-a^)dx by hyperbolic substitutions. 23 / • dx _ J{l+a?) ] 0^^(1+9?) X ■ 79] INTEGRATION. 187 ^'- \a dx _ ^(a!' — a?) a? J{a^ — a?) a'x ''■ /;/^-(^^-^»')^^('^^^^>- 79. Integration by Parts. The second method referred to in Art. 76, viz. that of ' integration by parts,' consists in aa inversion of the formula d . ^ dv du ,-, ^M=«^+«^ (!>' given in Art. 37. Integrating both sides, we find f dv , f du , uv= I u-j- dx + j v-j- dx, whence lu-j-dx = uv— Iv-j-dsa (2).* This gives the following rule : If the expression to be integrated consists of two factors, one of which (dv/dx) is by itself immediately integrable, we may integrate as if the remaining factor (u) were constant, provided we subtract the integral of the product of the inte- grated factor (v) into the derivative {dujdx) of the other factor. A very useful particular case is obtained by putting v = x, in (2). Thus \udx = xu— I a; ,- dx (3). The following are important applications of the method. 1°. \\ogxdx = x\ogx—\x.-dx — xlogx — x (4). 2°. To find J>^(w'-iif')dx. * If we write v for dvldx, and therefore D~% for v, this takes the form D-i {wv) = uD-h) - D-' {Du . D'^v). 188 INFINITESIMAL CALCULUS. [OH. V Putting u = i^{a? — ar") in (3), we have J V(a° -«?)dx=x V(a' - a^) +J ^^^, _ ^,^ (5). But /V(«^-«^)'^«'=/;^^)^ _ r da; _ r (ii?dx ~" JV(a''-a^')~iv(a''-«') = a=sin-^-f^P^, (6). Adding to the former result, and dividing by 2, we find fV(a° -x')dx = \x V(a' - «') + i «' sin"' - ... (7) ; cf. Art. 78, Ex. 1. In exactly the same way we should find fvCa" ■\-oi?)dx=\x sj{vi? + x") + Ja'' sinh"' - . . .. (8), fv(ar' - a») rfa; = ^a; V(«' - »') -i«' cosh-' - .. . . (9). 3°. To find the integrals P = /e"* cos ySajda;, Q=fe^ sin /Sirda; (10). Putting M = cos /3a;, v = -ef^ iu (2), we find 1 n P = ~ef^cosBx- I -e'^J-^ sin 8x)dx a J a ^ = -e"'cos/3a; + -Q (11). Similarly 1 n Q = - e"* sin ;8a; - - e"^ . ^ cos ^xdx a Jo. = ~e^sinl3x--P (12). 79-80] Hence and therefore INTEGRATION. aP -/8Q = e°* COS /3a!, 1 l3P + aQ = e"* sin ^oo J o 7 Ti iS sin iSaj 4- a COS 8x „, \ e^ cos 0xdx = P= ~ „ . ^„ '— e^ 0? + ^ ^ ■ n 7 /^ a sin 0SB — B COS ^se „, e^ sin ^xdx =Q= „ . '^„ — !— - e°* 189 .(13), (14). 80. Integration by Successive Reduction. Sometimes, by an integration ' by parts,' or otherwise, one integral can be made to depend on another of simpler form. 1°. Let Un=Jai^e'^dx (1). 1 f 1 We have u^ = - e"* . «" — I -e"". nx^^dx a ] a (2). 1 n : - a;"e°* — m„ If n be a positive integer, we can by successive applications of this formula obtain m„ in terms of '«o.= \e.'^''dx, =-< .(3). Un = fx^e~'"dx .. Un = -x^e-'' + nUn-i Ex. 1. Thus, if we have For example, Mj = - ar'e-"' + 3u^ = - ar'e-* + 3 (- sT'e""^ + 2u,) = — 3?e~''- 3ar'e-'"+ 6 (-xe'" + u^, or /a-' e-''dx = -{a?+ iii? + 6a; + 6) «-". 2°. Let (4), (5). We find Un = /«" COS ^xdx, \ Vn = /«" sin ^xdx ) ^ '" «" — I 3 sin ^x . nx'"''^dx M„ = ^ sin ^x 1 ■ /I „ w = 3Sin/3a;.a!"-^%_,. (7), 190 INFINITESIMAL CALCULUS. [CH. V and Vn = — a costal, x'^— If— -= cos y8a!J.7ia!"~' da; 1 n = --gCos/3a;.a5» + ^M^i (8). If w is a positive integer, these formulae enable us to express w„ and Vn in terms of either u^ or v„, which are known. Ex. 2. Thus, if ;8= 1, we have M„ = a!''sina! — «t)„_i, Vn = -x'^ coax + nu„_^ (9). For example, u^ = 3? sin X — 3i)2 = n? sin x—Z {—a? cos as + 2z«i) = a? sin X + Za? cos a; — 6 (aj sin x — vj, or Ja? cos xdx = (a? — 6a;) sin a; + (Zs^ — 6) cos x. 3°. If M„=/tan»0d0 (10) = /tan"-2,9(gec2^_l)d0 =/tan»-2 0d(tan 6')-/tan"-''^d^, we have «„= 7 tan"~> — «„_, (11). n— i. Hence if « be a positive integer, «„ can be made to depend either on t , we find A-C=0, whence ^ = 1. An equivalent method is to clear of fractions and equate the coefficients of «*. Again, we might assign some other special value to x ; for example, putting x = -l, we find -A+B+lC=l which, combined with the previous results, gives A = l. Hence / „,.'" — \=/(-+-j + .i )dx jaf(l-a!) J\x a? 1-xJ = loga:---log(l-a;) (5). ^^■'- '''^-' J ixXi^'-sy ^^ («)■ . 2a! + 1 A B C ,„, The short method of determining coefficients gives -4+1 _ 3 ^6+1 7 ^-^_2-3)2- 25' S + a^S" Also, multiplying by x, and then putting re = 00 , we find A+B = 0, or B = -^. 83-84] INTEGRATION. 197 The integral is therefore -^\log(a!+2) + ^«,log(a!-3)-g^^ (8). ^..3. Toiind /^, (9). We recall Art. 76, 3°. Regarding x^/(x'+ 1)^ as a function of ar*, we find (by inspection) a? _ {x'+l)-l _ 1 ]__ (x' + lf' (a;^ + l)^ ~x'+l {x' + lf „ f 3?dx Cxdx f xdx ^^"""^ ) (S^Ti7 = j^^Ti ~ ic^n? 84. Case of Quadratic Factors. The preceding methods are always applicable, but if some of the roots oi f{x) = are imaginary, the integral is obtained in the first instance in an imaginary form. If we wish to avoid altogether the consideration of imaginary expressions, we may proceed as follows. It is known from the Theory of Equations that a poly- nomial f(x) whose coefficients are all real can be resolved into real factors of the first and second degrees. Then, in the resolution of the function ^^ (1) into partial fractions, we have (a) for each simple factor x— a which does not recur, a fraction of the form ^a <2); (6) for a simple factor x-^ which occurs r times, a series of r fractions, of the form x-^'^(x-l3f'^'"^(x-^y ^'^^' 198 INFINITESIMAL CALCULUS. [CH. V (c) for each quadratic factor af'+px + q which does not recur, a fraction of the form 0^ + I> (4,). a^+px + q ^ ^' (d) for a quadratic factor a^+px+q which occurs r times, a series of partial fractions, of the form a?+px + q {a?+px+qy " (ar'+joaj + g)''"""^ ''■ It is easily seen that in this way we have altogether just sufficient constants at our disposal to effect the identification of the function (1) with the complete system of partial fractions, by the method of equating coefficients. *It only remains to shew how the indefinite integral of the partial fraction G,x + D, ,g, {a?+px + q)' ^ ' can be found. The case s=\ has been treated in Art. 74, and the general case can be reduced to this by a formula of re- duction. In the first place, we can find A,, /a so that G,x + D, .^ 2x+p ^ fi „, (a^+px + q)' (x'+px + q)' {a^+px + q)' '' viz. we have \ = \Ga, ii. = Di — \pGi (8). The integral of the first term on the right-hand of (7) is ~s^'{x^ + px + qy-^ ^^^' and it remains only to find J{a?+px+qy *"■ J(i^T^« ^^^^' where i = x+^p, c = q-\f (11). • The investigation which follows is given for the sake of completeness, hut it is seldom reqaired in piactice. The student will lose little by postpon- ing it. Another method of integrating expressions of the type (10) is indicated in Ex. 2, below. / 841 INTEGRATION. 11)9 Now, by differentiation, we find dt{t^ + cy-^ (f + c)'-^ ^ '(t' + c)' = -(2^ -3) (TCP -^(2*- 2) (^» (12)- Hence, integrating, f dt _ 1 «^ 28-3 1 r dt *"" j (?+7)» ~ (2s - 2) c (i'' + c)»-' "'" 2s - 2 ■ c j (d' + c)»-i (13). Returning to our previous notation, we have dx _ 1 ^+ ip {ix? +px + qy~ 2(8-1) {q- {}}') {aP +px + q)'-' 2s— 3 r dx ,^., "^ 2{s-l) (q-lf) J {x'+px + qy-'"-'^ '' which is the formula of reduction required. By successive applications of this result, the integral (10) is made to depend ultimately on x'+px + q ^ '' which is a known form (Art. 74). ^x.l. To find Z^,^ (16). The denominator has here two quadratic factors, ay' + x+ 1 and ai?-x+\, which are not further resolvable. We therefore assume, in conformity with the above rule, 1 _ Ax + B Gx^D a^ + !K^+l~a;'' + a; + l"^£B^-a; + l ^ '' or \ = {Ax + B){a?-x + \)\{Gx\B){^^x+\). Equating coefficients of the several powers of x, we have ^ + C = 0, -4 + C+^ + i) = 0, h dx 200 INFINITESIMAL CALCULUS. [CH. V Hence ^ = -C=J, B = D = \ (18). The integration can now be effected by the method of Art. 74. We have C dx _. f x+l , C x-1 _^f(2x+iy^ f{2x-l)-l -*) a?+x+l *" 4j x'-x+l ''* = J log (ar" + a: + 1) - 1 log (a;" -a: + 1) ,, oj' + aj+l 1 / ,2a: + l ^ ,2a!-l\ = ' ^°S ^Z^:^ + 2;;3 (tan- ^73- + tan- ^^) J, ar' + ai + l 1 ;^3a; Ex.2. To find J^^, (20). This conies under (14), but may be treated more simply as follows. If we put X = tan 0, we get = 1/(1 + cos 20) d9 = l9 + l sin 2d = itan-a! + i^ (21). l+a?" EXAMPLES. XXV. X 3. ja:(l-a?)-^°SV(l-'«')" U{x-l){x+2)'''" °^ a^ (a; + 2)* r a:^cZa; (a; -1) (a; -2) (a; -3) = ^log (a;- 1)- 4 log (a; -2) +|log (a; -3). 84] INTEGRATION. 201 4 r 2a;- 3 J (a3»-l)(2a! + 3) = I log (a? + 1) - ^ly log (a; - 1) - i^ log (2a! + 3). = f xdx a? -2 ^- ia?3^32=ii*'g^rn:- 6 ■ ( '^ 1 ^11 »:-l • ya,'' + 3a!^-4-37^T2)"^^^°S^:;r2- j(a:-a)(a:-6)(a;-c)-(a-6)(a_c)^°S;(«'-a) 8 r dx 1/1 « 1 x\ i(a^ + "^c^ 1 ai'i + as - log - - a') {a? + b') ~2(b'- a'} ""^ IfTb^ ' ■IQ /■ a!^cfe 1 / jg jg-. ■ i(a;= + a»)(a:^ + 6^) = S^Tj^ (^« tan-> - - 6 tan-> ^ j . ■, 1 r a;'fi?a; 1 ,„ r Jgt^a; 2 a; + 2 ■ y(a!+l)(a; + 2)=-~^^T2 + ^°S^:n:- '^- ](»,•+ l)(x + 2)^ = ST2+^°S(«'+l). 14 f ^^ 11 «=+ 1 1 1 Ifi f ^ 2a; -1 a; ^^- ;=^^(r3^=^irT^+2iog^— . ,„ [j^dx \ X x + \ Ifi i ^"^ iji '^-i 1 1 « 202 INFINITESIMAL CALCULTTS. [CH. V 19- 1 -^/ ^ ix = -5-a+ — log (a; +1) + log SB. f 3a!+2 , „, X 2 20. — j^ ^,cfo; = 21o.a; 7 + {x+\f "ai+l a; + l 2(a! + l)''' rdx_g..x-l3x 1 jB J(a!=^-1)»~"" ^^^+1 "^ 8a?^ ~i(ar'-l)»* /■ (ia; , , 1 + a: , , , nr r <^a! ,, a+xf 1 ^ _,2a:-l fajc^a; ,, l-aj + a;'' 1 , _,2a;-l 2^- /(r;^fT^=il°g(l + «=)+Tlog(l + a=»)-Jtan-a.. „„ /■ ar'da; , , x-1 J2 x 3°- /sriSr2=i°-"(^-^)-^^°^(^^^>- 32. I -J = — = rfo; = A log -5 zj . no f «^ X , 1 1 33. T-7, r\ = tan"' 33 + - - 5-5 . y a;* (1 + ar*) x 3af f a?dx _ l + 'ix' j(l+a^)»~"'4(l + ar')»" 85] INTEGRATION. 203 dx 2 + Sa:" r -J, 36, ^^- /^(fT^=-2f(ra)-**^'^''^- /• dx _ J^ 1 + x^2 + as' 1 , «! J2 85. Integration of Irrational Functions. The following are the leading results in this connection. 1°- In the case of an algebraic function involving no irrationalities except fractional powers of the variable, we may put a! = «™, dx/dt^mt^-'- (1), where m is the least common multiple of the denominators of the various fractional indices. The problem is thus reduced to the integration of a rational function of t. 2°. Any rational function of oc and X, where X = J{aa? + hx + c), the problem of finding JF{x,X)dx (5), where i'(a!, X) is a rational function of x and X, can also be reduced to the integration of a rational function. If a be positive, we may write X= Ja.^{x'+px + q) (6), where p = bja, q = cja. Now assume J{x^+px + q) = t-x, t^-q dx 'i{e + pt + q) ,„. whence ^ = 2«T7' Tt= {2t+pf ^''• and Jix^ + px + q)=t±f^ (8). It is evident that by these substitutions the problem is reduced to the integration of a rational function of t. If the factors oia?+px + q are real, say x'+px + q = (x — a){x — P) (9) we may also make use of the substitution x-p={x-o.)e (10), whence x = a + .^—j^, _=-^^— ^ (11) and ^{x'+px + q) = {x-a)tJ-^^ (12). If a be negative, we may write X=J{-a).J{q+px-a?) (13), where p = — hja, q = — cja. If the radical is to be real, the factors of q + px-a? must be real, for otherwise this expression would have the same sign for all values of x, and since it is obviously negative for sufficiently large values of x, it would be always negative. We have, then, q+px-3? = {x- a) {P-x) (14), These substitutions evidently render 85] INTEGRATION. 205 where a, j8 are real. If we assiune P-x = {x-a)f (15), we find a:-a. + ^^:^ dx Jiifi-c^ , wenna x-a+^^^„ ^^ - ^^^^^^ (It.;, and ^{q+pa^-^) = (a,-a)tJ^f^ (17). tly render Fi^x,X)% a rational function of t. The above investigations are of some importance, as shewing that functions of the given forms can be integrated, and that the results will be of certaia mathematical types ; but the actual integration, in particular cases, can often be effected much more easily in other ways*. We have had instances of this fact in the course of the Chapter ; and we add one or two further illustrations. Ex. 3. By rationalizing the denominator, we have = |(l+ai)t-|.rf. ^^•*- Ix.Z^-ir ^^^-^^^-'^'^'^ = ^x>-f^{x>-l)dx = ^x' — |a: J{x' - 1) + J cosh"' x. Otherwise, putting x = cosh u, the integral takes the form sinhwt^M r .. ■ L 7 ■ = le'^siimuau h cosh u + sinh u = 1/(1 - e-^) du = ^u + ie-'^, which may be easily shewn to differ from the former result only by an additive constant. * See especially the methods of Arts. 75, 76, 78. 206 INFINITESIMAL CALCULUS. EXAMPLES. XXVL 1. fxj{l+x)dx=^{l+x)i-%{l+x)i. 2- [-77 V 77 lx = -5-^—r^{i^ + 1')^-('o+b)^. J J(x + a)+ J{x + b) 3(a-6)^^ ' ^ ' ' ^- /;^^i = 2^+2 log (1-^4 j(l-a!)Va: ^l-V": « /(r7.i!a^,=-v^"-V(T)- 8. /^,*, = 2V..log^;. q [ ^=« _ 1 V(l + as) - 1 "■ JxJ(l+x) °^J{l+x)+l- ^"- ja;V(l+a;)~ x * '°^ V(l + ic) + T /vpTl) = ^^^ V('^ + 1 ) - i ®i°^~' «=• ar'cfo (r)V)t~~N/(l + ar') ,„ /• dx _J_ _, xJ2 11 ^ = TTi ^ + sinh-' X. 14. i (1 - a!»)7(l + a^) 2^2 ^ «V2 - ^(1 + a?) ' EXAMPLES. XXVn. 1. A particle moves according to the law ds prove that the space described before it comes to rest is u ^j^g. INTEGRATION. 207 2. If a point start from rest at time < = and move with a constant acceleration, and if v^ be the velocity after any interval and V the mean velocity in this interval, then «' = K- 3. If, with the same notation, the acceleration vary as f, then 1 4. A particle moves according to the law ds prove that the space described from time t=0 until it first comes to rest is vjn. 5. If the velocity of a particle moving in a resisting medium be given by prove that the particle never attains a distance vjk from its position when < = 0. 6. A particle moves according to the law -r. = '"i,e "' cos nt; prove that the space described from time t = until it first comes to rest is 7. If the angular velocity of a body rotating about a fixed axis be given by ^ = 2»i sech 7)t, at prove that ^ = 4 tan-^e"' — jr, supposing that 6 vanishes for t = 0. CHAPTEK VI. DEFINITE INTEGRALS. 86. Definition, and Notation. Let y, = {x)icc (4), 8a; standing for the increments h^, h^, h^ of x. The limiting value (when it exists) to which this sum converges, as the increments 8* are all indefinitely diminished, and their number in consequence indefinitely increased, is called the 'definite integral' of the function {x) between the limits a, and b*, and is denoted by I ydx or I <\>{x)dx (5), J a J a • It is a little unfortunate that the word 'limit' has to be used in several different senses. The word 'terminus' might perhaps be substituted in the present case. 210 INFINITESIMAL CALCULUS. [CH. VI the object of this notation being to recall the steps by which the limiting value was approached*. The connection of the notation (5) with the notation for an indefinite integral used in the preceding chapter has of course yet to be explained. See Art. 92. Problems in which we require the limiting value of a sum of the type (3) occur in almost every branch of Mathematics. The area of a curve has already been used as an illustration ; other simple instances are : the length of a curved arc, regarded as the limit of an inscribed (or circumscribed) polygon; the volume of a solid of revolution, and so on. These wUl be considered more particularly in Chap. vii. Again, in Dynamics, the ' impulse ' of a variable force, in any interval of time, is defined as the ' time-integral' of the force over that interval ; viz. if F be the force, considered as a function of the time t, the impulse in the interval t^ - <, is the limiting value of the sum F^T^^Fir^+...+F^t^ (6), where Ti, Tj, ..., t„ are subdivisions of the inteiwal t^ — t^, such that T1 + T2+ ... +T„ = «i-<„ (7), whilst ^1, F^, ..., F„ denote values of the force in these respec- tive intervals. Hence, in our present notation, the impulse is f'Fdt (8). Newton's Second Law of Motion asserts that the change of momentum of any mass (m) is equal to the impulse which it receives, or ■mv, — mv, f'-P'dt (9), J in where «„, v^ are the initial and final velocities. Again, the work done by a variable force is defined as the space-integral of the force. If F denote the force, regarded now as a function of the position («) of the body, the work done as s changes from «„ to Sj is rFds (10). * The symbol J is a specialized form of S, the sign of summation employed by the earlier analysts. The mode of indicating the range of integration was introduced by Fourier. 86-87] DEFINITE INTEGRALS. 211 For example, the work done by unit mass of a gas as it expands from volume v^ to volume i^i is fV« (11). if p be the pressure when the volume is v. This is seen by supposing the gas to be enclosed, by a piston, in a cylinder of sectional area unity. The graphical representation of the integral (10) or (11) is frequently employed in practice. Thus, in the case of (10), if a curve be constructed with s as abscissa and F as ordinate, the work is represented by the area included between the curve, the axis of s, and the ordinates corresponding to s„ and Sj. This is the principle of Watt's indicator-diagram*. 87. Proof of Convergence. Whenever the sum S has a definite limiting value, in the manner above explained, the function <^{x) is said to be ' integrable.' It may be shewn that every continuous function is integrable in this sensef, but as regards the formal proof we shall confine ourselves to the particular case where the range of the independent variable can be divided into a finite number of intervals within each of which the function either steadily increases or steadily decreases. This will be sufficient for all practical purposes. Before, however, introducing any restriction (beyond that of finiteness) we may note that two fixed limits can be assigned between which 2 must necessarily lie. For if A and B be the lower and upper limits (Art; 24) of the values which the function ^ (as) can assume in the interval 6 — a, it is evident that 2 will lie between A(hi + h + ... + h„), = A(b-a), and B{hi + h3+...+hn), = B{b-a). We will now suppose, for definiteness, that b>a, and * See Maxwell, Theory of Heat, c. v.; Eankine, The Steam-Enqine, Art. 43. t It is not implied that a mathematical formula for the integral can be found. 14—2 212 INFINITESIMAL CALCULUS. [CH. VI that ^(oo) steadily increases as x increases from a to b. Consider any particular mode of subdivision "1> "^> «7» of the range b — a, and let 2 = yA + y^h^ + ... + ynK- •a). •(2), where, as in Art. 86, y^ denotes some value which the function assumes in the interval A,. Now if in (2) we replace yi, ya, ... y^ by the values which the function has at the beginnings of the respective intervals, none of the terms will be increased ; and if the resulting sum be denoted by 2', we shall have S>2 (3). Again, if we replace y^, y^, ... y^ by the values which the function has at the ends of the respective intervals, none of the terms will be decreased ; hence if the resulting sum be S", we shall have %"■>(.% (4). Fig. 47. 87] DEFINITE INTEGRALS. 213 In Fig. 47 the quantity S' is represented by the sum of a series of rectangles such as PN, and S" by the sum of a series of rectangles such as SN. Hence the difference 2" — %' is represented by the sum of a series of rectangles such as SR. The sum of the altitudes of these latter rect- angles is KB — HA, or (j)(b)-(j> (a), and if k be the greatest of the bases, i.e. the greatest of the intervals (1), we shall have 2"-S>A;{<^(6)-.^(a)} (5). Now, considering allpossible modes of subdivision of the range b — a, the sums Z*, being always less than B(b — a), will have an upper limit, which we will denote by S', and the sums S", being always greater than A{b — a), will have a lower limit, which we will denote by S", and it is further evident that S" -^ S'. It follows, from (5), that the difference ;S" — S' must lie between and k {(b)- ,]>(a)} (6); hence by taking k small enough we can ensure that | S — /S | shall be less than any assigned quantity, however small*. A similar proof obviously applies if the function ^(x) steadily decreases throughout the range b — a. It follows that the final result also holds when the range admits of being broken up into a finite number of smaller intervals within each of which the function either steadily increases or steadily decreases. See Fig. 48. It has been supposed that b > a. If 6 < a, the intervals hi, hi, ..., hn will be negative, but the argument is sub- stantially unaltered. * The proof is a development of that given by Newton, Principia, lib. i., sept, i., lemma iii. (1687). It would be easy to eliminate all geometrical considerations and present the argument in a purely quantitative form. 214 INFINITESIMAL CALCULUS. [CH. VI Fig. 48. 88. Examples of Definite Integrals calculated ab The meaning of a definite integral may be further illustrated by the study of a few cases in which the limiting value can be calculated from first principles. The methods to which we are obliged to have recourse for this purpose will at all events enable the student to appreciate the enormous simplification which was introduced into the sub- ject by the invention of the special rule of the Integral Calculus, to which we afterwards proceed (Art. 92). Ex.1. Tofind I xdx (1). This is equivalent to finding the area of the trapezium PABQ in the figure, where OA = a, OB = 6, and OPQ is the straight line y = x. Fig. 49. 87-88] DEFINITE INTEGRALS. 215 Take Aj = Aj = = A„, =h, say, so that nh^b — a, and let y^, y^, y„, be the values which the function to be integrated has at the beginnings of the successive intervals, viz. 2/i = «. yi=a + h, y^ = a + 2h, y„ = a+{n-l)h. Then % = ah+ {a + h)h + {a + 2h) h+ ... + {a + {n~\)h]h = nah+{\ + 2+ ... + {n- l)}h^ = nah + ^{n- l)!i? = a{b-a) + i(l-lyb-af. When h = 0, we have n = co , and the limiting value of the above expression is a{b-a) + ^(b-aY=^ {b°' - a'), rb Hence I xdx--=^(b^-a') (2). Ja Ex.% Tofind I a?dx (3). Ja This is equivalent to finding the area included between the parabola y = a?, the axis of x, and the ordinates x = a,x = h. /) Fig. 60. Putting hi = hi= ...=hn, =h, yi = < yi = (a + hf, y, = {a + 2hy, y„ = {a + (n-1) A}", we have % = a,% + (a + hf h + (a + 2hy h + ... +{a + (n-l)h}' = na^h + 2{l + 2 + ...+{n-l)}aJi'+{l'+2^+...+{n- If} Jv> = na'h + n{n-\)ah^ + \{n-\) n (2n - 1) A' = «^(J-a)+(l-i)«(6-«r-.i(l_l)(l_i-)(J_a)3 Hence 216 INFINITESIMAL CALCULUS. [CH. VI The limiting value of this f or w = oo is a\h-a) + a{b- af + \{h- af, or J (6»- a'). \^a?dx = \{¥-a?) (4). Ja Ex.Z. To find \ ^dx (5). Ja This is the limit, f or w = oo , of S, = W» + ;te«=("+") + /te«<«+'*) + . . . + Ae*(»+("-W, where A. = (6 - a)/n. Now S^Ae^^O+e^^ + e^+.-.+ef"-')"*) /."»* _ 1 hh 1 = ^'"-V-^=e-^-|(^-^) (^)- And, by Art. 29, the limiting value of the first factor, for kh = 0, is unity. C" 1 Hence I e'"dx = j(^-^) (7). Ja * /■/3 ^a;. 4. To find I sinxdx (S). This is the limit of 2, = {sin (a + |A) + sin (a + |A) + . . . + sin (j3 - |/t) + sin (j8 - JA)} 7t, ■where A = (/8 - a)/«, and the values of sin a; at the middles of the respective intervals have been taken. Now - '^^ . 2 = 2 sin J/t sin (a + JA) + 2 sin,M sin (a + |7t) + ... + 2 sin JA sin (/? - f A) +^in JA sin (J3 - JA) = cos a — COS (a + A) + COS (a + h) — COS (o + 2A) + + COS (|8 - 2A) - COS (y8 - A) + COS (/8 - A) - cos /3 = cos a — cos /3 (9). 88-89] DEFINITE INTEGRALS. 217 Hence, proceeding to the limit {h = 0), we find sina;aa! = cosa- cosyS r .(10). 89. Properties of 1 4>{x)dx. J a 1°. If we compare the integrals I tj}(x)diB and I 4^{x)dx, Ja J b we see that they may be regarded as limits of the same summation, with this difference, that in one case the incre- ments h^, hi, ... hn o{ X, which make up the interval b — a (or a — b) have the opposite sign to that which they have in the other. Hence r4,{x) = -fcl,ix)dx (1). J b J a 2°. Again, it follows from the definition that re rb re I (x) dx, being intermediate in value to A{b — a) and B(b — a), must be equal to G(b — a), 218 INFINITESIMAL CALCULUS. [CH. VI If, as we ■where G is some quantity intermediate to A, B. suppose, ^(is) is continuous, it assumes within the range h — a all values intermediate to A, B. Hence there must be some value (c) of X, between a and b, such that (x)dx = (b — a)(})(a + 6b — a) (3). 4°. More generally, if u, v, y be three functions such that for values of x ranging from a to b, then the integral u>y>v. I ydx. J a ,.(4), .(5) ..(6). will be intermediate in value to j Mcte and I vdx , Suppose, first, that b>a. We have (•6 rb rb I udx — I ydx =1 (u — y)dx. J a J a J a In virtue of (4), every term of the sum, of which the latter integral is the limit, will be positive. Hence I ydx\ vdx (8). J a J a If 6 < a, the inequalities, in (7) and (8) must be reversed. EXAMPLES. XXVIII. 1. Prove by the method of Art. 88 that Ja 2. Prove from first principles that I cos xdx = sin /3 — sin a. 3. Shew from graphical considerations that I hl}(x)dx = k I (aj) dx, Ja Ja rb rb-a I (l>{x)dx= I (a! + a) dx, Ja Ja rb \ rkb I (kx) cfe = T / <^ (a) dx. Ja "^ Jka 4. By dividing the range b — a into n intervals such that the abscissae of the points of division are in geometric progression, and finally making n infinite, prove that I x'^dx = (6^+i-a^+i). (Wallis' method.) 90. Differentiation of a Definite Integral with respect to either Limit. Let '=( 4>{x)dx (1). J a Evidently, 7 is a function of the ' limits of integration ' a, b, and will in general vary when either of these varies. 220 INFINITESIMAL CALCULUS. [CH. VI Regarding a as fixed, let us form the derived function of / with respect to the upper limit h. We have rb+Sb I + BI= ^{x)div J a /b rb+Sb )dx (2), by Art 89, 2°. Hence /•M-Si S/ = J (x)dx = hh.4>{h + ehh) (3), by Art. 89, 3°. This shews that S7 vanishes with Sb, so that / is a continuous function of 6. Also, since g = ,/,(6 + 0S6) (4). we have, on proceeding to the limit (86 = 0), Fig. 53. In the figure, OA = a, OB = 6, Bff = Sb, and 87 is represented by the rectangle having BB' as its base. 90-92] DEFINITE INTEGRALS. 221 In the same way, if we regard the upper limit h as fixed, and the lower limit a as variable, we find that / is a continuous function of a, and that J^-^^'^) ('^- 91. Existence of an Indefinite Integral. We can now shew that any function {a!) (2). For if we write ^(^)=j'(x)dx (3), J a the expression on the right hand is, by Art. 87, a determinate function of ^, and the investigation just given shews that it satisfies the condition t'(^) = «^(a (4). The lower limit of integration in (3) is, from the present point of view, arbitrary, and the function -yfr (f ) is therefore indeterminate to the extent of an additive constant. For, by Art. 89, 2°, the substitution of a' for a, as the lower limit in (3), is equivalent to the addition of /* J a' (f) (x) dx to the right-hand side. Of Art. 71, 92. Rule for calculating a Definite Integral. Whenever the analytical form of a function yfr (x), which has a given function ^ («) as its derivative, has been dis- covered, the value of the definite integral I=\%(x)dx (1), J a 222 INFINITESIMAL CALCULUS. [CH. VI can be written down at once. For, if we regard a as fixed, we have, by Art. 90, -+■'(4) (2). by hypothesis. It follows by Art. 56 that I and yjr (6) can only differ by a 'constant,' i.e. a quantity independent of b; thus (''^(x)dx=ylr{b) + G (3). J a To find the value of G we may, since it does not vary with b, put b = a, whence i|r(a)+C7=[''^(a!)rfa; = (4). J a Hence (7 = — •^{a) (5). / J a This is the fundamental proposition of the Integral Calculus. It reduces the problem of finding the definite integral of a given function ^ («) to the discovery of the inverse function ■^ {x), or D~' ^ (a;). The reason why this inverse function is usually denoted by S{co)dx (6) is now apparent. The form (6) is simply an abbreviation for ^{<^)dx (7), J a where a is arbitrary. We have seen that a change in a is equivalent to the addition of a cdhstant. The notation \-^Q>) (8) is often used as an abbreviation for •^ (6) — -^ (a). Ex. I. To find \ xdx (9), Ja Here {x) = x, ij/ (x) = \a?, rb whence I x dx = i;, {b) - ij, (a) = ^ {b^ - a") (10). Ja 92-93] DEFINITE INTEGRALS. 223 Ex. 2. To find i* !>?dx (11). Ja Here ?dx = l{W-a?) (12). f ^dx (13). ja (x) = ^, xl,{x) = ^e'^, r^ 1 whence I e*^c?a; = T(e**-e*») (U). The above results agree with those obtained, by much greater labour, in Art. 88. 93. Cases where the function {x)dx (1), where co>a. If, as « is increased indefinitely, the integral tends to a definite limiting value, this value is denoted by J a {x)dx (2). The integral (1) is then said to be 'convergent' for (i) = qo. As might be anticipated from the theory of infinite series (Art.~^) it is not a sufficient condition for convergence that lim„=„^(a:) = (3); this condition is moreover not essential, for there may even be convergence when (ft (x) has no definite limiting value for a; = oo . 224 INFINITESIMAL CALCULUS. [CH. VI A similar definition of (x) = 00 . The general case can be reduced to this by breakihg up the range b — a into smaller intervals * If ^(a;) become infinite at the upper limit (only), we consider in the first place the integral (f>(x)da; (9), / J a where e is positive. If, as e is diminished indefinitely, this integral tends to a definite limiting value, this value is adopted as the definition of r» ^ (x) dx. I J a A similar definition applies to the case where (f)(a!) becomes infinite at the lower limit a. * It being assumed that (x) becomes infinite only at a finite number of isolated points. 93-94] DEFINITE INTEGRALS. 225 f.j^) (^°)- The function l/ij(l - ^) becomes infinite for a;= 1, but and as e is indefinitely diminished this tends to the limit 2. Hence /. '^'° =2 (12). J{l-=o) If {x) becomes infinite between the limits a, b, say for x = c, we consider the sum [" '(x)dx+i {x)dx*- J a The cases where (^) dx J a J c+e is ultimately infinite, whilst, if some special relation te assumed between the ultimately vanishing quantities e, e', the infinite elements of the two integrals cancel in such a way that the sum remains finite. The value of the sum will then depend on the nature of the assumed relation. The considera- tion of such cases is, however, beyond the scope of the present treatise. They do not often occur in physical problems. L. 15 226 INFINITESIMAL CALCULUS. [CH. VI Ex. I. I siua5cfa;= -cosa; =1 (1); I cos a; cfo; = sin a! =1 (2) j I sin X cos K cfo; = |- sin'' a; =^ (3). Ex. 2. By Art. 79, -we have /"» . „ , fa sin iSa; + iS cos fia; J]" j^ e-'^sinpxdx=-\_ ^^,^^ ^ e-J^ ^ asinffo) + ^COS^. As cu increases indefinitely, this value tends to the limit ^tt, so that /. /: dx - 1 ATT. .(6). Ex. 4. By Art. 77, 4° we have Now, as 6 increases from to ^ir, J{alP) . tan increases from 94] DEFINITE INTEGRALS. 227 to 00 , and we may therefore suppose that tan~^{^(a/y8) . tan6} increases from to ^tt. Hence f4" dd IT ,y. Jo Jo asin^e + pcos'e 2J{al3) The student may have remarked in the course of the preceding Chapter that when an 'indefinite' integration is effected by a change of variable (Arts. 76, 78) the most troublesome part of the process consists often in the transla- tion back to the original variable. This part is, however, unnecessary when the object is merely to find the definite in- tegral between given limits. It is then sufficient to substitute the altered limits in the indefinite integral as first obtained. Ex. 5. To find r^{a'-aP)dx (8). Jo We found (Art. 78), putting 03 = a sin 6, that J^{a^ - a?) dx = a? /cos^ edd = ^a? (6 + | sin 26). Now, if 6 increase from to ^ir, x will increase from to a. Hence r J{a?- x") dx = ^a? p + I sin 2$!^ =\ira' (9). EXAMPLES. XZIZ. 1. iijxdx=i, /;^^=2, t^j^yj^-^. 1/V3 dx dx TT a^+bV 2ab' ^ xdx [^ xdx [^ xdx n xdx Jo v(i-«^)~ ' Jo jii+x^y^' '• 6. r^^_iog2, f^-f^^ = ilog2. Ji x(l+x) " ji a;(l + a3^) ^ s 15—2 228 INFINITESIMAL CALCULUS. [CH. VI 7 r' d^ 2^ ft ri-a='^ 1 1 7. I -J rT=Q~7a- °- / 1 5«a: = i'r-l- ^0 x^ - a; + 1 3^3 J^ \ + a? " jo (=c''+a^)(K2 + 6-^) 2a6(a + 6)' - /■" a?clx _ IT Jo {3? + a')(s^ + lf)~2{a + V)' ■. f" xdx la h {«? + a") {a? + 6=) " ^«362 '°S 5 • Jo Jo 9. I sm25rfe = l, f cos2ede = 0. Jo Jo 20. f_^^e=f-5^^rffl=i.. J a 1 + sm^ 5 7o 1 + cos" 5 ' 21. I^se.c*ede = ^, j^ta.n*ede = iT-^. „„ f dd tr r n 23. f*"--i^=f*''-^^ ' jo 1 + cos e jo 1 ■ . + sin 6 24. f*" tanl9(f5=0, [*" secerffl = log(3 + 2V2). 95J DEFINITE INTEGRALS. 229 25. / logxdx=-l, I xlogxdx = -^. 26. / sin~'a;tfo!= Jtt- 1, / tan~' a; cfos = Jir - J log 2. 27. I esmede = i, l^''e 1, the first part vanishes, since sin = 0, cos ^TT = 0. Hence / cos'^dde= / cos^'-^e d9 (3). Similarly, from Art. 81 (6), f %in"6'd0 = ^^ [\ia«~''ede (4). 230 INFINITESIMAL CALCULUS. [CH. VI JO If « be a positive integer, we can, by successive applica- tions of (3), express fi,r cos" dd0 '0 in terms of either ('^ cos0d0, = l, or f dd.^^TT (5), Jo •'0 according as « is odd or even. . In the same way sin" 0d9 can be made to depend either on I sin^d^, = 1, or on | dO,—^Tr (6). Jo Jo Ex. 1. [*" cos= 6de = I [*" cos' 6de = ^.^j^oos6d0 = -^. After woi'king out one or two examples in this way, the student will be able to supply the successive steps mentally, and write down at once the factors of the result ; thus /. \os<'ed6=^.i.hi^=^%^- The general values of the preceding integrals can be written down without difficulty. Thus, if n be odd, we have j'\os^ede= rsin^ede J^-p^\-^^-\ ..in Jo Jo n(n-2)...S whilst, if n be even, P%os«^rf^=f%in»^d^ = 1, the expression in [ ] vanishes at both limits, and we have [ " sin"* 8 cos" ede = -^^ f sin™ cos"^ 0d0...(l'^)- Jo m + nJo In the same way from Art. 81 (11) we obtain, if m > 1, sin™0cos''^ci^=-^^^ sin"»-^0cos"^cZ^...(12). Jo m + nJo By means of these formulae, either index can be reduced by 2, and by repetitions of this process we can, if m, n be positive integers, make the integral (9) depend on one in which each index is 1 or 0. The result therefore finally involves one or other of the following forms: I sin^cos^ci^, = 4; d0, = iTr; I •'o h I ....(13). [*" sin 0d0, = 1 ; [*" cos 0de, = 1 £x. 2. We have f^'sin'^cos^ec^^-lf " sin' e cos' ecZ(9 = 4. 1 [ "sin^cos'^t^^, Jo Jo Jo by (12). Again, by (11), j 'smecos'edO = l.l ''smecosed0 = l.^. Hence ( " sin" 6 cos' 6d0 = ^ . § . ^ . ^ = ^\. After a little practice, the result can be written down immedi- ately. Thus f "sin" e cos^ 6»c^^ = I . f . i . i . ^TT = , Jo _1 Xt, , 232 INFINITESIMAL CALCULUS. [CH. VI The formulae (11) and (12), as well as (3) and (4), are often required in practice, and should be remembered. Again, the algebraic integral I a;™ (1 -«)»(?« (14), Jo is reduced by the substitution x = sin'' 6 to the form 2 ['sm''»'+'dcoa«'+^ed0 (1-5), Jo and can therefore be evaluated by means of the formulsB given above, whenever 2m + 1 and 2re + 1 are positive integers or null. Similarly, if we put a; = sin 6, the integral [ a;™(l-a;^)»da; (16) J takes the form f ''sin™0cos="+'^de (17). J Ex. 3. f ar= (1 - x)idx = 2 ( sin" 9 cos," 6dQ Jo Jo _5! 4, 2 3 1 1 _ X8 Ex. 4. f a? {I- a?)^ = f "sin'' 6 cos^ OdO Jo Jo .18 1 1^_ 1 _ 96. Related Integrals. There are various theorems concerning definite integrals which follow almost intuitively from the definition of Art. 86. For example, ra ra I (f) (x) dx = I ^{a — x)dx (1). This is proved by writing x = a — x', dx= — dx', 95-96] DEFINITE INTEGRALS. 233 the new limits of integration being «' =a, x = 0, correspond- ing to x = 0,x = a, respectively. Thus ra ro ra I (a — »') dso' = j ^(a — x) dx, Jo J a Jo the accent being dropped in the end, as no longer necessary. This process is equivalent to transferring the origin to the point x = a, and reversing the direction of the axis of x. The areas represented by the integrals in (1) are thus seen to be identical. An important case of (1) is f ^/(sin 0) dO = f *"/(cos e)dd (2). Jo Jo Ex.\. Thus 1^' sin' ed6= I'' cos^ede. Hence each of these integrals = J ( "(sin^6l + cos=6l)c?6l = J [^ dd = W. Jo Jo Again, if ^ (x) be an ' even' function of x, that is {-x) = 4>{x) (3), /« ra 4>(x)dx = 2 4){x)dx (4), -a JO the area represented by the former integi-al being obviously bisected by the axis of y. 234 INFINITESIMAL CALCULUS. [CH. VI On the other hand, if (x) be an ' odd' function of x, so that we have {x). r <^{x)dx = 0.. J ~a •(5). .(6), since in the sum, of which the definite integral is the limit (Art. 86), the element cf) (x) Sx is cancelled by the oppositely- signed element (— x) 8x. Ex. 2. We have J—iir Jo whilst f " sin' 6 cos'' OdO^^O, since sin' changes sign with 0. For. similai^ reasons, if {a-x) = (x)dx=2l (j>(x)da; (8); Jo Jo whilst if ^(a-x) = -(j){x) (9), we have ( 6{x)dx = (10). Jo As a particular case of (8), we have rf{sm0)d0 = 2f f{sm0)de (11), Jo Jo since sin {tt — 6) = sin d. 96] BEFINITE INTEGRALS. 235 Ex. 3. f "sin' e oos" Odd = 2 [^ sin' cos'Sde = 2 .^.^.1=^; Jo Jo whUst , " sin" 6 cos' OdO = 0. Jo EXAMPLES. XXX. 1. Write down the values of the following integrals : (1) j^sin^ede, (^ coa' edO, l^sin^edd. (2) I ''sin" 6 cose d$, j " sin^B cos^edO, j sin" 6 cos« e«f6l. (3) r&in^ede, ("co^ ed9, j" sia'doos^Ode. (4) (^ sm*ed0, [^ siu'dde, p" cos' Ode, [ sin' e cos' ed9. J-h' 2. Prove from first principles that f x"'(l-xYdx= [ x" (1 - x)'^ dx. Jo Jo Prove that the common value is (m + w + 1) I 3. Prove that ( ^{3?)dx=2 r^ix') dx, r (j> (x") xdx = 0. J-a Jo J-a 4. Prove that /a ra {- x)}doc, -a Jo r {/(l - «==) ~ ^^"^ 10. f " esin0coserfe = j7r, i' esin^ede = o. J -in J-in 11. r0sinecoa^6de = ^7r, j" esin'Ocosede = -j. jo sin^ ^ jo sine ^ f i" sin3g ,„ , /■*" sin iO sin^ *13. Prove that r dx _ 2m- 3 p (^a; Jo (l+xy ~ 2/1-2 io (1 +«;=)''-' ■ [Put a; = tan e.] 14. Prove that, if w> 1, r ___dx n Jo {a!+^(a;»+l)}»~»?^- [Put X = sinh M.] 15. Prove that [ (1 + a;)™ (1 - a;)"rfa! = 2™+"+^ f " sin«'+'e cos'"^! ed0. at p t^M _ ra-2 r Jo cosh"M~ n-1 Jo 16. Prove that du n — 2f'° du cosh"~^ M * [Put cosh u = sec 6.\ * The Examples which follow are of a more difficult character, and may be passed over on a first reading of the subject. DEFINITE INTEGRALS. 237 17. Prove from first principles that f ^ sin^+i ede< [^ sin» Ode. Jo Jo Hence show that Jtt lies between 2. 2. 4. 4. 6. 6. ..2w. 2m 1.3.3.5.5.7...(2ra-l)(2»i-<-l) and the fraction obtained by omitting the last factors in the numerator and denominator. (Wallis.) 18. Prove from first principles that ( " tan»+i ed$<( tan" 6 dO. Jo Jo Hence, using the result of Ex. 5, shew that /: tsLii^ede Jo lies between 1 and 2(»i-l) 2(n+l)' 19. Shew that I sin dd6 and j cos 6d6 Jo Jo are indeterminate. 20. Shew from graphical considerations that is finite and determinate. 21. Prove that if (^ (aj) be finite and continuous for values of X ranging from to a, except for x — 0, when it becomes infinite, the integral i: (x) dx Jo will be finite, provided a positive quantity m can be found, less than unity, and sucli that lim^j 9* (x)dx Jo will be finite, provided a quantity m can be found, greater than unity, and such that liinj,_„a!'" (oj) is finite. [Put « = <-".] 23. Prove that I cosK°rfa! and j sina^dx Jo Jo are finite and determinate. 24. Prove that I xe-'''dx = l, j xe-'^dx = 0. 25. Prove that the integral Jo (where to>— 1) is finite and determinate. 26. Prove that, if ra > 1, ("x'^e-'^dx = i(n - 1 ) f " a;"-='e-*' {x) retain the same sign, throughout the interval from x = ato x = b, then / f{x)^{x)dx=f\a + 6{b — a)\ j ^ > 0. 29. Shew how it follows from the equality fdx , that the sum of n terms of the harmonic series lies between log (w + 1) and 1 + log n. Shew that the sum of a million terms of this series lies between 13 '8 and 14-8. 30. Shew from graphical considerations that if /(x) steadily diminishes, as x increases from to o) , the series be finite. Apply this to the series /(l) + /(2)+/(3) + ... ts sum lies bi i = / f{x) dx, is convergent, and that its sum lies between I and /+/(!), provided the integral 1 1 1 {n+\f (n + 2f (n + 3f EXAMPLES. XSSI. 1. Prove that if the pressure {p) and volume {v) of a gas be connected by the relation jo« = const., the work done in expanding from volume v^ to volume w, is 240 INFINITESIMAL CALCULUS. [CH. VI 2. Prove also that if the relation be jBi)^= const., the work done is T (^o^o-yA)- 3. If the tension of an elastic string vary as the increase over the natural length, prove that the work done in stretching the string from one length to another is the same as if the tension had been constant and equal to half the sum of the initial and final tensions. 4. Prove that the work done by gravity on a pound of matter, as it is brought from an infinite distance to the surface of the Earth, is n foot-lbs., where n is the number of feet in the Earth's radius. [Assume that the force varies inversely as the square of the distance from the Earth's centre.] CHAPTER VII. GEOMBTKICAL APPLICATIONS. 97. Definition of an Area. In Euclid's Elements a system of propositions is developed by means of which we are able to give a precise meaning to the term ' area,' as applied to any figure bounded wholly by straight lines. In particular it is shewn that a rectangle can be constructed equal to the given figure, and having any given base, say the (arbitrarily chosen) unit of length. The ' area ' of the figure in question is then measured by the ratio of this rectangle to the square on the unit length. This process obviously does not apply to a figure bounded, in whole or in part, by curved lines, and we require therefore a definition of what is to be understood by the ' area ' in such a case.* To supply this, we imagine two rectilinear figures to be constructed, one including, and the other included by, the given curved figure. There is an upper limit to the area of the inscribed figure, and a lower limit to that of the circum- scribed figure, and these limits can be proved to be identical. The common limiting value is adopted, by definition, as the measure of the ' area ' of the given curvilinear figure. Thus, in the case of a circle, if, in Fig. 4, p. 7, PQ be the side of an inscribed' polygon, the area of the polygon will be it {ON. PQ). Now OR is less than the radius, and X (PQ) is less than the perimeter, of the circle. Hence the upper limit to the area of an inscribed polygon cannot exceed L. 16 242 INFINITESIMAL CALCULUS. [CH. VII ^a X 2Tra, or ira', where a is the radius. Similarly we may shew that the lower limit to the area of a circumscribed polygon cannot be less than Tra". Moreover, the difference between the area of an inscribed polygon, and that of the corresponding circumscribed polygon, is represented by S (FN. NT), and is therefore less than 2 (PN) . e, where e is the greatest value of NT. Since this can be made as small as we please, the upper and lower limits aforesaid must be equal, and each is therefore equal to Tral In the same way we may prove that the area of any sector of a circle of radius a is ^a'0, where is the angle of the sector. 98. Formula for an Area, in Cartesian Coordi- nates. If the equation of a curve in rectangular coordinates be y = 'f>i^) (1), the area included between the curve, the axis of cc, and the ordinates x = a, x = h, is I (f){a!)dso or I ydx (2), J a J a it being assumed that a, and that the ordinate ^ (x) is positive throughout the range of integration. If we drop these restrictions, it is easily seen that the integral I ^{x)dx (1) J a is equal to + 8, where 8 is the area included between the curve, the axis of x, and the extreme ordinatcs; the sign being -|- or — according as the area in question lies to the right or left of the curve, supposed described in the direction Y P^-— -v. + 3 o \ ^^^-^1— X Fig. 58. from P to Q, where PA, QB are the ordinates corresponding to x = a, x = b, respectively*. If the curve cuts the axis of x between A and B, the integral gives the excess (positive or * It is assumed here that the axes of x and y have the relative directions shewn in the figures. In the opposite case, the words 'right' and 'left' must be interchanged. 246 INFINITESIMAL CALCULUS. [CF. VII negative) of the area which lies to the right over that which lies to the left. Even with these generalizations, the formula '' <^{x)dx=±8 (2) / J a still applies in strictness only when there is a unique value of y, or (/) («), for each value of x within the range b — a. If however we replace x, as independent variable, by a quantity t such that, as t increases, the corresponding point P moves in a continuous manner along the curve*, the formula /; .(3) will give in a generalized sense the area included between Y / + ) O X Fig. 59. the curve, the axis of x, and the ordinates of the points P^, Pi for which t = to, t=ti, respectively, viz. it will give the excess of those portions of the area swept over by the ordinate y as it moves to the right over those swept over as it moves to the left, or vice versd, according as y is positive or negative. If, for a certain value of t, P return to its former position, having described a closed curve, the integral dx > dt dt. .(4), taken between proper limits of t, will give the area included by the curve, with the sign + or — , according as the area lies to the right or left of P, when this point describes the curve in * For instance, we may take as the new variable the arc « of the curve, measured from some fixed point on it. 99-100] GEOMETRICAL APPLICATIONS. 247 J to accordance with the variation of t*. If the curve cut itself, the formula (4) gives the excess of those portions of the included area which lie to the right over those which lie to the left. (See Fig. 59.) It is sometimes convenient, in finding the area of a curve, to use y as independent variable, instead of x. The area included between the curve, the axis of y, and the lines y = h, y =k, is evidently given, with the same kind of qualification as before, by / 33% (5)- Jh The more general formula, analogous to (3), is M'^ (6)= but it will be found on examination that the words ' left ' and ' right ' must now be interchanged in the rule of signs. 100. Areas referred to Polar Coordinates. If the equation of a curve in polar coordinates be r={6) (1), the area included between the curve and any two radii vectores 6 = a, = /3 is given by the formula i|%^(^^ oxii^\{e)Yde (2). For we can construct, in the manner indicated by the figure, an including area 8, and aa included area S', each built up of sectors of circles. The area of any one of these sectors is equal to \r^^6, where r is its radius, and hd its angle, and the sum of either series of sectors is therefore given by a series of the type ^^o ^ ' Fig. 60. * Thus, in the indicator-diagrams referred to on p. 211, the area enclosed by the curve gives the excess of the work done by the steam on the piston during the forward stroke over the work done by the piston in expelling the steam during the back stroke, and so represents the net energy communicated to the piston in a complete stroke. 248 INFINITESIMAL CALCULUS. [CH. VII Hence either series has the unique limit denoted by (2). It is here assumed that /3 > a and that each radius vector through the origin intersects the arc considered in one point only. If however we introduce a new independent variable t, such that, as t increases, the corresponding point P moves in a continuous manner along the curve, the ex- pression ■Jt'^' (*> will give the net area swept over by the radius vector as t varies from <„ to t^, i.e. the (positive or negative) excess of those parts which are swept over in the direction of 6 in- creasing over those swept over in the contrary direction. Moreover if, as t increases, P at length returns to its original position, having described a closed curve, the expression dt (5), ih'%' taken between suitable limits of t, gives in a generalized sense the area enclosed by the curve ; viz., it represents the excess of that part of the area which lies to the left of P (as it describes the curve in accordance with the variation of t) over that part which lies to the right. Cf. Art. 99. Ex. 1. The area of the circle r = 2asin0 (C) fsec Fig. 39, p. 126) is given by J r r'dO = 20.^ Tain' Ode = 7ni,^ (7). £x. 2. The equation of an ellipse in polar coordinates, the centre being pole, is 1 cos''^ sin=e r a? 0- ^ ' Hence the area is 2 ^0 a" siu^ e + V co&^ 6 Jo a' sin" e + b^ cos« (9)- 100-101] GEOMETRICAL APPLICATIONS. 249 The value of the latter integral has been found in Art. 94 to be ^irjiab). Hence the required area is vab. 101. Area swept over by a Moving Line. The area swept over by a moving line, of constant or variable length, may be calculated as follows. Let PQ, P'Q' be two consecutive positions of the line, and let their directions meet in G. Let iJ, R' be the middle points of PQ, P'Q', and let RS be an arc of a circle with centre G. Then if the angle PGP' be denoted by Bd, we have, ultimately, area PQQ'P' = AQGQ' - APGP' ^^GQ'.Be-^GPKSe = PQ.:k{GP + CQ)8e = PQ.GR.Be = PQ.RS. Fig. 61. Hence, if we denote the length PQ by u, and the elementarj' displacement of jB, estimated in the direction perpendicular to the moving line, by Bcr, the area swept over may be represented by Iud, where w is the inclination of the axes. [Put x=a sin* 6, y = h cos* d.] 9. The area swept over by the radius vector of the parabola 2a If =z 1 + cos is equal to the difference between the initial and final values of a? (tan J6I + J tan' |(9). 10. A curve AB is traced on a lamina which turns in its own plane about a fixed point through an angle 6. Prove that the area swept over by the cur^'e is i^ {OA^ ~ OB^) e. 11. Prove that, with a proper convention as to sign, the area of a closed curve is given by dy dx\ "" dt y dt)"^^' provided the total variation of t corresponds to a complete circuit of the curve. 12. The formula (Art. 79 (2)) for integration by parts may be written judv = uv — jvdu ; interpret this geometrically in terms of areas. 254 INFINITESIMAL CALCULUS. L<-'H. VII 13. The area common to the two ellipses is 4a6tan~' hja. a" = 1, 4 = 1 as" 14. The area common to the two parabolas is ie„a f = : 4aa;, a? = iay 15. Prove by integration that the area of an ellipse is trap sin m, where a, ^ are the lengths of any pair of conjugate semi-diameters, and o) is the angle between these. 16. Prove by transformation to polar coordinates that the area of the ellipse is ^IJ{AB-H'-). 17. A weightless string of length I, attached to a fixed point 0, passes through a small ring which can slide along a horizontal rod ^ 5 in the same vertical plane with 0, and the lower portion hangs vertically, carrying a small weight F. Find tlie locus of P, and prove that the area between this locus and AB ia where h is the depth of AB below 0. 18. Prove directly from geometrical considerations that the area included between two focal radii of a parabola and the curve is half that included between the curve, the corresponding perpendiculars on the directrix, and the directrix. 19. What is indicated by the record of the wheel in Amsler's Planimeter when the bar FQ (Fig. 63) makes a com- plete revolution whilst the point Q traces out the closed curve 1 20. Tf S^, S„, Sg be the areas of the closed curves described by three points A, B, C on a. bar which moves in one plane, and returns to its original position without performing a complete revolution, prove that BC.S^ + CA.S^ + AB.St, = 0, 103] GEOMETRICAL APPLICATIONS. 255 where the lines BG, CA, AB have signs attributed to them according to their directions, and the signs of Sj^, S^, Sg are determined by the rule of Art. 99. 21. If P be a point on a bar AB which moves in one plane, and returns to its original position after accomplishing one revo- lution, prove that '^ a + b where a = AP, h = PB, and the meanings of ')dx (2), taken between suitable limits of x. Ex. 1. Thus, in the case of a cone (or a pyramid), right or oblique, on any base, we take the origin U at the vertex, and the axis of X perpendicular to the base. If A be the area of the base, and h the altitude, the area of a section at a distance x from will be f{x) = (^^\A (3), since similar areas are proportional to the squares on correspond- ing lines. Hence the volume, being equal to i^r^ oi?dx, or ^hA (4), is one-third the altitude into the area of the base. Ex. 2. The volume of a tetrahedron is ^haa' sin a (5), where a, a' are any pair of opposite edges, h their shortest distance, and a the angle between their directions. Divide the tetrahedron into laminae by planes parallel to the edges a, a', and therefore perpendicular to the shortest distance h. It is evident, on reference to Pig. 65, that the section made by a plane of the system at a distance x from the edge a is a parallelogram whose sides are X , , h-x ■J- . a and — ; — a, h n and whose area is therefore aa a '^ IT, \ • -p- x(h,-x) sm u.. Hence the volume = ^-r- sin a / x(h — x)dx V Jo ' ' which reduces to the value given above. I. 17 258 INFINITESIMAL CALCULUS. [CH. VII a' Pig. 65. 105. Solids of Kevolution. Let the equation of the generating curve be 2/ = <^(^) (1), the axis of x being that of symmetry, and let the solid be bounded by plane ends perpendicular to a;. In this case, the area /(oo), being that of a circle of radius y, is Tryl Hence the required volume is I-^y'dx (2), taken between proper limits. Each element of the sum, of which this integral is the limit, represents, in fact,, the volume of a circular plate of thickness Sx and area -n-y^. Ex. 1. The equation of a circle, referred to a point on its circumference as origin, is y^ = x{;ia-x) (3). Hence the volume of a segment of a sphere, of height h, is ir \ X (2a — x)dx = ir\ aa? - ^a? ^■kIv' {a-\h) (4), 105-106] GEOMETRICAL APPLICATIONS. 259 a being the radius of the sphere. For the complete sphere, we have h = 2a, and the volume is f ira', or two-thirds the volume (wa" X 2 a) of the circumscribed circular cylinder. Ex. 2. The volume of a segment, of height h, of the parabo- loid generated by the revolution of the curve y^ = iax (5) about the axis of x, is I y^dx = iira[^af\ =2irah? (6). If 6 be the radius of the base, we have 6' = 4aA. Hence the volume is \irV . h, or one-half that of the cylinder of the same height on the same base. Ex. 3. To find the volume of the 'anchor-ring,' or 'tore,' generated by the revolution of the circle x'+(y-af=^V (7) about the axis of x\a> 6]. See Fig. 70, p. 272. For each value of x (< h) we have two values of y, say 2/. = o+V(6»-a^), y, = a-^{b'-x') (8). The area of a section of the ring by a plane perpendicular to x is therefore ^i^-7ry^^ = i7raj{b^-x'} (9) and the required volume is 47raf J{b^-a?)dx = 2Tr'ab^ (10); by Art. 94, Ex. 5. This is the same as the volume of a cylinder whose section (trb^) is equal to that of the ring, and whose length (2ira) is equal to the circumference of the circle described by the centre of the generating circle. 106. Some other Cases. We give some further examples of the general formula (2) of Art. 104. 17—2 260 INFINITESIMAL CALCULUS. [CH. VII Ex. 1. The section of the elliptic paraboloid 2x=t.+ t. (1) by a plane x = const, is an ellipse of semi-axes ij(2px) and J(2qx), and therefore of area 2TrJ{pq) x. Hence the volume of the segment cut off by the plane a; = A is ^^J{P9) I xdx=nj{pq)h'' (2). Jo This is one-half the volume of a cylinder of the same height h on the same elliptic base. £lx. 2. In the ellipsoid i-i-i-' (')' the section by a plane x = const, is an ellipse of semi-axes and therefore of area -'^{'-i) (')■ The volume included between any two planes perpendicular to x therefore = '^6«/(l-5)'^«= W' taken between the proper limits of x. For the whole volume the limits of X are + a, and the result is ^irabc. 107. Simpson's Rule. Most of the preceding results are virtually included in a general formula applicable to all cases where the area of the section by a plane perpendicular to a; is a quadratic function of x, say /(^)=^-f5| + 0| (1). Integrating this between the limits and h, we find /: f(a;)dx = h{A + ^B + iG) (2). 107] GEOMETRICAL APPLICATIONS. 261 Now A is evidently the area of the section a; = 0; and denoting the areas of the sections x = ^h, and x = h, respec- tively, by A' and A",Yie have A+i,B-\r\G=A', A + B + G = A" (3), whence pf(x) dic = lh(A+4!A' + A") (4). Jo This gives the volume included between two parallel sections, in terms of the areas of these sections and of the section half-way between them. The formula (1) is obviously applicable to the case of a cone, pyramid, or sphere, and also to the case of a paraboloid, ellipsoid, or hyperboloid, provided the bounding sections be perpendicular to a principal axis. The student who is familiar with the theory of surfaces of the second degree will easily convince himself, moreover, that the latter condition is not essential. Another case coming under the present rule is that of a solid bounded by two parallel plane polygonal faces and by plane Fig. 66. lateral faces which are triangles or trapeziums. We may even include the case where some or all of the lateral faces are curved surfaces (hyperbolic paraboloids) generated by straight hnes moving parallel to the planes of the polygons, and each inter- secting two straight lines each of which joins a vertex of one polygon to a vertex of the other (see Fig. 67). And since the number of sides in each polygon may be increased indefinitely, the rule will also apply to a solid bounded by any two plane 262 INFINITESIMAL CALCULnS. [CH. VII parallel faces and by a curved surface generated by a straight line which meets always the perimeters of those faces. Fig. 67. EXAMPLES*. XXXra. 1. The volume generated by the revolution of one semi- undulation of the curve ij = h sin xja, about the axis of x is one-half that of the circumscribing cylinder. 2. The volume of a frustum of any cone, with parallel ends, is \h {A^ + J{AJ.^ + A^], where A^, A^ are the areas of the two ends, and h is the perpen- dicular distance between them. 3. In the solid generated by the revolution of the rectangular hyperbola 3? -y^ = a? about the axis of x, the volume of a segment of height a, measured from the vertex, is equal to that of a sphere of radius a. * See the footnote on p. 252. 6KOMETRICAL APPLICATIONS. 263 4. The volume of a segment of a sphere bounded by two parallel planes at a distance h apart exceeds that of a cylinder of height h and sectional area equal to the arithmetic mean of the areas of the plane ends, by the volume of a sphere of diameter h. 5. If a segment of a parabola revolve about the ordinate, the volume generated is -^ of that of the circumscribing cylinder. 6. The volume of the solid generated by the revolution of a parabola about the tangent at the vertex is ^ that of the circumscribing cylinder. 7. The volume of a frustum of a triangular prism cut off by any two planes is where Aj, Aj, h^ are the lengths of the three parallel edges, and A is the area of the section perpendicular to these edges. 8. If h be the radius of the middle section of a cask, and a the radius of either end, prove that the volume of the cask is ^TT (3a» + 4a6 + 86=) h, where h is the length, it being assumed that the generating curve is an arc of a parabola. 9. An arc of a circle revolves about its chord ; prove that the volume of the solid generated is ^Tra? sin a + f-rrfls' sin a cos^ a - 2wa? a cos a, where a is the radius, and 2a is the angular measure of the arc. 10. The figure bounded by a quadrant of a circle of radius a, and the tangents at its extremities, revolves about one of these tangents ; prove that the volume of the solid thus generated is (M)' 11. The axis of a right circular cylinder of radius 6 passes through the centre of a sphere of radius a(> S) ; prove that the volume of that portion of the sphere which is external to the cylinder is ■ {a? - 6^)t 264 INFINITESIMAL CALCULUS. [CU. VII 12. The volume enclosed by two right circular cylinders of equal radius a, whose axes intersect at right angles, is -^-a'. If the axes intersect at an angle a, the volume is -^ a' cosec a. 13. If the hyperbola revolve about the axis of x, the volume included between the surface thus generated, the cone generated by the asymptotes, and two planes perpendicular to x, at a distance h apart, is equal to that of a circular cylinder of height h and radius b. 14. A right circular cone of semi-angle u. has its vertex on the surface of a sphere of radius a, and its axis passes through the centre. Prove that the volume of the portion of the sphere which is exterior to the cone is ^Tra' cos* a. 108. Rectification of Curved Lines. The perimeter of a rectilinear figure is the length obtained by placing end to end in succession, in a straight line, lengths equal to the respective sides of the figure. But since a curved line, however short, cannot be super- posed on any portion of a straight line, we require some definition of what is to be understood by the 'length' of a curve. The definition usually adopted is that it is the limit to which the perimeter of an inscribed polygon tends as the lengths of the sides are indefinitely diminished. It will appear that, under proper conditions, this limit is unique ; and it can also be shewn that it coincides with the corresponding limit for a circumscribed polygon. If (,«, y) and (on + Ba!,y + By) be the rectangular coordinates of two adjacent points P, Q on a curve, the length of the chord PQ is ^/{iSxy + iSyy}. It has been shewn, in Art. 56, that if y and dy/dx be finite and continuous, the ratio Sy/Bx is equal to the value of the derived function dy/dx for some point of the curve between P and Q. Hence, with a properly chosen value of dyjdx, we have FQ-^ '-(lA^ 108] GEOMETRICAL APPLICATIONS. 265 The limiting value of the perimeter of the inscribed polygon is therefore V{'+©> «■ taken between proper limits of oo. The fact that this limiting value is unique has been established in Art. 87. If X be regarded as a function of y, the corresponding formula is WHt^V" <^)- If we denote by s the arc of the curve measured from some arbitrary point (a;„), and if as in Art. 53, we put dy /dx = tan yfr, the formula (1) becomes := i sec -^jr dx (3). There is of course a similar transformation of the formula (2). Hx. 1. In the catenary f = ccosh - (4), : I cosh - = |cosh-tfo! = csinh-. Since this vanishes with x, the arc (s) measured from the lowest point is given by s = c sinh — (5). Hx. 2. In the parabola y'' = 4:ax (6), '">•" /■/{'* (l)"h^/v/C-^>' » This may be integrated by the method of Art. 75, first rationaliz- ing the numerator, or we may put x = a siny u, 266 INFINITESIMAL CALCULUS. [CH. VU and obtain 2a I cosh" udu = a 1(1+ cosh 2m) du = a (m + J sinh 2m) (8). Since u vanishes with x, this gives the length of the arc measured from the vertex. For example, at the end of the latus-rectum we have sinh M = 1, cosh u = J2, u = log (1 + ^^2), whence we find that the length of the arc up to this point is a {log (1 + V2) + ^2} = 2-296a. 109. Generalized Formuls. It is a consequence of the definition above given that any infinitely small arc PQ of a curve is ultimately in a ratio of equality to the chord PQ. This may be verified immediately by differentiating the formula (3) of the preceding Art. with respect to the upper limit (x) of the integral. We thus find limg^ = secf (1). Since, when Q is taken infinitely near to P, seci|f is the limiting value of the ratio of the chord PQ to Sx, we have, ultimately, linipQ = l (2). This leads to several important formulae. In the first place, if the coordinates cc, y of any point P on the curve be regarded as functions of the arc s, we have, in Fig. 26, p. 66, and therefore cosi/r = -5-, sin-\|r = -^ (3). The former of these results might have been written down at once from (1). It follows that ©■-(1)'- (^)- 108-109] GEOMETRICAL APPLICATIONS. 267 Again, if x, y be functions of any other variable t, we have andlherefo™ K™ ^ = VfCtY + ff VKS) U V WdtJ \dt or svKsy-d)'} (»'• He.ce .= l./)(|)V(f •d« (6). This may be regarded as a generalization of Art. 108 (1). The formula referred to was obtained on the supposition that there is only one value of y for each value of x, within the arc considered. The result (6) is free from this restriction ; all that is essential is that as t increases, the point P should describe the curve con- tinuously. In the same way, the formulae (3) may be taken to apply to any rectifiable curve, provided we understand by \j/ the angle which the tangent, drawn in the direction of s increasing, makes with the positive direction of the axis of x. The formulae (5) and (6) have an obvious interpretation in Dynamics. If x, y be the rectangular coordinates of a moving point, regarded as functions of the time t, then dxjdt and dyjdt are the component velocities parallel to the coordinate axes, and if V be the actual velocity, we have Jent to Udt (8). The formulae (5), (6) are thus equivalent to ds Jr''' Ex. In the ellipse x—as\ntj>, y = 6 cos ^ (9) 268 INFINITESIMAL CALCULUS. [CH. VII where e is the eccentricity. Hence the arc s, measured from the extremity of the minor axis, is i = a I* Jil-e'sin" 4>)d (10). This cannot be expressed (in a finite form) in terms of the ordinary functions of Mathematics. The integral is called an ' elliptic integral of the second kind,' and is denoted by E (e, ). It may be regarded as a known function, having been tabulated by Legendre*. The whole perimeter of the ellipse is expressed by 4a (^ J(,l -e»sin'<^)<^^ (11). Jo The integral in this expression is denoted by E(e, ^ir), or more shortly by £1^(6). It is called a 'complete' elliptic integral of the second kindt. The quantity e is called the 'modulus' of the integral. The calculation of the integral (11) by means of a series will be treated in Chap. xiii. 110. Arcs referred to Polar Coordinates. Let OP, OP' be two consecutive radii of the curve, and let PN be drawn perpendicular to OP'. If we write OP = r, OF=r + Sr, Z POP' = 80, then, as in Art. 55, PN will diflFer from rS6, and NP' from Sr, by quantities which are infinitely small in comparison with PN, NP respectively. Hence PP', or V(-P-^' + NF% is ultimately in a ratio of equality to It follows that, if 6 be the independent variable. de --^Vi--©] <'>■ * Traite des Fonctions Elliptiques, t. ii. (1826). + The elliptic integral of the ' first kind ' is d0 /: and is denoted by F{e, ip). The corresponding 'complete' integral (with ^ir as the upper limit) is denoted by F^ («). 109-110] GEOMETRICAL APPLICATIONS. 269 and therefore -M-©V^ «' provided the integral be taken between the appropriate limits of 6. Fig. 68. Ifr and be given as functions of an independent variable t, we have S=-^V{©'-(S)] <')• and therefore -/y{©'--(i)> (^)' which includes (2) as a particular case. Ex. In the circle r=2asin6 (5), we have '^ "*" ( ivfl ) ~ ^'^^' and therefore ds d0 = 2a, « = 2a0 + const. .(6), as is otherwise obvious. See Fig. 39, p. 126. Again, if r, be regarded as functions of the arc s, and if ^ denote the angle which the positive direction of the 270 INFINITESIMAL CALCULUS. [CH. VII radius vector makes with the tangent-line, drawn in the direction of s increasing, we have NP' PN cos NFP = Jp , sin NP'P= ^ , aud therefore, in the limit, , dr . dO -Kv cos./, = ^, sin,^ = r^ (7). These results, again, have a dynamical illustration. If v denote the velocity of a moving point, the component velocities along and transverse to the radius are dr ds dr "'"'^'t'^dsdrd-f •(8), . , rd9ds rd6 and V sm = 2wab U{l-e'siji')d{sin) (9), by Art. 109. If we put esin<^ = sin0 (10), this = ^jcos'ed9 = '^[e+smecose] (ii). To find the whole surface we take this between the limits ij) = + ^ir, or 6 = + sin~^e. The result is ^Trab , e {sin-' e + 6^(1 -e'')}, 1-1^ or 27r6''+27ra6?HL_? (12). By a similar process we find, for the surface generated by the revolution of the ellipse about its minor axis, the value ^{sinh-e' + eV(l+«'% where e' = J{d' - b^)jb, or 2^a^+2^ab^^^^^ (13). e This may also be put into the form 27ra!' + ->rP.-logl^ (U). e ° 1—e ^ ' L. 18 274 INFINITESIMAL CALCULUS. [CH. VII EXAMPLES*. XXXIV. 1. The length of a complete undulation of the curve of sines y = b sin xja is equal to the perimeter of an ellipse whose semi-axes are ^{a' + U') and a. 2. Prove the following formula for the length of the perpendicular (jp) from the origin on any tangent to a curve. dy dx Also prove that the orthogonal projection of the radius vector on the tangent is dx dy dr a; -=- + w -r or r-=- . as as as 3. The surface generated by the revolution about the direc- trix of an arc of the catenary y = c cosh x\o, commencing at the vertex, is IT (ex + ys), where x, y, s refer to the extremity of the arc. 4. The curved surface cut off from a paraboloid of revolution by a plane perpendicular to the axis is where h is the length of the axis, and h the radius of the bounding circle. 5. The curved surface generated by the revolution about the axis of x of the portion of the parabola y'=4aa; included between the origin and the ordinate a; = 3a is ^ira^ 6. The segment of a parabola included between the vertex and the latus-rectum revolves about the axis ; prove that the curved surface of the figure generated is 1-2 19 times the area of its base. * See the footnote on p. 252. 112] GEOMETRICAL APPLICATIONS. 275 7. A circular arc revolves about its chord ; prove that the surface generated is 4ira' (sin a - a cos a), where a is the radius, and 2a the angular measure of the arc. 8. A quadrant of a circle of radius a revolves about the tangent at one extremity ; prove that the area of the curved surface generated is tt (tt - 2) a". 9. A variable sphere of radius r is described with its centre on the surface of a fixed sphere of radius a ; prove that the area of its surface intercepted by the fixed sphere is a maximum when r=^a. 112. Approximate Integration. Various methods have been devised for finding an approximate value of a definite integral, when the indefinite integral of the function involved cannot be obtained. For brevity of statement, we will consider the problem in its geometrical form ; viz. it is required to find an approximate value of the area included between a given curve, the axis of X, and two given ordinates. The methods referred to all consist in substituting, for the actual curve, another which shall follow the same course more or less closely, whilst it is represented by a function of an easily integrable character. The simplest, and roughest, mode is to draw n equi- distant ordinates of lihe curve, and to join their extremities by straight lines. The required area is thus replaced by the sum of a series of trapeziums. If h be the distance between consecutive ordinates, and yi, yn-.., yn the lengths of the ordinates, the sum of the trapeziums is i (2/1+2/2) A+i(2/s+ys) A + . . . + i (2/,j_a+y^i) h + i(3/»_i+2/») h = (I2/1 + 2/2 + 2/s+ ••• + 2/»-2 + y»-i + i2/») A...(l) ; that is, we add to the arithmetic mean of the first and last ordinates the sum of the intervening ordinates, and multiply the result by the common interval h. 18—2 276 IKFINITESIMAL CALCULUS. [CH. VII The value thus obtained will obviously be in excess if the curve is convex to the axis of x, and in defect ia the opposite case. Fig. 71. Another method, originally given by Newton and Cotes*, is to assume for y a rational integral expression of degree n—1, thus y = J.o + 4ia; + A«'H-.-.+4n-ia;"-' (2), and to determine the coefficients A^, A^, ... -4,i_i so that, for the n equidistant values of x, y shall have the prescribed values yi, y^, ..., yn- The area is then given by jydx,=Aoai + iA^x' + ^A^si^+... +-An-iX^ (3), h taken between proper limits of x. Thus, in the case of three equidistant ordinates, taking the origin at the foot of the middle ordinate, we assume y=A + ^i|+^2(|) (4), • See the latter's tract Be Methodo Differentiali, printed as a supplement to the Harmonia Mensurarum, Cambridge, 1722. y= Vi, y-. for x= — h, 112] GEOMETRICAL APPLICATIONS. 277 with the conditions that ,: t ) «■ respectively. These give so that A.^Vi, Ai = i(y3-yi), A^ = i(y,+ya-2y,)...(7). Hence I ydx=2 {A,, + ^A.^ h = M2/i + % + 2/3)A (8). The method here employed is equivalent to replacing the actual curve by an arc of a parabola having its axis vertical ; and the result represents the difference between the trapezium i(2/i + 2/3).2A and the parabolic segment l-{H2'i + 2'3)-2/2}.2/i; see Art. 98, Ex. 4. In the case of four equidistant ordinates a similar process leads to the formula 1(2/1 + 32/. + 32/3 +2/4) A (9), whilst tor five ordinates we get :j% (7yx + B2y, + 12y, + 322^4 + 1y,) h (10). With an increasing number of ordinates the coefficients in this method become more and more unwieldy*. A simple, but generally less accurate, rule was devised by Simpson f. Taking an odd number of ordinates, the areas between alternate ordinates, beginning with the first, are calculated * The coefficients for the oases n=3, i, 5, ...11, were calculated by Cotes ; see also Bertrand, Galcul Integral, Art. 363. t Mathematical Dissertations (1748). 278 INFINITESIMAL CALCULUS. [CH. VII from the 'parabolic' formula (8), and the results added. We thus obtain hiVi+'^yi + ys + 2/3 + %4 + y» +2/6 + 43/8+2/7 + + yan-i + ^Vm + Vm+y} h = i{(2/i + 2/2«+i) + '2 (2/3 + 2/6 + ... + 2/an-i) + 4 (2/2+2/4+ ••■+2/271)} A (11). That is, we take the sum of the first and last ordinates, twice the sum of the intervening odd ordinates, and four times the sum of the even ordinates, and multiply one-third the aggregate thus obtained by the common interval h, Ex. To calculate the value of ir from the formula i-rifi <>')■ Dividing the range into 10 equal intervals, so that h= -1, we find j/j =1, 2/2 = -9900990, 2/8 = -9615385 2/1 = -9174312, 2/5 = •8620690, 2/6 = -8000000, 2/7= •7352941, 2/8 =-6711409, 2/8 =-6097561. 2/u = -5, 2/10= -5524862, Hence yi + 2/ii=l'5, 2/3 + ^6 + 2/7 + 2/9 = 3-1686577, ys + ^4 + 2/6 + 2^8 + yio = 3-931 1573. The formula (11) then gives iir = ^ (1-5 + 6-3373154 + 15-7246292) = -78539815, ■whence, retaining only seven figures, 7r= 3-141593. This is correct to the last figure. 112-113] GEOMETRICAL APPLICATIONS. 279 The formula (1) would have given iff= -78498150, T= 3-139926, which is too small by about one part in 2000. 113. Mean Values. Let 2/i, y^, y^ be the values of a function <^{oa) corresponding to n equidistant values of x distributed over the range h — a, say to the values of x which mark the middle points of the n equal intervals Qi) into which this range may be subdivided. The limiting value to which the arithmetic mean -(2/i+y!!+ — +2/») (1) tends, as n is indefinitely iacreased is called the ' mean value' of the function over the range h — a. Since h = {b — a)jn, the expression (1) may be written yih + y^h+ ...+y„A b-a and the limiting value of this for n = oo , A, = 0, is r* (f) (x) dx -r (2)- 0— a In the geometrical representation the mean value is the altitude of the rectangle on base h — a whose area is equal to that included between the curve y = ^ {x), the extreme ordinates, and the axis of x. See Kg. 52, p. 218. The theorem of Art. 89, 3° may now be stated as follows : The mean value of a continuous function, over any range of the independent variable, is equal to the value of the function for some value of the independent variable within the range. In applying the conception of a mean value it is essential to have a clear understanding as to what is the independent variable to which (in the first instance) the equal increments are given. Thus, in the case of a particle descending with a constant accelera- tion g, the mean value of the velocity in any interval of time t^ from rest is f J a 1 /•*■ 1 /-'i - vdt=j \ gtdt^yt^; h Jo - h Jo 280 INFINITESIMAL CALCULUS. [CH. VII i.e. it is one-half the final velocity. But if we seek the mean velocity for equal infinitesimal increments of the space (s), we have, since v^ = 2gs, l[\dsJ^rM = %{2,s,)^; »1 Jo «1 Jo i.e. it is two-thirds the final velocity. Ex. 1. The mean value of sin 6 for equidistant intervals of 6 ranging from to tt is 1 (-T 2 - Bm0de = -=-Q3&(x,y) (1). This may be interpreted, geometrically, as the equation of a surface (Art. 45). Take any finite region S in the plane xy, and let a cylindrical surface be generated by a straight line which meets always the perimeter of S, and is parallel to the axis of z. We consider the volume (F) included between this cylinder, the plane xy, and the surface (1). See Fig. 72. Fig. 72. If the region S be divided into elements of area 8^1, S^2, 8^3 and if Zi, z^, z,, ... be ordinates of the surface (1) at arbitrarily chosen points within these ele- ments, then, the coordinate axes being supposed rectangular, the sum zMi + zM!>-^zMi+ (2). will give us the total volume of a system of cylinders, of altitudes Zi,Zsi,Zs,---, standing on the bases BAi, SA^, SA^, 284 INFINITESIMAL CALCULUS. [CH. VII And if the function (/> (x, y) be subject to certain generally satisfied conditions*, the above sum will, when the dimensions of hAi, hAi, hAs, ... are taken infinitely small, tend to a unique limiting value, viz. the aforesaid volume V. If the subdivision of the region S be made by lines drawn parallel to the axes of 00 and y, the elements 8.4i, SAi, SAs, ... are rectangular areas of the type SxSy, and the sum (2) may be denoted by XtzBxSy (3), where the sign S is duplicated because the summation is in two dimensions. The limiting value of this sum is denoted by JJzdccdy (4), and we have the formula V=Jf4>(x,y)da!dy (5). The expression on the right hand is called a ' double integral'; it is of course not determinate unless the range of the variables as, y, as limited by the boundary of S, be specified. The volume V may, however, be obtained in another way. If f(x) denote the area of a section by a plane parallel to yz, whose abscissa is x, we have, by Art. 104, >dx (6), F=/V(*), ■I a where a, b are the extreme values of x belonging to the area S. But, by Art. 98, f{x)=j\dy (7), where a, ^ are the extreme values of y in the section /(«), and are therefore in general functions of x. Hence we have V = ^^^^y{x,y)dy^dx (8), * The already stipulated condition of continuity is sufficient, but the proof is simplified if we introduce the additional condition that (x, y) shall have only a finite number of maxima and minima within any finite area of the plane xy. Cf. Art. 87. /: 114] GEOMETRICAL APPLICATIONS. 285 or, as it is more usually written, r=rrcl,(a,,y)da!dy (9)* J a J a If the limits of both integrations be constants, i.e. if the region S take the form of a rectangle having its sides parallel to X and y, the volume V is expressed also by {x,y)dydx (11). __ X J a J a This is illustrated by Fig. 30, p. 96. In other cases the limits of the respective integrations require to be adjusted when we invert the order. The above explanations have been clothed in a geometrical form, but this is not of the essence of the matter. The same principles are involved, for example, in the calculation of the mass of a plane lamina, having given the density at any point (x, y) of it, and in many other physical problems. See Chapter viii. Another mode of decomposition of the area S is often useful. Taking polar coordinates r, 6 in the plane xy, we may divide the area into quasi-rectangu|ar elements by means of concentric circles and radii. The area of any one of these elements may be denoted by rW . 8r, if r be the arithmetic mean of the radii of the two curved sides. The formula (8) is then replaced by V=S^zrd6dr (12), where z is supposed given as a function of r and 0. After what precedes, the meaning of a ' triple-integral,' III' whether the axes be rectangular or oblique. The suminations % are supposed to include every particle of the system ; for example S (mx) stands for rn^aci + m^i -t- . . . -f- m„a;„. * The simplest and best being the vector definition proposed by Qrass- mann, Ausdehnungslehre (1844). 19—2 292 INFINITESIMAL CALCULUS. [CH. VIII If the origin be at the centre of mass we have S(wia;) = 0, 2(my) = 0, 2(m^) = (2). If the masses twi, m^, ... m„ be all equal, the formulae (1) reduce to x = -2(x), y = --Z(y), z^-t{z) (3). n n n If the masses wii, m^, ... m„, though unequal, be com- mensurable, then each may be regarded as made up by- superposition of a finite number of particles, all of the same mass, and the formulae (3) will then still apply. And since incommensurable magnitudes may be regarded as the limits of commensurable magnitudes, the formulae (1) may in all cases be interpreted as expressing that the distance of the centre of mass from any plane is in a sense equal to the mean distance of the whole mass from that plane. An important principle, constantly made use of in the determination of centres of mass, is that any group of particles in the system may be supposed replaced by a particle equal in mass to the whole group, and situate at the mass-centre of the group. This is an easy deduction from (1). In the case of a continuous distribution of matter, if we denote by hM an element of mass situate about the point {x, y, z), the formulae become S(^gJf) . H^ S(ygilf) 2(.8ilf) *^''" 2(Sif) ' ^~^"" S(8if) ' ^^'"" S(8if) (4). or . _ /Xr»/3 dxdydz _ _ SSJyp dx dydz _ _ fjjzp dxdydz ~ Jfjpdxdydz ' ^ ~ JJfpdxdydz ' ^ ~ Jjfpdxdydz (5). In the application to particular problems or classes of problems the integrations can be greatly simplified. This is illustrated in the following paragraphs. 116-118] PHYSICAL APPLICATIONS. 293 117. Line-Distributions. If ju, be the line-density, and 8s an element of the arc, we have '"- i fids' y~7]2s ^^^• We consider only plane curves, and take the axes of x, y always in the plane, so that 1=0. Ex. In the case of a uniform circular arc, if the origin be taken at the centre, and the axis of x along the medial line, we have y = 0, by symmetry. Also, putting /t=l, as we may do, without loss of geiierality, we have, writing x = a cos $, 8s = aW, I a cos, 6 . add I cos &dd r adO "■ sm a . a= . a... .(2), a if 2a denote the angle which the whole arc subtends at the centre. As a increases from an infinitely small value to tt, x decreases from a to 0. For the semicircle, we have a = ^w, and x = ~a = '637a. TT 118. Plane Areas. If the surface-density be uniform, and if the area in question possess a line of symmetry, then taking this line as axis of X, we have, if y be the ordinate of the bowiding line, '"- jydx ' y " ^^>- the integrals being taken between the proper limits. Ex. 1. For an isosceles triangle, taking the origin at the vertex, we may write y = mx, and therefore /■ft x^dx 7K =1/' (2), xdx Jo h being the altitude. 294 INFINITESIMAL CALCULUS. [CH. VIII Ex. 2. For a semicircular area of radius a, we find I xj(a^ — sc") dx = Jo Jo .(3). dx The integrations are exactly the same as in Art. 115, Ex. 1 ; and the result is jE=i-a = . 4244a (4). oir Ex. 3. For a segment of the parabola 1/^= ^ax (5) bounded by the double ordinate x = h, we have (•ft I xydx I tc^cfe -^' \ = fA (6). eh rn I ydx Jo Jo 3^dx The formute (1) will apply also to the case of oblique axes, provided the axis of x bisect all chords drawn within the area parallel to the axis of y. For instead of an ele- mentary rectangle y^x we have now an elementary parallel- ogram yhx sin w, where w is the inclination of the axes. The constant factor sin oj, occurring both in the numerator and in the denominator of the expression for x, will cancel. Thus, with the same integrations as in Exs. 1, 2, 3, above, we ascertain that the mass-centre of a triangle is in any median line, at a distance of § of the length, from the vertex; that the mass- centre of a semi-ellipse bounded by any diameter lies in the conjugate semi-diameter, at a distance 4/37r of its length from the centre; and that the mass-centre of any segment of a parabola is in the diameter bisecting the bounding chord, and divides the breadth in the ratio 3 : 2. In the case of any area included between a curve y = ^ (x), the axis of x, and two bounding ordinates, the 118] PHYSICAL APPLICATIONS. 295 mass-centre of an elementary parallelogram ySx sin w will be at the point {x, ^y). Hence x = - '- /, , y = - ,j (7), Jydx ^ Sydx the factor sin « cancelling as before. Ex. 4. The mass-centre of the area included between the parabola 2/'' = 4aa; (8), the axis of x, and any ordinate x = h, is given by /•ft ai / xdx ^ = T^ y=-J^ = f(aA)i = |A (9), Jo if A be the ordinate corresponding to a; = h. In polar coordinates, we may resolve any sectorial area into elementary triangles ^'S6. The mass-centre of any one of these is ultimately at the point (|r, 6). Hence referring to rectangular axes, of which the axis of x coincides with the origin of 0, we have _ __ J^r cos e . ^r'de ^ yr" cos OdO ^ ''- Jir'de - ^fr^de '[ ^ _ fir smd .^'de _ i J 1^ sin 9de [" ^ ^' y- Jir'dd - Ur'de j the integrals being taken between proper limits of 0. Ex. 5. In the case of a circular sector of angle 2a, taking the origin at the centre, and the initial line along the bisector of the angle, we have y = 0, and *»'/: cos 6(19 „ sm a . . = 1 a (11). w a a If the surface-density be not uniform, it is sometimes convenient to take as the element of area the area included between two consecutive lines of equal density {ra\ and therefore x = ^a (10) as before. For a spherical lune, of angle 2a, we project on the plane perpendicular to the central radius. The two bounding meridians project into the two halves of an ellipse whose semi-axes are a and a sin a. Hence S=4:aa'', 5 = ■to' sin a, and therefore , sma = i7r.---.a (11), as above. 121. IVIass-Centre of a Solid. In the case of a homogeneous solid, if the area of a section by a plane perpendicular to os be denoted as in Art. 104 by f(x), the ^-coordinate of the mass-centre of the volume included between two such sections is obviously given by the formula J xf{x)d(c lf{x)dx ^'^' taken between the proper limits of oc. It will sometimes happen that the mass-centres of a system of parallel sections lie in a straight line ; in this case, taking the straight line in question as axis of x, we have ^ = 0, and = 0. In the case of a solid of revolution, taking the axis of x coincident with the axis of symmetry, we have f{x) = -n-f; if y be the ordinate of the generating curve. Hence Jxy^dx JyHx ^''^- 304 INFINITESIMAL CALCULUS. [CH. VIII Ex. 1. Tn the case of a right circular cone, the origin being at the vertex, /(a;) oc a?, so that rh j a?dx Jo x = a?dx Jo if h be the altitude. i:- -ih (3), Ex. 2. For the segment of an elliptic paraboloid > 2? 2x = y- + - (4) p q cut off by a plane x = h, since/(a!) oc x, as in Art. 106, Ex. 1, we have 1 a?dx ^ = '-jl -^^ (5). xdx Ex. 3. For a hemisphere of radius a, putting y^ = a^ — a?, we have /"a X {a^ - a?) dx 1: = l« (6). (a^ - 33^) dx Jo The same formula gives the position of the mass-centre of the half of the ellipsoid - + ^' + - = 1 (7) which lies on the positive side of the plane yz, sincey (x) in this case also varies as a" - a;". See Art. 106, Ex. 2. Ex. 4. In the case of the more general formula fix) = A+Bl + c'^, (8), the aj-coordinate of the mass-centre of the volume included between the planes x = and x = his, by (1), ._ iA + ^B + it! 2A' + A" ''- A+^B + iC^'A + iA' + A"" ^^'' 121] PHYSICAL APPLICATIONS. 805 where, as in Art. 107, A, A', A" denote the areas of the sections £c = 0, x = ^h, x = h, respectively. The distance of the centre of mass from the middle section is therefore ^-^^^ 2(i + 4^V I^)^ (1°)- This result has the same degree of generality as that of Art. 107. The application of the formula (1) is easily extended to the case of oblique axes. Denoting hjf(x) the area of a section parallel to the plane yz, the appropriate element of volume is f(x) Sx sin \, where X is the inclination of the axis of x to the plane yz. The constant factor sin \, occurring both in the numerator and in the denominator of the expression for ad, -will cancel, and we are left with the same form as before. Ux. 5. In the case of a cone, or a pyramid, on a plane base, taking the origin at the vertex, and the axis of 'a; along the line joining to the mass-centre G of the area of the base, the area of any section parallel to the base wiU vary as the square of its intercept a; on OG. We are thus led as before to the result where h now = OG. Hence the mass-centre of the pyramid, or cone, is at a point ^^^ in OG, such that OH=^OG. In a similar manner the investigations Exs. 2, 3, above, can be modified so as to apply to any segment of a paraboloid, and to a semi-ellipsoid cut off by any diametral plane. For special forms of solid other methods of decomposition into elements will suggest themselves. Ex. 6. Thus in the case of a 'spherical sector,' i.e. the portion cut out of a solid sphere by a right circular cone having its vertex at the centre, the volume of a thin spherical stratum of radius r is proportional to r'Sr. Also the distance of the mass-centre of this stratum from the vertex is, by Art. 120, Ex. 1, ^{r + r cos a), = r cos'' Ja, L. 20 306 INFINITESIMAL CALCULUS. [CH. VIII ■where a is the semi-angle of the cone. Hence the distance of the mass-centre of the sector from the vertex is 77 X = —^ cos' Ja = fa cos' Ja (11), r^dr Jo ■where a is the external radius. For the hemisphere we have a = Jtt, and x = ^a, as in Ex. 3, above. Ex. 7. To find the mass-centre of a ■wedge cut from a solid sphere by two planes meeting in a diameter. Let a be the radius of the sphere, and 2a the angle between the planes. Divide the -wedge, by planes through the aforesaid diameter, into elementary wedges of infinitely small angle, and let X be the distance from the diameter of the mass-centre of any one of these. Transferring the mass of each elementary wedge to its centre of mass, we obtain, as in Art. 120, Ex. 2, a uniform circular arc, -whose centre of mass will be at a distance from the centre equal to (x sin a) jo. Since, for o= ^ir, this must equal •§(1, we infer that Sir in agreement with Art. 118, Ex. 6. Hence the distance, from the edge, of the mass-centre of the given wedge of angle 2a, is Sir sin a T6--V--" x=—. .a (12). 122. Solid of Variable Density. In a solid of variable density, if the surfaces of equal density be parallel planes, and if f(x) be the area of the section made by one of these planes, supposed expressed in terms of the intercept on the axis of x, we have, in place of Art. 121,(1), jocpf{x)dx_ , ''- Spf{x)dx ^'^- For example, in the case of a solid of revolution, in which the surfaces of equal density are planes perpendicular to the 121-123] PHYSICAL APPLICATIONS. 307 axis (»), we have /(«) = Try", where y refers to the generating curve, and therefore _ Ipxy^dx ,a\ '"--hh ^'^- Ex. 1. Thus in the case of a hemisphere whose density p varies as the distance {x) from the bounding plane, writing y' = 0^ — 11?, we have ra I aj" {a' — a?) dx S-ys = A» (3). I xiaF — x^) dx Jo For other laws of density, other methods of decomposition may suggest themselves. For example, when the density is a function of the distance from a fixed point, a decom- position into concentric spherical shells is indicated. 123. Theorems of Pappus. 1°. If an arc of a plane curve revolve about an axis in its plane, not intersecting it, the surface generated is equal to the length of the arc multiplied by the length of the path of its centre of mass. Let the axis of x coincide with the axis of rotation, and let y be the ordinate of the generating curve. The surface generated in a complete revolution is, by Art. Ill, equal to lirlyds, the integration extending over the arc. But if y refer to the mass-centre of the arc, we have ^ Ids ' by Art. 117. Hence 2Tr Jyds = 2iry X Jds (1), which is the theorem. 2°. If a plane area revolve about an axis in its plane, not intersecting it, the volume generated is equal to the 20—2 308 INFINITESIMAL CALCULUS. [CH. VIIl area multiplied by the length of the path of its centre of mass. If SA be an element of the area, the volume generated in a complete revolution is Iim2(27r2/.S4). But if y refer to the centre of mass of the area, we have by Art. 116. Hence limX(2Try.SA) = 2Tryxlim'Z{SA) (2), which is the theorem*. The revolutions have been taken to be complete, but the restriction is obviously unessential. Ux. 1. The ring generated by the revolution of a circle of radius b about a line in its own plane at a distance a from its centre. The surface is 2ir6 x 2ira, = iir^ah ; and the volume is ir6° x 2ira, = 2ir°a6'. Of. Art. 105, Ex. 3, and Art. Ill, Ex. 2. Ex. 2. A segment of the parabola y'' = iax, bounded by the double ordinate x = h, revolves about this ordinate. If 2k be the length of the double ordinate, the area of the segment is ^hk, by Art. 98 ; and the distance of the centre of gravity from the ordinate is fA, by Art. 118. Hence the voliime generated is ^hkx^Th = ^nh^k. The theorems may be used, conversely, to find the mass- centre of a plane arc, or of a plane area, when the surface, or the volume, generated by its revolution is known indepen- dently. * These theorems are contained in a treatise on Mechanics by Pappus, who flourished at Alexandria about a.d. 300. They were given as new by Guldinus, de centra gravitatU (1635 — 1642). (Ball, History of Mathematics,) 123-124] PHYSICAL APPLICATIONS. 309 Ex. 3. Thus, for a semicircular arc revolving about the diameter joining its extremities, we have iraxlny - ^ira?, ■whence y = - a. Again, for a semicircular area revolving about its bounding diameter, ^TTci? X 27ry = ^ira?, whence y=-^a. Cf. Arts. 117, 118. 124. Extensions of the Theorems. A similar calculation leads to a simple formula for the volume of a prism or a cylinder (of any form of cross-section) bounded by plane ends. In the first place we will suppose that one of the ends, which we will call the base, is perpendicular to the length. Let P be any point of the base, and let z be the length of the ordinate PP' drawn parallel to the length, to meet the opposite end in P', and let z be the ordinate of the mass- centre of the oblique end. If ^A, ZA' be corresponding elements of area at P and P', we have ^-^™ 2(8iL') -^™ 2(S4) ' since hA, being the orthogonal projection of ZA' , is in a constant ratio to it. Hence the volume of the solid = %{z.hA) = zxt{hA) (1); that is, it is equal to the area of the base multiplied by the ordinate of the mass-centre of the opposite face. It is easily seen (Art. 131) that this is the same as the ordinate drawn through the mass-centre of the base. A prism or a cylinder with both ends oblique may be regarded as the sum or as the difference of two prisms or cylinders each having one end perpendicular to the length. 310 INFINITESIMAL CALCULUS. [OH. VIII We infer that in all cases the volume is equal to the area of the cross-section multiplied by the distance between the mass-centres of the two ends. Ex. Tlie volume of the wedge-shaped solid out oflf from a right circular cylinder by a plane through the centre of the base, making an angle o with the plane of the base, is 4 ^TTfls" X ^a tana =§a" tana; cf. Art. 114, Ex. 1. The theorems of Pappus may be generalized in various ways; but it may be sufficient here to state the following extension of the second theorem. If a plane area, constant or continuously variable, move about in any manner in space, but so that consecutive positions of the plane do not intersect within the area, the volume generated is equal to SSda. (2), where 8 is the area, and da- is the projection of an element of the locus of the mass-centre of the area on the normal to the plane. If ds denote an element of this locus, and 6 the angle between ds and the normal to the plane, the formula may also be written JScosOds (3). This theorem is the three-dimensional analogue of the proposition of Art. 101, relating to the area swept over by a moving line. It is a simple coroUarj'^ from the theorem above proved. EXAMPLES. XXXVIII. 1. A quadrant of a circle revolves about the tangent at one extremity; prove that the distance of the mass-centre of the curved surface generated, from the vertex, is STSa. 2. The mass-centre of either half of the surface of an anchor-ring cut oflf by the equatorial plane is at a distance 2bjir from this plane, where b is the radius of the generating circle. 124] PHYSICAL APPLICATIONS. 311 3. Two equal circular holes of angular radius a are made in a uniform thin spherical shell, and the angular distance of their centres is 2/8. Prove that the distance of the mass-centre of the remainder from the centre of the sphere is ^a sin^ a sec a cos p, where a is the radius. 4. A portion of a paraboloid of revolution is bounded by two planes perpendicular to the axis. Prove that the distance of the centre of mass of the solid thus defined from the middle point of its axis is la^_6^ where h is the length of the axis, and a, b are the radii of the two circular ends. 5. The distances from the centre of a sphere of radius a of the centres of mass of the two segments into which it is divided by a plane at a distance a from the centre of figure are Ha±cf 4 2a ±c 6. By dividing a tetrahedron into plane laminae parallel to a pair of opposite edges, as in Art. 104, Ex. 2, prove that the mass-centre bisects the line joining the middle points of these edges. 7. The figure formed by a quadrant of a circle of radius a and the tangents at its extremities revolves about one of these tangents ; prove that the distance of the mass-centre of the solid thus generated from the vertex is •869a. 8. A solid ogival shot has the form produced by rotating a portion APN of a parabolic area, where A is the vertex, and PiV an ordinate, about FN ; prove that the mass-centre divides the axis in the ratio 5:11. 9. AP is an arc of a parabola beginning at the vertex, and PN is a perpendicular on the tangent at the vertex ; prove that the mass-centre of the solid generated by the revolution of the figure APN about AN is at a distance from A equal to ^AN. 312 INFINITESIMAL CALCULUS. [CH. VIII 10. A right circular cone is divided into two halves by a plane through the axis ; prove that the distance from the axis of the mass-centre of either half is aj-ir, where a is the radius of the base. 11. The mass-centre of the volume included between two equal circular cylinders, whose axes meet at right angles, and the plane of these axes, is at a distance from this plane equal to I of the common radius. 12. The mass-centre of a hemispherical shell whose inner and outer radii are a and 6 is at a distance 3 (a + 6) («" + &') 8 a^ + ah + V from the centre. 13. The mass-centre of a hemisphere of radius a whose density varies as the nth power of" the distance from the base is at a distance (n + 1) (n + 3) (M + 2)(n+4)" from the centre. 14. If the ellipse -3+T2 = 1 revolve about the axis of x, the mass-centre of the curved surface generated by either of the two halves into which the curve is divided by the axis of y is at a distance 2 g" + a& -I- 6° a 3 a + b ' b + a (sin"' e)/e from the centre, where e is the eccentricity, it being supposed that b a. 15. Apply the theorems of Pappus to find the volume and the curved surface of a right circular cone, and of a frustum of such a cone. 16. A groove of semicircular section, of radius b, is cut round a cylinder of radius a ; prove that the volume removed is TT'ab' - ^-kI^. Also that the surface of the groove is 27r' ab-i-n-bK 125] PHYSICAL APPLICATIONS. 313 17. A screw-thread of rectangular section is cut on a cylinder of radius R. Prove that the volume of one turn of the thread is i-rahR + TTob^, where a, b are the sides of the rectangle, b being that side which is at right angles to the surface of the cylinder. 18. The mass-centre of either half of the volume of an anchor-ring cut off by the equatorial plane is at a distance Abj Sir from the plane, where 6 is the radius of the generating circle. 125. moment of Inertia. Radius of Gyration. If in any system of particles the mass of each particle be inultiplied by the square of its distance from a given line, the sum of the products thus obtained is called the ' moment of inertia' of the system with respect to that line. In symbols, if mi, OT2, W3, ... be the masses of the several particles, j»i, ^2, p,, ... their distances from the line, and if / denote the moment of inertia, we have 7 = mipi" + ma^a^ -I- rrispa' -f . . . = S (mpO (1). In the dynamical theory of the rotation of a solid body about a fixed axis it is shewn that the moment of inertia as above defined is the proper measure of the inertia of the body as regards rotation, just as the mass of the body measures its inertia in respect of translation. Thus if M be the mass of a body moving in a straight line with velocity u, its momentum is Mu ; and if F be the extraneous force, we have |W = ^ (2)- In like manner, if I be the moment of inertia of a body rotating about a fixed axis with angular velocity {«=) (o)> the axis of x, and two bounding ordinates, we may divide the area into elementary strips yhx. The square of the radius of gyration of a strip is ^y^, by Ex. 1, above. Hence ^^^_ !W-ydoo ^ hStdx fydx Jydx '' the integrals being taken between the proper limits of x. Ex. 5. To find the radius of gyration of an isosceles tri- angular area, about its line of symmetry. Taking the origin at the vertex, and the axis of x along the line of symmetry, the equations of the two sides will be a 316 INFINITESIMAL CALCULUS. [CH. VIII where a is half the base, and h the altitude. The radius of gyration for the whole triangle is evidently the same as for either half, whence ^ah ~ ' 1«2 = ia .(7). The same result obviously holds for the radius of gyration of a rhombus about a diagonal. Ex. 6. To find the radius of gyration of the area bounded by the ellipse 5-^S=i w. about the axis of x, we have /a fdx -a ^=^F^ W If we put x = a cos ^, y = h sin <^, this becomes A2=i.62 [^\in* J = lb^ (10). OTT Jo Similarly, for the radius of gyration about the minor axis, we should find ^^ = K (11). 127. Three-Dimensional Problems. The following, problems in three dimensions are im- portant. Sx. 1. To find the radius of gyration of a uniform thin spherical shell of radius a about a diameter. Take the origin at the centre, and the axis of x along the diameter in question. If the shell be divided into narrow zones by planes perpendicular to x, the area of anyone of these may be denoted by 2TraSx, by Art. Ill, Ex. 1, and its radius of gyration by y. Hence /a y^ . 2-7raJx ^ "" - r{a'-(^)dx = ^a' (1). *'- A. ^ 126-127] PHYSICAL APPLICATIONS. 317 Ex. 2. To find the radius of gyration of a uniform solid sphere about a diameter. Dividing the volume into thin concentric shells, and using the result of Ex. 1, we have ra /-5). tSc / a^ (l Adx -'•^^ ^- _ %lc =*"' (!«)' and similarly y' = ^6', z' = ic''. Hence A„» = ^ (6^ + c^), V = i(«' + «'). V = i(«^+ i^)...(19). 129. Comparison of Moments of Inertia about Parallel Axes. The following theorems give a simple means of comparing moments of inertia about different parallel axes. 1°. The mean square of the distances of the particles of a system from any given plane exceeds the mean square of the distances from a parallel plane through the centre of mass by the square of the distance between these planes. Let «!, oc^, sOi, ... be the distances of the particles mi, TTia, TO3, ..., respectively, from the first-mentioned plane; and let x denote the distance, of the centre of mass. If we write Xi^X + ^i, a;a=« + fa, X3 = X + ^3,...,...(1), we have - _ S (ma;'-) _ 'Z{m(x + ^y} ^- S(to) S(m) _ S (m) . ^ + 2^ . S(m^) + S (mp) "" . SCm) Now t (mf ) = 0, by Art. 1 1 6 ; and the fraction S (mf )/2 (m) is denoted, in accordance with our previous notation, by I". Hence ^=f + «" ..(2). I.. 21 322 INFINITESIMAL CALCULUS. [CH. VIII 2°- The mean square of the distances of the particles of a system from a given line exceeds the mean square of the distances from a parallel line through the centre of mass by the square of the distance between these lines. If the first-mentioned line be taten as axis of z, and if the coordinates be rectangular, we have, by the preceding case, ^+^=^+^=f+V« + (^ + ^) (3). Hence if k denote the radius of gyration of the system about an axis through the centre of mass, and V the radius of gyration about any parallel axis, we have li/^ = k^ + h'' (4), where A is the perpendicular distance between the two axes. This is a very important result in the dynamical appli- cation of the subject. J'aj. 1. The radius of gyration of a rectangle about a side is given by k'^ = a? + ^a'=*a'' (5), if 2a be the length perpendicular to that side. This may be easily verified by direct integration. Sx. 2. With the same notation as in Art. 128, Ex. 4, the radius of gyration of a rectangular parallelepiped about an edge (2c) is given by k'' = (a'' + b'') + i(a^ + ¥) = :^{a' + h') (6). Bx. 3. The radius of gyration of a uniform circular disk of radius a about an axis through a point on the circumference, normal to the plane of the disk, is given by k'' = a^+^a''=^^ (7). Similarly, the radius of gyration about a tangent line is given by >fc'2 = a2+i„2^|„2 ^8) 130. Application to Distributed Stresses. The calculations of radii of gyration of plane areas have an application in the theory of stresses distributed over plane areas. 129-130] PHYSICAL APPLICATIONS. 323 Thus, in deturmining the centre of pressure of an area in contact with a homogeneous liquid, if the axis of y be the line in which the plane of the area cuts the free surface, the a!-coordinate of the centre of pressure is, by Art. 119, ^=^^°^2Vs:z) • In our present notation, we have ultimately, and therefore r=J (2). Ex. In the case of a circular area, having its centre at a distance h from the line in which its plane meets the surface, we have tt? = h^ + \a^, x = h, a? and therefore ^=h + \j (3). Again, in the theory of flexure, referred to in Art. 119, the intensity of the force at any point of the cross-section of a beam is equal to f «. where y is the distance from a certain line in the plane of the section, called the 'neutral line,' B is the radius of the curve into which the beam is bent, and ^ is a certain coefficient depending on the miaterial. If hA be an element of area, the total force across the section is the limit of |s(j.8il) (.5). In a pure flexure, this force is, by hypothesis, zero; hence, by Art. 116, the neutral line will pass through the mass- centre of the section. The stresses on the cross- section now reduce to a couple. The moment of this couple about the neutral line (or about any Une parallel to it) is got by multiplying the force on each element 21—2 324 INFINITESIMAL CALCULUS. [CH. VIII 8^ by its distance from the neutral line. In this way we get, as the value of the ' flexural couple,' |xlimS(2/»8il) (6), or EAI^jR (7), where A is the area of the cross section, and k is its ra,diu8 of gyration about the neutral line. The ratio of the flexural couple to the curvature (1/22) is called the ' flexural rigidity ' of the bar. For bars of the same material it varies as Alt'. 131. Homogeneous Strain in Two Dimensions.* Taking first the case of two dimensions, let us suppose that, in any plane figure, the rectangular coordinates of a point (x, y) are changed to (a/, 'i/), where x' = ax, y' = ^y ••■■0), a and ^ being given constants. The resulting deformation is of the kind called ' homogeneous strain ' ; the coordinate axes are called the ' principal directions ' of the strain ; and the constants a, /8 are called the ' principal ratios.' A particular case is the method of ' orthogonal projection.' If the axis of x be the common section of the two planes, we have a=l, p = cos6, where 6 is the inclination of the plane of the original figure to the plane of projection. Since the substitution (1) is of the first degree, it follows that straight lines will transform into straight lines. Also, since infinitely distant points transform into infinitely distant points, parallel straight lines will transform into parallel lines, and therefore parallelograms into parallelograms. Hence, further, equal and parallel straight lines will transform into equal and parallel straight lines; so that lines having originally any given direction are altered in a constant ratio, the ratio varjdng however (in general) with the direction. The new direction of a straight line is of course in general different from the original direction. * This is the same as Eankiue's ' Method of Parallel Projection,' Applied Mechanics, Arts. 61, 82, 580. 130-131] PHYSICAL APPLICATIONS. 325 Again, any algebraic curve whatever transforms into a curve of the same degree. In particular, a circle x^ + y'' = a-' (2), Fig. 73. transforms into an ellipse where a' = aa, V = fia; . .(3), (4), 326 INFINITESIMAL CALCULUS. [CH. VIII and it is evident that by a proper choice of the ratios a, ^ a circle can be transformed into an ellipse of any given dimen- sions, and vice versd. Also since a system of parallel chords, and the diameter bisecting them, transform into a system of parallel chords, and the diameter bisecting them, it is evident that perpendicular diameters of the circle transform into conjugate diameters of the ellipse. Further, areas are altered by transformation in the constant ratio a/8. For this is evidently true of any rect- angle having its sides parallel to the principal directions of the straia ; and any area whatever can be approximated to as closely as we please by the sum of a system of rectangles of this type. JEx. 1. Thus, the area of the ellipse (3) is o^ times that of the circle (2); and so = a)8 . ira' = ir . aa . /8a = na'b'. Again, a chord cutting off a segment of constant area from a circle touches a fixed concentric circle. Hence, a chord cutting oflf a segment of constant area from an ellipse touches a similar, similarly situated, and concentric ellipse. Again, centres of mass of areas, considered as sheets of matter of uniform surface-density, transform into centres of mass. For, if BA, SA' be corresponding elements of area, we have since BA' = a^BA. Hence, and by similar reasoning, a! = ax, i/ = ^y (5). Ex. 2. The centre of mass of a semicircular area is on the radius perpendicular to the bounding diameter, at a distance 4/3ir of . its length from the centre. Hence, the mass-centre of a semi-ellipse, bounded by any diameter, lies on the conjugate semi-diameter, at a distance of 4/37r of its length from the centre. 131-132] PHYSICAL APPLICATIONS. 327 Finally, mean squares of distances from the axes of x, y transform into mean squares of distances. Thus Hence, and by similar reasoning, «^=aW, y^ = /3y (6). Ex. 3. The mean squares of the distances of points within the circle (2) from the coordinate axes are Hence, for the ellipse (3), ^ = iaV=ia'^ ^=i^V = i6'=' (7). The radius of gyration of an elliptic area about a line through the centre normal to the plane of the area is therefore given by l^=\{a■' + h^^) (8), where a', V are the principal semi-axes. 132. Homogeneous Strain in Three Dimensions. There is a similar method of transformation in three dimensions, the formula of transformation being now x' = ajc, y' = Py, «' = 7« (1), where the axes are supposed rectangular. It is easily seen that parallel planes transform into parallel planes ; and equal and parallel straight lines into equal and parallel straight lines. Also, the sphere x" + y'' + z" = a? (2), transforms into the ellipsoid a? y"" z^ _ ^.+ j^ + 3^-l (^). where a' = ao, b' = ^a, c' = 7a (4) ; and a set of three mutually perpendicular diameters of the sphere transform into a set of conjugate diameters of the ellipsoid. 328 INFINITESIMAL CALCULUS. [CH. VIII Again, volumes are altered by the transformation in the constant ratio a/87. ^^^ ^^^^ ^ obviously true of any rectangular parallelepiped having its edges parallel to the coordinate axes; and any volume whatever can be approxi- mated to as closely as we please by the sum of a system of such parallelepipeds. Ex. 1. The volume of the ellipsoid (3) is a;8y.|™» = |W6V. Again, a plane cutting off a segment of constant volume from an ellipsoid touches a similar, similarly situated, and concentric ellipsoid. By reasoning similar to that employed in the preceding Art., we learn that centres of mass of volumes, considered as occupied by matter of uniform density, transform into centres of mass. Also that mean squares of distances from the coordinate planes transform into mean squares of distances. < Ex. 2. The mass-centre of a uniform solid hemisphere is on the radius perpendicular to the bounding plane, at a distance of |- of its length from the centre. Hence the mass-centre of a semi-ellipsoid cut off by any diametral plane is on the radius conjugate to that plane, at a distance of f of its length from the centre. Ex. 3. The mean squares of the distances of points within the sphere (2) from the coordinate planes being assumed to be ^=i«^ ?=i'^% «~=K (5), it follows that, for the ellipsoid (3), ^ = i«'^ ¥ = W, ^ = i«" (6). The radii of gyration about the principal axes of the ellipsoid are therefore given by 132] PHYSICAL APPLICATIONS. 329 EXAMPLES. XXXIX. 1. The squares of the radii of gyration of a rhombus about its diagonals, and about an axis through its centre normal to its plane are respectively, where 2a, 26 are the lengths of the diagonals. 2. The radius of gyration, about the axis, of the area of a parabolic segment cut off by a double ordinate 26, is given by 3. The radius of gyration of the same segment about the tangent at the vertex is given by where h is the length of the axis of the segment. 4. The square of the radius of gyration of a semicircular area of radius a, about an axis through its centre of mass perpendicular to its plane, is 16 (*^£) a'. 5. The radius of gyration, about the axis, of a segment of a paraboloid of revolution, cut off by a plane perpendicular to the axis, is given by where 6 is the radius of the base. 6. Find by direct calculation the radii of gyration of the volume and surface of an anchor-ring about its axis. 7. The square of the radius of gyration of a uniform circular arc of radius a and angle 2a, about the middle radius, is J 2 /^ sin 2a\ 8. The radius of gyration of a uniform circular arc of radius a and angle 2a about an axis through the centre of mass, perpen- dicular to the plane of the arc, is given by *■"■ ('-"?)i 330 INFINITESIMAL CALCULUS. [CH. VIII and the radius of gyration about a parallel axis through the middle point of the arc is given by /fc» = 2a=(l-?^"). 9. The square of the radius of gyration, about the axis, of a solid ring whose section is a rectangle with the sides parallel and perpendicular to the axis, is i{a,' + ¥), where a, b are the inner and outer radii. 10. The mean square of the distance, from the centre, of points within an ellipse of semi-axes a, b, is iK + 6^). 11. The mean square of the distance, from the centre, of points within an ellipsoid of semi-axes a, b, c, is iia' + b' + c'). 12. The mean square of the distance,, from an equatorial plane, of the surface of an anchor-ring is ^b', where 6 is the radius of the generating circle. 13. The mean square of the distance, from the same plane, of the volume of the ring, is ^¥. 14. Explain how the method of homogeneous strain can be applied to simplify the determination of centres of pressure in certain cases ; and employ it to find the centre of pressure of a semi-ellipse, bounded by a principal axis, when this axis is in the surface of a liquid. 15. The centre of pressure of an elliptic area is in the diameter P'GP which bisects the horizontal chords, and is at a distance \GF^IGH from the centre C, where H is the point in which PP' produced meets the surface of the liquid. PHYSTCAT. APPLICATIONS. 331 16. The flexural rigidity of a beam of rectangular section varies as the breadth and as the cube of the depth. 17. The flexural rigidity of a beam of circular section is to that of a beam of square section as 3 : ir, if the areas of the sections be equal. 18. If the thickness of a semi-circular lamina of radius a vary as the distance from the bounding diameter, the square of the radius of gyration with respect to this diameter is fa^. 19. If ds be an element of arc of an ellipse, and /8 the parallel semi-diameter, the value of the integral (ds taken round the curve, is 2ir. CHAPTER IX. SPECIAL CURVES. 133. Algebraic Curves with an Axis of Sym- metry. The method of tracing algebraic curves of the type 2/=/W (1)> where /(a;) is a rational function, including the determination of asymptotes, maximum and minimum ordinates, and points of inflexion, has been illustrated in various parts of this book ; see Arts. 14, 15, 50, 68. The study of algebraic curves in general is beyond our limits, but a little space may be devoted to the discussion of curves of the type y'=f{^) (2). Two points of novelty here present themselves. Since the equation gives two equal, but oppositely-signed, values of y for every value of x, the curve will be symmetrical with respect to the axis of x ; also since y'^ must be positive, there can be no real part of the curve within those ranges of x (if any) for which /(«) is negative. Thus if /(«) contain a simple factor x — x^, so that the equation is of the form y^={x-x;){x) (3), the right-hand member will change sign as x passes through 133] SPECIAL CURVES. 333 the value ao^. Hence on one side of the point (a?i, 0) the ordinate is imaginary. Also, we have, at this point, \hx) {x — sciy x — Xi' and therefore, dyjdx = oo . The tangent is therefore perpen- dicular to Ox. If, on the other hand, f{x) contain a double factor, say y'' = (x-x,y4>(x) (4), the right-hand side does not change sign as x passes through the value Xi. Hence the ordinate is real on both sides of the point («,, 0), or imaginary on both sides. In the former case we have two branches of the curve intersecting at an angle and forming what is called a 'node'; in the latter case (xi, 0) is an isolated or 'conjugate' point on the locus. The directions of the tangent-lines at the node are given by \dxj (x-Xif ^^ ^' l{f(x) contain a triple factor, say y' = ia;-x,y(j>(x) (5), the right-hand side changes sign at the point (x^, 0); the curve is therefore imaginary on one side of this point. Also since dyjdx here = 0, the curve touches the axis of x. We proceed to some examples ; beginning with cases where f{x) is integral as well as rational. Ex. 1. In the cases where f(x) is of the first or second degree, say f = Ax + B, y'' = Ax'+Bx + G (6), the curve is a conic having the axis of a; as a principal axis. Ex. 2. The cubical curves f = Aa?+Ba? + Cx + D (7), include some interesting varieties. 334 INFINITESIMAL CALCULUS. [CH. JX (a) If the linear factors of the right-hand side be real and distinct, we may write ay' = (x-a){x-p){x-y) (8), and there is no loss of generality in supposing that a is positive and a < )8 < y. The ordinates are then imaginary for xa, but vanishes for x = p. The point (j8, 0) is here a node ; it niay be regarded as due to the union of the oval in the former case with the infinite branch. (d) If, however, the two smaller of the quantities a, /8, y coalesce, so that af=^{x-af{x-y) (11), y will be imaginary for xa. See Fig. 77. The curve is known as the ' witch ' of Agnesi. Mx. 5. y2 = a;2|±| (15). There is a node at the origin, and the curve cuts the axis of x again at (- a, 0). For x>b, and x<-a, y is imaginary. The line a; = 6 is an asymptote. See Fig. 78, ^^■^- y'=^^ (16). This is obtained by putting a = in (15). The loop now shrinks into a cusp; see Fig. 79. The curve is known as the 'cissoid. 1. 22 338 INFINITESIMAL CALCULUS. [CH. IX Fig. 78. Fig. 79. Ex. 7. y» = a!»^i^ (17). x — a Since y is imaginary for a > a; > — o, except for x = 0, the origin is an isolated point. To find the oblique asymptotes we have -i(l*^43--) W Hence the lines y=±(a' + o) (19) are asymptotes. See Kg. 80. 133J SPECIAL CURVES. 339 '^. Fig. 80. EXAMPLES. XL. 1. Trace the curves y^ = 4iX {I - x), y^ = 3? + x + \. 2. Trace the curve a^ = x'(a — x), and shew that it forms a loop of area ]^a'. Find where the breadth of the loop is greatest. 3. Trace the curve a^t^ = a^ (a^ — a?), and shew that it forms two loops, each of area |a'. [a; = fa] 22—2 340 INFINITESIMAL CALCULUS. [CH. IX 4. Trace the curves ^=a!(a!' — 1), y'=a!'(l - a;). 5. Trace the curve 0^ = 0^ {a? -a?), and shew that it encloses an area \ira\ 6. Trace the curve and shew that it encloses an area |^a". 7. The length of an arc of the curve (Fig. 75), from the vertex to the point whose abscissa is x, is 2l^(9tc+4a)»-^«. 8. The mass centre of the area included between the curve ay' = a? and the line a; = A is at the point (-f-A, 0). 9. If the curve ay'' = a? revolve about the axis of x, the volume included between the surface generated, and any plane perpendicular to the axis, is one-fourth that of a cylinder of the same length on the same circular base. 10. Trace the curves y~x' y~x{\-xy 11. The area included between the curve y^ _a — x a° X (Fig. 77) and its asymptote is Tra". If the same curve revolve about its asymptote, the volume of the solid generated is \i^a\ 12. Trace the curves . CB^ + l , !>?-l SPECIAL CURVES. 341 13. Trace the curves » a^ + 1 a? ' " a? 14. Trace the curves a: „ a; Determine the maximum and minimum ordinates (if any), and the points of inflexion. 15. The area included between the curve (Fig. 79) and its asymptote is ^Tca\ If the same curve revolve about its asymptote the volume of the solid generated is ^-n'd?. 16. Trace the curve and shew that the area included between its two branches and either asymptote is 2a\ 17. Shew that the area included between the curve 2 ,» + « " a-x' (Fig. 78) and its asymptote is J (tt + 4) cs". 18. Trace the curve and shew that the area included between the curve and either asymptote is \ira\ 19. Trace the curve and shew that it forms a loop of area J (tt — 2) cP, 342 INFINITESIMAL CALCULUS. [CH. IX 20. Trace the curve and shew that it encloses an area ^iraK 134. Transcendental Curves; Catenary, Tractrbc. We proceed to the discussion of some important curves, mainly transcendental, which are most conveniently defined by equations of the type already referred to in Art. 64<, viz. «' = ]■ 343 ■(5). y = asec'Jr = a cosh - , ^ a s = a tan ^Ir = a sinh — a .(6). O T N G Fig. 81. Some further properties follow easily from a figure. If Plf be the ordinate, PT the tangent, PG the normal, NZ the perpendicular from the foot of the ordinate on the tangent, we have MZ = y cos -«|r = a, PZ = a tan i^ = s. 344 INFINITESIMAL CALCULUS. [CH. IX Since PZ is equal to the arc of the catenary, it is easily seen that the consecutive position of Z is in ZN; in other words, ZN is a tangent to the locus of Z. Hence this locus possesses the property that its tangent ZN is of constant length. The curve thus characterized is called the ' tractrix,' from the fact that it is the path of a heavy particle dragged along a rough horizontal plane by a string, the other end (iV) of which is made to describe a straight line {OX). The curve has a cusp at A, and the axis of x is an asymptote. Many properties of the tractrix follow immediately from the constancy (in length) of the tangent. For example, since two consecutive tangents make an angle Si/r with one another, the airea swept over by the tangent is given by taken between the proper limits. The whole area between the curve and its asymptote is thus found to be ^ira". 135. LissajouB' Curves. These curves, which are of importance in Acoustics, result from the composition of two simple-harmonic motions in perpendicular directions. They may therefore be repre- sented by x = acos{nt + e), y = bcos{n't + e') (1), and it is further obvious that we may give any convenient value to one of the quantities e, e', since this amounts merely to a special choice of the origin of t. 134-135] SPECIAL CURVES. 345 When the periods 2Tr/w, iir/n' are commensurable, we can by elimination of t obtain the relation between x and y in an algebraic form. Ex. 1. In the case n' =n, we may write x = a cos (nt + e), y = b cos nt . whence r cos e = - sin ni sin e, y sin « = cos nt sin e. a Squaring, and adding, we find a? ^xy y^ . „ —„ =2 cos e + f5 = sm" e a' ah W •(2), .(3). / \ \ ^ S ,/ ^ / '' / / V / / '\ / y ^"•. •„,„__., & Fig. 83. This represents an ellipse. In the special case of € = or e = x, the ellipse degenerates into a straight line X _y . - + r = a .(4). If the equality of periods be not quite exact, the figure described may be regarded as an ellipse which gradually changes 346 INFINITESIMAL CALCULUS. [CH. TX its form owing to a continuous variation of the relative phase (e) of the two component motions. When the ellipse (3) is referred to its principal axes, the coordinates of the moving point take the forms x = a cos (nt + e), y = b sin (nt + e) (5). We identify nt+e with the 'eccentric angle'; and since this increases uniformly with the time it appears that the point (pe, y) moves like the orthogonal projection of a point describing a circle of radius a with a constant velocity was. Since in the transition from the circle to the ellipse any infinitely small chord is altered in the same ratio as the radius parallel to it, we see that in the elliptic motion the velocity at any point P will be n . CD, where CD is the semi-diameter conjugate to GP, G being the centre. The type of motion here considered is called 'elliptic har- monic' Ex. 2. If n' = 2ra, we may write x=a cos nt, y=b cos (2nt + e) (6). Here y goes through its period twice as fast as x, and the point (0, — b cos e) is passed through twice as nt increases by 27r. The curve therefore consists in general of two loops. For c = + Jit, the curve is symmetrical with respect to both axes, the algebraic equation being i'4('-S) (')■ When £ = 0, or ir, the curve degenerates into an arc of the parabola N(^5-') » When the relation of the periods is not quite exact the curve oscillates between these two parabolic arcs as extreme forms*. * A method of constructing Lissajous' curves is indicated in Fig. 83, where the vertical and horizontal lines, being drawn through equidistant points on the respective anxiliar; circles, mark ont equal intervals of time. There are numerous optical and mechanical contrivances for producing the curves. For a description of these, and for specimens of the curves described, we must refer to books on experimental Acoustics. 135-136] SPECIAL CURVES, 347 136. The Cycloid. The 'cycloid' is the curve traced by a point on the circumference of a circle which rolls in contact with a fixed straight line. It evidently consists of an endless succession of exactly congruent portions, each of which represents a complete revolution of the circle. The points (such as J. in the figure) where the curve is furthest from the fixed straight line or 'base' (BD) are called 'vertices'; the points (B) half- way between successive vertices, where the curve meets the base, are the 'cusps.' A line (AB) through a vertex and perpendicular to the base is called an 'axis' of the curve. It is evidently a line of symmetry. It is convenient to employ the circle described on an axis AB as diameter as a circle of reference. Let IPT be any Fig. 84. other position of the rolling circle, I the point of contact with the base, G the centre, T the opposite extremity of the diameter through /, and let P be the position of the tracing- point. Draw PMN parallel to the base, meeting TI and AB 348 INFINITESIMAI, CALCULUS. [CH. IX in M and N respectively, and the circle of reference in Q. If AT, AB be taken as axes of x and y, the coordinates of PwiUbe x = NP = BI+MP, y='AN=GT-GM. Let a be the radius of the rolling circle, and 6 the angle (POT) through which it turns as the tracing point travels from A to P. We have, then, BI = ad, PM = a ain 6, GM= a cos 6, and therefore X = a {6 + sva. d)\ , . y = a(l-cos^)J ^ '' From these equations all the properties of the curve can be deduced. Thus if i^ denote the inclination of the tangent to AT, or of the normal to BA, we have tan i/r = -/ = -5^ / j^ = -— ^ = tan^^, ^ dx dd I dO 1 + cosp ^ whence -^ = ^6 (2). Since the angle TIP is one-half of TCP, it follows that IP is the normal, and PTthe tangent, to the. curve at P. Cf Art. 164, below. Again, to find the arc (s) of the curve, we have (S" "*" @y " ''' ^^^ "^ °°^ ^^' "*" ^'''' ^^ " ^'^^ ''*'^' *^' whence, by Art. 109, s = 2a/cos^^d^ = 4a sin J0, or, in terms of ■<^, s = 4asin'\^ (3), no additive constant being required, if the origin of s be at A. This relation is important in Dynamics. Since TP=TI sin yfr, we have arc ^P = 22^=2 chord 2 Q (4). In particular, the length of the arc from one cusp to the next is 8a. 136] SPECIAL CURVES. 849 00 350 INFINITESIMAL CALCULUS. [CH. IX If we put y' = IM = a{l + cos0) (5), the area included between the curve and the base is given by Jy'dx = a>/(l + cos 0yd0 = ^aPf cos* ^0dd = Sa'Jcos* yfr dyjr. Taking this between the limits + ^tt, we find that the area included between the base and one arch of the curve is three times the area of the generating circle. The curve traced by any point fixed relatively to a circle which rolls on a fixed straight line is called a 'trochoid.' If, in Fig. 84, the tracing point be in the radius CF, at a distance k from the centre, its coordinates will be aj = ffl5 + Asin 6| . y = a-kcoB6 ) ^ '' When k>a we have loops, which in the particular case (k = a) of the cycloid degenerate into cusps. When k . The inclination of GF to OA will be ^ + <^. Hence if we Fig. 86. take as origin of rectangular coordinates, and OA as axis of x, we find, by orthogonal projections, that the coordinates of P are or, since a; = {a + b) cos 6 + b cos {0 + <^), y = {a + b) sinO + b sin (d + = (a — b) cos — b cos — j— i ,(5). y = (a~b)sin0 + b sin — =— 6 The verification is left to the reader ; see Fig. 87. In the hypocycloids we have a > 6, in the pericycloids a' whence, integrating, log r = cot a + const., or r = ae* •="' " (2). Fig. 100. As 6 ranges from — oo to + oo , r ranges from to oo , See Fig. 100. 140] SPECIAL CURVES. 367 Since, by Art. 110, we have clrjds = cos a, it appears that the length of the curve, between the radii r^, r^, is /, — dr= (r^ — ri) sec a. . .(3). 2°. The ' spiral of Archimedes ' is the curve described by a point which travels along a straight line with constant velocity, whilst the line rotates with constant angular velocity about a fixed point in it. In symbols, whence if a = m/k. r = ut, 6 = nt, r = aO ■(4), Fig. 101. Fig. 101 shews the curve. The dotted branch corresponds to negative values of 6. Another mode of generation of this curve has been explained in Art. 138. 368 INFINITESIMAL CALCtJLUS. [CH. IX 3°. The ' reciprocal spiral ' is defined by the equation r = aie (5). If y be the ordinate drawn to the initial line, we have sind f = r sin 6 = a" 6 As approaches the value zero, r becomes infinite, but y approaches the finite limit a. Hence the line y = a is an asymptote. Y Fig. 102. The dotted part of the curve in Fig. 102 corresponds to negative values of 0. 141. The Llma^on, and Cardiold. If a point on the circumference of a fixed circle of radius |a be taken as pole, and the diameter through as initial line, the radius vector of any point Q on the cir- cumference is given by r = acos^ (1). If on this radius we take two points P, P' at equal constant distances c firom Q, the locus of these points is called a ' lima9on.' Its equation is evidently r' = acos ^ + c (2). This includes the paths both of P and of P', if 6 range from to 2ir. 140-141] SPECIAL CURVES. 369 If c < a, the curve passes through when 6 = COS"' (— cja), and forms a loop. See the curve traced by the points P^, Pa' in Fig. 103. If c>a, r cannot vanish ; see the curve traced by Pj, Pj' in the figure. Pig. 103. In the critical case of c = ft, the loop shrinks into a cusp. The locus is now called a 'cardioid' or heart-shaped curve. Its equation is r- = a(l+cos^) (3). See the curve traced by Pj, Pi' in the figure. Also Fig. 89, p. 355. L. 24 370 INFINITESIMAL CALCULUS. [CH. IX 142. The curves r" = a"cosn0. A number of important curves are included in the type r'" = a^cosn0 : (1). Thus if w = + 1, we have the circle r = acos (2) and the straight line rcos 6 = a (3). If n = + 2 we have the ' lemniscate of Bernoulli ' 7^ = 0=003 2^ (4), and the rectangular hyperbola /■I cos 20 = a" (5). The equation (4) makes r real for values of 6 between ■t Jir, imaginary for values between ^ and fir, and so on. Also r" is a maximum for ^ = 0, 6 = ir, etc. It follows that the lemniscate consists of two loops, with a node at the origin. See Fig. 113, p. 389. If n = + ^, we have the cardioid ri = aicos^O, or r=Ja(l + cos0) (6), and the parabola 2a r^eosi0 = a^, or r = q ;: (7). 1 + cos p ^ ^ The curves corresponding to equal, but oppositely-signed, values of n, are 'inverse' to one another; see Art. 145. If we differentiate (1) logarithmically, we find, if ^ denote the angle between the tangent and the radius vector, 1 dr cot = -^^ = -taiine (8), or = ^Tr + n0 (9). The student should examine the meaning of this result in the various special cases mentioned above. 142-143] SPECIAL CURVES. 371 143. Tangential-Polar Equation. If p be the perpendicular from the origin on any tangent, and r the' radius vector of the point of contact, p will in general be a function of r. The equation expressing this relation is called the 'tangential-polar' equation of the curve. If the ordinary polar equation be given, the tangential- polar equation is to be found by eliminating d and between the formulae p = rsm^, -^=cot(j> (1) (for which see Art. 55) and the given equation. From (1) we obtain i,= -(l+cot^^) = - + -(|) (2). It is occasionally convenient to employ the reciprocal of the radius vector instead of the radius itself If we write 1 , du 1 dr ,„. M = -, we have -j^ = — „-j^ (3), r da r^dO ^ and the formula (2) takes the shape y-'" where the focus is the pole, we find ^ = Jtt - Jfl, p = r cos ^0, whence p^^ar (9). This is a well-known property of the curve. 24-2 372 INFINITESIMAL CALCULUS. [CH. IX This example, like the preceding, is included in a general result embracing all curves of the type »-» = a»cos7ie (10). By Art. 142 (9) we have p = rsin a, is ■ir{<^ + ^a'). 12. Prove geometrically that if two straight lines, touching two fixed circles, make a constant angle with one another, their intersection traces out a limagon. 13. The whole area of the lemniscate r^ = a' cos 26 is a^. SPECIAL CURVES. 375 14. The perimeter of either loop of the same curve is h V(l-2sin»e)" Prove that, in the notation of elliptic integrals (Art. 109), this is equal to ^'<-ji) 15. The mass-centre of the area of either loop of the lemniscate is at a distance |-^2ira from the pole. 16. Shew that the area included by one loop of the epicyclic r = fls sin m6 is Tra^jim. 17. Trace the curve r' = a^cos9. 18. Prove the following properties of the ■ solid of greatest attraction' (viz. the figure generated by the revolution of the curve ifi = a? cos Q about the initial line) : (1) The volume is -j^Tros' ; (2) The greatest breadth is r2408as, at a distance • 4389a from the pole ; (3) The mass-centre of the volume is at a distance ^a, from the pole. 19. If the 'polar subtangent' of a curve be defined to be the length intercepted by the tangent, on a perpendicular drawn to the radius vector from the pole, prove that it is equal to Prove that in the reciprocal spiral the polar subtangent is constant. 20. The tangential-polar equation of the involute of a circle of radius a is the centre being pole. 21. Shew that in the spiral of Archimedes (Fig. 101) a^'rr' 376 Infinitesimal calculus. [oa. ix 22. Shew that in the reciprocal spiral (Fig. 102) 1 1 1 23. Shew that in the curve a cos mO ' 1 l-wP m^ 24. Shew that in the curves a a cosh mO ' ' sinh mO ' 1 1 + m'' _ m'' respectively. 25. Prove that, in the epicycloid (Art. 137), ^ (a + 26)'^- What is the corresponding formula for the hypocycloid 1 26. Prove the formula rdr 7; for the arc of a curve whose tangential-polar equation is given. 27. Prove the formula pds = r'dO, and give its geometrical interpretation. Hence shew that if the area swept over by the radius vector of a moving point increase uniformly with the time, the velocity will vary inversely as the perpendicular from the origin on the tangent to the path. 144. Associated Curves. Similarity. There are several methods of associating with a given curve another curve connected with it by a definite relation. 144] SPECIAL CURVES. 377 The simplest relation is that of similarity. Two plane figures are said to be ' similar ' when they differ only in scale. More precisely, it is implied that to every point P of one figure corresponds a point P' of the other, and that the distance PQ between any two points of the one bears a constant ratio to the distance P'Q' between the corresponding points in the other. It follows that if P, Q, R be any three points of the one figure, and P', Q', R' the corresponding points of the other, the triangles PQR and P'Q'R' will be equiangular to one another. Hence straight lines in one figure will correspond to straight lines in the other, and angles in the one figure to equal angles in the other. If we take rectangular axes OX, OF in the one figure, and the corresponding axes O'X', O'Y' in the other, the coordinates x, y of any point P in the one figure will be connected with the coordinates x , y' of the corresponding point P' of the other by the relations x' = mx, y' = my (1), where m is a constant ratio. If by a displacement of either figure in its own plane O'X' and O'Y' can be made to coincide with OX and OT, respectively, the two figures are said to be ' directly ' similar. In the new position, any two corresponding points P, P' are in a straight line with ; moreover OP' = m . OP, and any two corresponding straight lines are parallel. The same statements hold if either figure be turned about through two right angles, until OX' coincides with XO produced, and OY' with YO produced. In either of these two relative positions the figures are said to be ' similarly situated,' and the origin is called the ' centre of similitude.' If r=f(6) (2) be the polar equation of any curve in the one figure, the equation of the corresponding curve in the other will be r= + m/(^) (3). It may happen, however, that when O'X' is brought into coincidence with OX, the line O'Y' coincides, not with OY 378 INFINITESTMAT- CALCULUS. [CH. IX but with YO produced. The two figures may be said, in this case, to be ' perversely ' similar. To the curve (2) would now correspond r = mf{— 6) (4). We shall not have occasion to consider this kind of relation. Ex. 1. Conies of the same excentricity are similar. Let the two conies be placed so as to have a common focus, and the perpendiculars on the corresponding directrices coincident in direction. The polar equations will then be of the forms I V -=1 +ecos5, - = 1 +6C0s5 (5), T T whence it follows that radii drawn in the same direction are in the constant ratio I ; V . Ex. 2. All catenaries are similar curves. For the equation y = c cosh x/c (6) is unaltered when x, y, c are all altered in any the same ratio. 145. Inversion. If from a fixed origin we draw a radius vector OP to any given curve, and in OP take a point P' such that OP.OP'=±le (1), where A; is a given constant, the locus of P' is said to be the ' inverse ' of that of P. The point is called the ' centre,' and ± h? is called the ' constant,' of inversion. Pig. 104. A curve and its inverse make supplementary angles with the radius vector. For if P, Q be consecutive points of a 144-145] SPECIAL CtJRVES. 379 curve, and P', Q' the corresponding points on the inverse curve, we have OP .OP'=OQ.OQ', and therefore OP: OQ=OQ':OP'. .(2). Hence the triangles POQ, Q'OP' are similar, and the angles OPQ, OP'Q' are supplementary. In the limit, when Q is infinitely close to P, these are the angles which the respective tangents make with the radius vector. The curves obtained from a given curve, with the same centre but different constants of inversion, are similar. For if we have r'ri = const., rr2 = const (3), r*! : /"a = const (4). Ex. 1. The inverse of a straight line is a circle through the centre of inversion, and vice versd. First let the constant of inversion be equal to the square of the perpendicular distance OA of the origin from the given straight line. If P be any point on the line, and HOP meet the circle on OA as diameter in P', we have OF.OP' = OA' = k' (5), that is, the straight line inverts into the circle. Fig. 105. If the constant of inversion be changed we get a similar curve, which will still be a circle through the centre of similitude 0. 380 INFINITESIMAL CALCULUS. [CH. IX Ex. 2. More generally, the inverse of any circle is a circle. Fig. 106. Let be the centre of inversion, C the centre of the given circle, a its radius ; and let le' = OG-'-a' (6). If, then, we draw any chord OPP' through 0, it is known from Geometry that OP. OP' = 00'- a? ^k'' (7) Hence P' traces out the inverse of the locus of P ; i.e. the circle inverts into itself. And by changing the constant of inversion we get a similar curve, and therefore a circle. If, as in the right-hand figure, be within the given circle, the constant in (7) is negative. This means that P' and P are now on opposite sides of 0. 146. Mechanical Inversion. There are various devices by which the inverse of a given curve can be traced mechanically. 1°. Peaucellier's Linkage. This consists of a rhombus PAQB formed of four rods freely jointed at their extremities, and of two equal bars con- necting two opposite corners A,B to a, fixed pivot at 0. It is evident that, whatever shape and position the linkage assumes, the points P, Q will always be in a straight 145-146] SPECIAL CURVES. 381 line with 0. If N be the intersection of the diagonals of the rhombus, we have OP.OQ=ON^ ~ PN'=OA^ ~ ^P = const. ... (1). Fig. 107. Hence if P (or Q) be made to describe any curve, Q (or P) will describe the inverse curve with respect to 0. In particular if, by a link, P be pivoted to a fixed point 8, such that SO = SF, the locus of P is a circle through 0, and consequently the locus of Q will be a straight line perpendicular to OS. This gives an exact solution of the important mechanical problem of converting circular into rectilinear motion by means of link-work. 2°. Hart's Linkage. This consists of a ' crossed parallelogram ' A BCD formed H-- Fig. 108. 382 INFINITESIMAL CALCULUS. [CH. IX of four rods jointed at their extremities, the alternate sides being equal. A point in one side AB is made a fixed pivot, and P, Q are points in AD and BG such that AP : PD=GQ • QB = AO : OB, = m : w,say. Evidently 0,P, Q will lie in a straight line parallel to ^C and BD. If H, K be the orthogonal projections of J., C on BD, and if N be the middle point of BD, we have AC.BD = 2NH. 2NB = DH^ - BH' = AD'- AB\ Now OP:BI) = AO:AB = m:m + n, and OQ:AG = BO:AB = n:m + n. Hence OP.OQ= , ^^ ., (AD' - AB") = const (2). Hence P and Q describe inverse curves with respect to 0. As before, by connecting P to a fixed pivot jS' by a link PS equal to SO, we can convert circular into rectilinear motion. 147. Pedal Curves. If a perpendicular OZ be drawn from a fixed point on the tangent to a curve, the locus of the foot Z of this perpendicular is called the ' pedal ' of the original curve with respect to the origin 0. Thus : the pedal of a parabola with respect to the focus is the tangent at the vertex. The pedal of an ellipse or hyperbola with respect to either focus is the ' auxiliary circle.' If OZ = p, and if i|r be the angle which OZ makes with any fixed straight line, then p, ■y^ may be taken to be the polar coordinates of Z with respect to as pole. Hence if the relation between p and ■\fr can be found, the polar equation of the pedal can be at once written down. Ex. 1. If the origin be at the centre of the conic i4-' (1)' and i/r be the angle which p makes with Ox, it is shewn in books on Conic Sections that p' = a^cos^^p±Vs.\n^ijf (2). 146-147] SPECIAL CURVES. 383 Hence the polar equation of the pedal is r' = a!'coa''e±Psm^e... In the case of the rectangular hyperbola .(3). -r ■ w the pedal is the lemniscate r° = a°cos 2& ,(5). Eos. 2. In the case of a circle of radius a, the pole being at a distance c from the centre G, and (.he line OC being the origin of tj/, we have at once from a figure p = a + c cos i[/ (6). Hence the pedal is the limagon r=a + c cos 6 (7). If be on the circumference, we have c = a, and the pedal is the cardioid r=a(l +cose) (8). The angle which the tangent makes with the radius vector at corresponding points is the same for a curve and its pedal. For let OZ, OZ' be the perpendiculars from on two consecutive tangents PZ, PZ', and let Of7 be drawn perpendicular to ZZ' produced. The points Z, Z' lie on the circle described on OP as diameter. Hence the exterior angle OZU of the quadrilateral OZZ'P is equal to the interior and opposite OPZ'. In the limit these are the angles which OZ and OP make with the tangent to the pedal, and with the tangent to the orio;inal curve, respectively. Also, by similar triangles, we have OU:OZ=OZ':OP (9). Hence if r be the radius vector of the original curve, p the 384 INFINITESIMAL CALCULUS. [CH. IX perpendicular from on the tangent, and p' the per- pendicular from on the tangent to the pedal, we have, ultimately, p'lp=p/r, or p'=p'/r (10). Again, if OZ' meet PZ in N, we may write, OZ =p, OZ'=p+ Bp, Z ZOZ' = Z ZPZ' = Syjr. Neglecting small quantities of the second order, we have Sp = NZ' = PZ'Syjr. Hence, proceeding to the limit, when PZ' coincides with PZ, we obtain an expression for the projection of the radius vector on the tangent to a curve, viz. ^' = ^ (11)- This result enables us easily to solve the problem of ' negative pedals,' viz. to find the curve having a given pedal. Taking as origin, and the initial line of ^fr as axis of x, the coordinates of the point of contact P are given by x = OZ cos i/r — ZP sin ■>}r, y = OZ sin i|r + ZP cos ■>/r, or a; = 2> cos i/r — -5^ sin i/r,| . ,dl , <12)- y =p sm y + jy cos y I Ex. 3. To find the curve whose pedal is the cardioid r = a(l+cose) (13). Writing p = a (1 + cos i/r) (14), the formulse (12) make x = a cos i/f + a, y = a sin i/f, whence {x-aY + ^ = a? (15), a circle through the origin. 148. Reciprocal Polars. The locus of the pole of the tangent 10 a curve 8, with respect to a fixed conic 2, is called the ' reciprocal polar ' of 147-148] SPECIAL CURVES. 385 S. It is proved in books on Conies that if S' be the locus of the poles of the tangents to 8 then S is the locus of the poles of the tangents to 8'. This explains the use of the word ' reciprocal.' We shall here only notice the case where the fixed conic S is a circle. If be the centre of this circle, and k denote its radius, the pole P' of any tangent to the curve 8 is found by drawing OZ perpendicular to this tangent, and by taking in OZ a point P' such that OZ.OP' = k' (1). Hence the reciprocal polar is in this case the inverse of the pedal of the given curve, with respect to the point 0. By the reciprocal property above cited, the original curve must be the inverse of the pedal of the locus of P'. This is easily verified ; for if P be the point of contact of the tangent to the original curve, and if OP meet the tangent to the locus of P' in Z', the angles OP'Z' and OPZ will be equal, by Art. 147. Hence OZ'P' is a right angle, and Z' traces out the pedal of P'. And, since PZP'Z' is a cyclic quadri- lateral, we have OP.OZ' = OZ.OP' = k^ (2). Hence P describes the inverse of the locus of Z'. L. 25 386 INFINITESIMAL CALCULUS. [CH. IX Ex. 1. The reciprocal polar of a circle with respect to any origin is a conic having the origin as focus. As in Art. 147, Ex. 2, the formula for the pedal of the circle is ^ = fls + ccosi/f (3). Writing 6 for ^, and h^jr for p, we get the equation of the reciprocal polar in the form -=a + c cos 6 (4), r which represents a conic, having its focus at the origin, of eccentricity cja. Hence the conic is an ellipse, parabola, or hyperbola, according as the origin is inside, on, or outside the circle. Ex. 2. The pedal of the conic S4=' <»). ■with respect to the centre, is given by j9' = a''cos''i/^±6''sin''i/r (6). Hence the reciprocal polar is -^=a!'cos^e±V'sm^e (7), or o»ar'±6V = ^ (8), a concentric conic. 149. Bipolar Coordinates. A curve may be defined by a relation between the distances (r, /) of any point P on it from two fixed points, or foci, 8, S' ; thus .fir,r') = (1). If we denote the angles PSS', PS'8 by 9, 6', respectively, and the angles which the radii r, r' make with the tangent by ^, ^', we have, as in Art. 110, dr , dr' ,,' dd . , ,de' . ,, (2). 148-149] SPECIAL CURVES. 387 We have, in addition, the relations rsmd = r'smd', rcosd + r' cos6' = 2o (3), where c = ^SS'. y^,9' ^ e}^ G' c o S Kg. 111. Bx. 1. In the ellipse we have r + r' = 2 + cos ' - 0, or =-ir — (5). The focal distances therefore make supplementary angles with the curve. Similarly, in the hyperbola r~r' = 2a (6), we find cos^ = cos^' (7), or, the focal distances make equal angles with the curve on opposite sides. Ex. 2. To find the form which a reflecting or refracting surface must have in order that incident rays whose directions pass through a fixed point S may be reflected or refracted in directions passing through a fixed point iS". The case of reflection is merely the converse of Ex. 1. The surface must have the form generated by the revolution of an ellipse or hyperbola about the line joining the foci (S, S'). 25—2 388 INFINITESIMAL CALCULUS. [OH. IX In the case of refrcMtwn, we have, if /«. and /*' be the refractive indices of the two media, /xsinx=/i'sinx' (8), where X = ±(i''-^)» x' = ± (i'^ " "^') W- Hence ^ cos i^ + /i' cos <^' = (10), or |(;xr±;aV) = : (11). Integrating, we have /*r + /tV' = const. (12). These curves, in which the sum (or difference) of given multiples of the two radii is constant, are called 'Cartesian ovals,' after Descartes, by whom the optical problem was first discussed. Fig. 112. When the lower sign in (12) is taken, the family includes the wrch rl't' = \i.\i^ (13). See Fig. 112. 149] SPECIAL CURVES. 389 The ' ovals of Cassini ' are defined by rr' =:T^ Ex. 3. ■ (14). h being a given constant. Since for a point P in SS' the greatest value of rr' is > y = as sin (13), 151] CURVATURE. 399 •we have -r; = ^ (■'■^)> where p is the semi-conjugate diameter. For the element of arc is altered from aS<^ to ^8, and the parallel radius from a to )8. Also since J/S^Si/' and Ja'St^ represent corresponding elements of area, we have yS'^Si/f = - x a^h, #=^ (^^>- Hence P = df=d^df^ab ^^^> If p be the perpendicular from the centre on the tangent-line, ■we have pfi = ah, so that our result may also be written fp' °' P^f (^^>- Since p^ = o" cos^ ij/ + ¥ sin* i/r = a* (1 - e" sin* if/), the last form is equivalent to '' = "(l-/sin*^)« (^^)- This formula leads to an important result in Geodesy. The figure of the Earth being taken to be an ellipsoid of revolution, the expression for the radius of curvature in terms of the latitude ^, is, if we neglect e*, ^ = a(l-e* + |6*sin*,/,) = a(l-Je-f£C0s2.^)...(19), where e = (a — h)ja = Je° ; that is, € denotes the ■ ellipticity ' of the meridian. Integrating (19) we find, for the length of an arc of the meridian, from the equator to latitude i/r, s = a (1-|€) if- !«£ sin 2i/r (20). Ex. 6. In the equiangular spiral (Art. 140), we have ^ = e + a (21), whence di^jds = dOjds = (sin o)jr, or P = -^ (22). ^ sin a ^ ' Hence the radius of curvature subtends a right angle at the origin. 400 INFINITESIMAL CALCULUS. [CH. X 152. Formulae for the Radius of Curvature. The expression dyjrlds for the curvature is easily translated into a variety of other forms. 1°. In rectangular Cartesian coordinates, we have *-^=l W' and therefore sec .d±_d^(dy\_^d^_ d^ ^ ds ~ds\.dx)~dx^ ds~ ^da?' J = sinh5, S = icosh?, l+f^^^cosy?. dx a dar a a \dxJ a d?y whence -= . ,j ,.,n. (2). This form shews, again, that the curvature vanishes at a point of inflexion. Ex. 1. In the catenary y = acoshx/a (3) we have whence p = a coah^ x/a = i/'ja (4). Since y = asec i/r, this agrees with Art. 161, Ex. 1. When dyjdx is a small quantity the formula (2) gives, approximately, p-da? ^^'' the proportional error being of the second order. The form (5) is an ohvious transcript of difz/ds, since when yj/ is small we may write dyjdx (= tan i/f) for i/', and djdx for djds. The approximate formula (5) has many important practical applications, e.g. to the theory of flexure of bars (see Art. 130). If the axis of x be parallel to the length, and if y be the lateral 152] CURVATtTRE. 401 deflection at any point, the ' bending moment,' or 'flexural couple,' is ^^^^g («)• 2°. It was proved in Art. 147 that the projection (i) of the radius on the tangent is given by '-'4 <')■ If OU, OU' be the perpendiculars from the origin on two consecutive normals PC, P'G, and if OU' meet PG in N, we have, ultimately, 0U'-0lT^U'N=GN8f, or 8« = CiV8.|r. The limiting value of 0?7 or GN is therefore dt/dyfr, whence p = GP=^OZ+GU=p + ^=p + §^, ^(8). c /if- Fig. 115. 3° With the notation of Arts. 110, 147 we have ds (9>- t , dr - = COS Pi Pa where pi, pa are the radii of curvature of the given curve on the two sides of P, and TOj , m^ are the limiting values of the ratios PQjQR and PBjQB, respectively. 44. The acute angle which a chord PQ of a curve makes with the tangent at P, when Q is taken infinitely close to P, is ultimately equal to ^8s/p, where Ss is the arc PQ and p is the radius of curvature at P. 45. Prove that if the tangents at the extremities of an infinitely small arc PQ meet in T, then TP and TQ are ultimately in a ratio of equality. Why does it not follow that the line joining T to the middle point of PQ will be ultimately perpendicular to PQ ? 46. Assuming that the radius of the circumcircle of a triangle ABG is equal to \djsaxA, shew that it follows from Ex. 44 that the osculating circle coincides with the circle of curvature. 47. Prove that when the resultant force on a particle is in the direction of motion the tangent to the path is ' stationary.' 155. Envelopes. Suppose that we have a singly-infinite system, or family, of curves differing from one another only in the value assigned to some constant which enters into their specifi- cation. Two distinct curves of the system will in general intersect; and we consider here, more particularly, the limiting positions of the intersections when the change in 414 INFINITESIMAL CALCULUS. |.CH. X the constant (or 'parameter,' of the system, as it is some- times called), as we pass from one curve to the other, is infinitely small. On each curve we have then, in general, one or more points of ' ultimate intersection ' with the con- secutive curve of the system. The locus of these points of ultimate intersection is called the ' envelope ' of the system. Ex. 1. A system of circles of given radius, having their centres on a given straight line. The parameter here is the coordinate of the centre. If C, C" be the centres of two circles of the system, the line joining their intersections bisects GC at right angles. Hence the points of ultimate intersection of any circle with the consecutive circle are the extremities of the diameter which is perpendicular to the line of centres. The envelope therefore consists of two straight lines parallel to the line of centres, at a distance equal to the given radius. Fig. 120. Ex. 2. A straight line including, with the coordinate axes, a triangle of constant area (k^). If AB, A'B' be two positions of the line, intersecting in F, the triangles APA', BPB' will be equal, whence PA.PA' = PB.PB'. Hence, ultimately, when AA' is infinitely small, P will be the middle point of AB. If x, y be the coordinates of P, and w the inclination of the axes, we have, then, OA = 2x, OB = 2y, and therefore 2xy sin m^Ii?. The envelope is therefore a hyperbola having the coordinate axes as asymptotes. Mg. 121 illustrates the case of m= Jtt. 165-156] CURVATURE. 415 156. General Method of finding Envelopes. The equation of any curve of the system heing cl,(x,y,a) = (1), where a is the parameter, then at the intersection with another curve (2), (3). ^ (a?, y. a') = we have, evidently, ^ fe y, a') - ^ {«', V'") ^Q a' — a When the variation a' — a of the parameter is infinitely small, this last equation takes the form ^4>{x,y,a) = (4), where 9/9a is the symbol of partial differentiation with respect to a. See Art. 45. The coordinates of the point, or points, of ultimate intersection are determined by (1) and (4) as simultaneous equations, and the locus of the ultimate intersections is to be found by elimination of a between these equations. 416 INFINITESIMAL CALCULUS. [CH. X Ex. 1. The circles considered in Art. 155, Ex. 1 may be represented by {x-af + y'' = a? (5). Differentiating with respect to a, we find x-a=0 (6). Eliminating a between (5) and (6) we get y = ±a (7), the envelope required. Ex. 2. If a particle be projected from the origin at an elevation 0, with the velocity ' due to ' a height h, the equation of the parabolic path is y = a;tane-i^sec«e (8), where the axes of x, y are respectively horizontal and vertical. Writing a for tan 0, we get X- y = aa,-i^(l + a^) (9). To find the envelope of the paths for different elevations, and therefore for different values of o, we differentiate (9) with respect to a, and find «'-ix=° (1*^)- This is satisfied either by x = 0, or by ax = ^h. The former makes y — 0, and shews that the origin is part of the locus, as is otherwise obvious. The alternative result leads, on elimination of a, to a?=ih(h-y) (11), a parabola having its axis vertical, its focus at the origin, and its vertex at an altitude h*. 157. Algebraical Method. If in the equation (a!,y,a) = (1), ^ be a rational integral function of a, the rule of the preceding Art. may be investigated otherwise as follows. * This problem is interesting historically as being the first instance in which the envelope of a family of curved lines was obtained (Bernoulli). The general method of finding envelopes appears to be due to Leibnitz. 156-157] CURVATURE. 417 If we assign any particular values to x, y, the equation determines a, that is, it determines what curves of the system pass through the given point {x, y). If the equation be of the nth degree in a, the number of these curves (real or imaginary) will be n, and these n curves will in general be distinct. But if the point in question be at the inter- section of two consecutive curves, two of the values of a. will be coincident. Now it was shewn in Art. 49 that the con- dition for a double root of the equation in a is ^ ^^^{x,y,a) = (2). The ultimate intersections are therefore determined as before by (1) and (2) as simultaneous equations, and the envelope by elimination of a between them. If the equation (1) be of the first degree in a, only one curve of the system passes through any assigned point, and there is of course no envelope. Examples of this are furnished by the parallel lines Ix + my = a (3), and by the concentric circles x' + y' = a (4). If (1) be a quadratic in a, say Pa? + 2Qa+R = (5), where P, Q, R are given functions of x and y, the condition for equal roots is PB = Q' (6). This is therefore the equation of the envelope. Ex. 1. If the straight line a'-r^ <^) include with the coordinate axes a triangle of constant area k% we have a/3sinu) = 2Ar= (8), L. . 27 418 INFINITESIMAL CALCULUS. [CH. X where id is the inclination of the axes. Heuce, ieliminating ^, the equation of the variable line is found to be a''2/sin{x,y,ct) = (1), and ^<}>(a!,y,a) = :...(2), determine x, y as functions of a, say x = F{a), y=/(a) <3), and the latter pair of equations define the envelope. If we substitute from (3) on the left-hand side of (1) we obtain a 157-158] CURVATURE. 419 function of a which must vanish identically, and the result of differentiating this function with respect to a must also be zero. Hence, by the rule of Art. 62, 1°, we must have d±dsB_^d^dy^d^da^Q (4), dx da dy da dada which reduces, in virtue of (2), to d^dso^d^dy^Q .gv dasda dy da ' dy d- da dy N6w, by Art. M, the left-hand side of this equality is the value of dyjdx for the envelope; and the right-hand side is, by Art. 62 (10), the value of dyjdx for the curve (1). Hence at the point, of ultimate intersection the curve (1) and the envelope have a common tangent line. The geometrical basis of the theorem may be indicated as follows : Let the figure represent portions of two curves of the system, corresponding to values Oq, aj of the parameter a, Fig. 122. and intersecting in P. Let P,, and Pj be the corresponding points on the envelope ; viz. Po is the limiting position of P when, a,, being fixed, Mj is taken infinitely nearly equal to a^ ; 27—2 420 INFINITESIMAL CALCULUS. [CH. X and Pi is the limiting position of P when, ai being fixed, Oo is taken infinitely nearly equal to a^. Since these variations of a are in opposite senses, and since the coordinates of P are as a rule symmetric functions of «», Mj, the corresponding displacements of P, viz. PPo and PPi, will in general, when I Hi — Oo I is very small, be in nearly opposite directions, and PjPPi will be a very obtuse-angled triangle. Hence, ulti- mately, when I «! — Oo I is infinitely small, the chords PoPi and PoP will coincide in direction; i.e. the tangent to the en- velope is identical with the tangent to the variable curve The foregoing investigations break down in certain cases. As regards the analytical proof, it is plain that no inference can be drawn from (5) whenever at the point in question we have s=«. |=« w. simultaneously ; i.e. when the value of dyjdx for the curve (1) is not uniquely determinate. This peculiarity occurs at a ' singular point,' whether it be of the nature of a node, a cusp, or an isolated point (see Art. 133). It appears that the locus of the singular points of the given family, when such a locus exists, is included in the result of eliminating a between (1) and (2), but this locus does not in general ' touch ' the given curves, in any proper sense of the word. The full investigation of this matter is beyond our limits*, but a simple example may be given. Consider the family a{y-af = a?{x + h) (8). It appears from Art. 133 that there is a node, a cusp, or an isolated point at (0, a), according as 6 is positive, zero, or negative. The process for finding the envelope gives y — a = and therefore a?{x + h) = (9). The line a; = gives the locus of the singular points, and does not touch the original curves ; the line x = — b on the other hand does so (unless 6 = 0). In the geometrical view of the matter it was assumed that there is no other intersection of the curves o„ a^ in the immediate neighbourhood of P. In the case of a node we have usually two * It is usually given in books on Differential Ec^uations, under the head of 'Siogolar Solutions.' 158-159] CURVATURE. 421 adjacent intersections, ■whose a:-coordinates (for instance) are of the formsy(ao, Oj) andy"(ai, a,), respectively; buty'(a„, oi) is not a symmetric function of a„, Oi. The argument does not there- fore apply to the node-locus. Again, in the case of a cusp the displacement of the point P in Fig. 122, due to an infinitesimal variation of a, or a^, is found not to be of the first order; and the points P„, Pi are as a rule on the sams side of P. In the neigh- bourhood of an isolated point there is no real intersection of consecutive curves. 159. Evolutes. The 'evolute' of a curve is the locus of its centre of curvature. Since the centre of curvature is (Art. 150) the intersection of two consecutive normals, the evolute is also the envelope of the normals to the given curve. Hence the normals to the original curve are tangents to the evolute*. Ex. 1. In the parabola y^ = ^ax (1), we have a; = as cot^ i/r, 2/ = 2ascoti/f (2), and (by Art. 151, Ex. 4) p = -2a/sm^f (3). The coordinates of the centre of curvature are therefore i = x- pahxtj/ = 3x + 2a,'\ ... ■q = y + p cos i/f = — y^/ia^) Hence rf = y'jUa^ = ia^ja = A (f - ^af/a. The evolute is therefore the semi-cubical parabola af = .^\{x-2af (5). Otherwise : it is shewn in books on Conies that the equation of the normal is of the form y = m(x — 2a) — am^ (6). To find the envelope of this we differentiate partially with respect to m, and obtain x-2a = Sam', y = 2am' (7). The elimination of m leads again to the result (5). The curve is shewn in Pig. 123. * It being evident that the exceptional cases noted at the end of Art. 158 cannot present themselves in the envelope of a straight line. 422 INFINITESIMAL CALCULUS. [CH. X C Fig. 123. Ex. 2. The normal at any point of the ellipse a: = acos^, y = 6sin<^ (8) i, - _J^ = „»_6^ (9). COS0 sin^ ' DiflFerentiating with respect to <^ we find ax hy . ,,». — 3:7 = -^-fjL' =^>say (10). cos'<^ sin'^ ^ ' Substituting in (9), we have \=a'-h\ (11). Hence the coordinates of the centre of curvature are x = cos'<^, y = 7 — sin'<^ (12); Of o and the evolute is (ax)i + {by)i = {a^-b'')i (13). 159] CURVATURE. 423 This curve, which may be obtained by homogeneous strain from the astroid, is shewn in Fig. 124. Pig. 124. The centres of curvature at the points A, B, A', B are E, F, E', F', respectively. Ex. 3. To find the evolute of a cycloid. At any point P on the cycloid APB (Fig. 125), we have, by Art. 151, Ex. 2, p = '2PI (14). Let the axis AB be produced to D', so that BD' = AB; and produce TI to meet a parallel to BI, drawn through D', in /'. If a circle be described on //' as diameter, and PI be produced to meet the circumference in P', we have P'I=PI, so that P' is the centre of curvature of the cycloid at P. And since the arc P'i' is equal to the arc TP, and therefore to BI or D'l', the locus of P' is evidently the cycloid generated by the circle IP'I', supposed to roll on the under side of DT, the tracing point starting from D'. That is, the evolute is a cycloid equal to the original cycloid, and having a cusp at D'. It appears, further, from Art. 136 (4), that the cycloidal arc PD is equal to 2/P', or PP. Hence arc i>'P' + P'P = const (15). 424 INFINITESIMAL CALCULUS. [CH. X The lower cycloid iu Fig. 125 is therefore an * involute ' (Art. 161) of the upper one*. It may be noted that whenever a curve is defined by a relation between p and ■ CURVATURE. 431 15. Find the envelope of the ellipses x = a sin (^ — a), y = b cos 0, where a is the parameter. 16. The envelope of the catenaries y = c cosh x/c, where c is the variable parameter, consists of two straight lines. 17. The envelope of the ellipses where a + )8 = A, is the ' astroid ' a;* + y* = A . 18. The envelope of the straight line which makes on the coordinate axes intercepts whose sum is k is the parabola >Jx+ Jy= Jk. 19. From any point on the ellipse a' b' perpendiculars are drawn to the coordinate axes ; prove that the envelope of the straight line joining the feet of these perpen- diculars is the curve ©'^(D* 20. Find the locus of ultimate intersections of the curves ay^ = x{x + a)", where a is the parameter ; and examine the result. 21. If a circle of constant radius has its centre on a given curve, the envelope of the circle consists of two parallel curves. 22. If a circle of given radius touch a given curve, its envelope consists of two parallel curves. 23. If the equation of a curve be given in the form that of any parallel curve is of the form r^ =/(j> -c) + 2cp - (?. 432 INFINITESIMAL CALCULUS. [CH. X 24. Prove that the problem of negative pedals (Art. 147) is equivalent to finding the envelope of the straight line X cos ilf + y sin <^=p, where ^ is a given function of the parameter i^. Verify that this leads to the formulae (12) of Art. 147. 25. Shew that the negative pedal of the parabola •j^ = 4aa; with respect to the vertex is the curve 26. Prove by the method of envelopes that the negative pedal of a circle is an ellipse or hyperbola according as the pole is inside or outside the circle. 27. Prove geometrically that the radius of curvature at any point of an equiangular spiral subtends a right angle at the pole. 28. The evolute of an equiangular spiral is an equiangular spiral of the same angle. 29. The coordinates of the centre of curvature at any point of the curve ay^ = a? are | = -a!- | -, oj = 4y + |^. Shew that near the origin the evolute has the form of the parabola y'' = ~ ^O'^- 30. Shew that if a curve has a point of inflexion the evolute has an asymptote. Shew that the part of the evolute of the curve which corresponds to the part of the curve near the origin may be represented approximately by the hyperbola scy = ^a\ 31. The evolute of the hyperbola x = a cosh u, y = b sinh u is {axf~{hy)i={a^ + Vf. 162] CURVATURE. 433 32. If rays emanating from a pcant be reflected at any- given curve, the reflected rays are all normal to a curve which is similar to the pedal of the given curve with respect to 0, but of double the dimensions. 33. Hence shew that the caustic by reflection at a circle will be the evolute of a limagon; and that in the particular case where the luminous point is on the circumference of the given circle the caustic is a cardioid. 34. Prove that the caustic by reflection at any curve is the evolute of the envelope of a system of circles described with the various points of the curve as centres, and all passing through the luminous point. What is the corresponding theorem for the case of refraction 1 162. Displacement of a Figure in its own Plane. Centre of Rotation. Suppose that we have two plane figures, each of in- variable form, and that a curve fixed in one rolls without sliding on a curve fixed in the other. Any point of either figure will then describe a curve relatively to the other ; a curve so described is called a ' roulette.' The cases where the rolling curves are circles have been considered in Arts. 136 — 138. The general theory of roulettes is of some importance in Geometry and in Kinematics, owing to the fact that any continuous motion whatever of a figure in its own plane may be regarded as consisting in the rolling of a certain curve fixed relatively to the figure on a certain curve fixed in space. This theorem will be proved in Art. 167 ; but it is necessary, as a preliminary, to recall one or two propositions of elementary Kinematics, which are, moreover, of independent interest. In the first place, any displacement whatever of a figure in its own plane is equivalent to a rotation about some finite or infinitely distant point. If A, B be any two points of the figure in its first position, and A', B' the same points in the second position, the new position P' of any third point originally at P is found by constructing the triangle A'P'B' congruent with L. 28 434 INFINITESIMAL CALCULUS. [CH. X APB. Hence the positions of two points are sufficient to determine the position of the moveable figure. Now, considering any point whatever of the figure, let P be its initial and Q its final position ; and let R be the Fig. 129. final position of that point of the figure which was origin- ally at Q. Since PQ and QR are two positions of the same line, they are equal. Hence if / be the centre of the circle PQR, the triangles PIR, QIR are congruent ; that is, l represents the same point in the two positions. The dis- placement is therefore equivalent to a rotation about /. This point is called the 'centre of rotation.' It may happen that PQ, QR are in a straight line. The displacement is then equivalent to a translation of the figure, without rotation ; or, we may say, the centre of rotation is at infinity. 163. Instantaneous Centre. Next considering any continuous motion of a plane figure in its own plane, let us fix our attention on two consecutive positions. The figure may be brought from the first of these to the second by a rotation about the proper centre. The limiting position of this point, when the two positions are taken infinitely close to one another, is called the 'instan- taneous centre.' If P, P' be consecutive positions of any the same point of the figure, and SB the corresponding angle of rotation, 162-163] CURVATURE. 435 the centre of rotation (/) is on the line bisecting PP' at right angles, and the angle PIP' is equal to hO. Hence, ultimately, the infinitesimal displacement of any point P at a finite distance from / is at right angles to IP and equal to IP. W. If we introduce the consideration of time, and denote by ht the interval that elapses between the two positions, the limiting value of hOjht, viz. dOjdt, is called the 'angular velocity' of the figure. The velocity of that point of the figure which coincides with the instantaneous centre / is zero, that of any other point P is at right angles to IP, and equal to IP . d&jdl. The fact that in any motion of a plane figure (of in- variable form) the normals to the paths of the various points all pass through the instantaneous centre is often useful in geometrical questions. If we know the directions of in- finitesimal displacement of two points of the figure, the instantaneous centre is determined as the intersection of the normals at these points to the respective directions. We can thence assign the directions, and relative magnitudes, of the displacements of all other points. Ex. 1. The extremi- ties of a straight line AB of constant length describe two straight lines OX, OT at right angles to one another. It is known that any point P of the line de- scribes an ellipse whose principal axes are along OX, OY*. The above theorem now gives us a construction for the nor- mal to this ellipse at P; viz. if we draw AI, BI perpendicular to OX, OY, respectively, / is the in- stantaneous centre, and IP the required normal. Fig. 130. * Viz. if BP=a, AP=b, L OAB = .ON (2). ,.0Q.'^- OQ 163] CUEVAT0EE. 437 Fig. 132. Again, considering any line (straight or curved) in the moving figure, it is evident that the point or points of ultimate intersection of this line with a consecutive position are the feet of the normals drawn to it from the instan- taneous centre. For any other point of the line is moving in a direction making a finite angle with it. Ex. 4. Thus in the case of a line AB oi constant length moving (as in Art. 163, Ex. 1) with its extremities on the axes Fig. 133. 438 INFINITESIMAL CALCULUS. [CH. X of coordinates, the point of ultimate intersection with a consecutive position is at the foot Z of the perpendicular from the instan- taneous centre /. Now if AB = h, lOAB = ^, the coordinates of Z are given by X = BZcoa <^ = 57cos° ^ = A cos' ,\ y = AZ sin. ili = AI sin' = k sin' ) and the envelope oi AB is therefore the astroid - a? + y^ = k^ , see Fig. 134. Cf. Art. 138, Ex. 4. (3) •W; Fig. 134. 164:. Application to Rolling Curves. When one plane curve rolls upon another, which is regarded as fixed, the instantaneous centi'e is at the point of contact. We will suppose, in the figure, that it is the lower curve 163-164] CURVATUEE. 439 which is fixed. Let A be the point of contact, and let equal in- finitely small arcs AP, AP' (= Ss, say) be measured off along the two curves. Let the normals at P and P' meet the common normal at A in the points and 0'. Then ultimately we have OA = E, 0'A=B', where R, R' are the radii of curva- ture of the two curves at A. After an infinitely small displacement, P'O' will come into the same straight line with OP, the two curves being then in contact at P. Hence the angle (8^) through which the rolling curve will turn, being equal to the acute angle between OP and P'O', is equal to the sum of the angles at and 0', so that ^^-M »■ Fig. 135. ultimately. Again, the chords AP, AP' are ultimately equal, and they include an infinitely small angle at A. Hence the distance PP' is ultimately of the second order in 8s. It follows that when hs is inde- finitely diminished the limiting position of the centre of rotation (/) coincides with A, for if it were at a finite distance from this point, the displacement of P', being equal to IP' . 80, by Art. 163, would be of the first order in Ss. It follows that when a curve rolls upon a fixed curve, the normals to the paths of all points connected with the moving curve pass through the point of contact. We have already had instances of this result in the cycloidal and trochoidal curves discussed in Arts. 136, 137. Again if a straight line roll on a curve, it is normal to the path traced out by any of its points (Art. 161). Further, if we consider any line (straight or curved) which is carried with the rolling curve, the points of ultimate 440 INFINITESIMAL CALCULUS. L^H. X intersection of the carried curve with its consecutive position are the feet of the normals drawn to it from the point of contact. And the envelope of the carried line is the locus of these feet. Ex. 1. If a circle roll on a fixed straight line, any diameter envelopes a cycloid. " Fig. 136. Let G be the centre of the rolling circle, / the point of contact, IZ the perpendicular on the diameter PQ. Since Z is on the circle whose diameter is GI, it is easily seen that if this circle be supposed to roll always with twice the angular velocity of the large circle, it will always have the same point of contact with the fixed line, and the point Z will move as if it were carried by the small circle. Its locus is therefore a cycloid. Ex. 2. Similarly if a circle (A) roll on a fixed circle (B), the envelope of any diameter of A is an epi- or hypo-cycloid which would be generated by the rolling of a circle of half the size of A on the circumference of B. 165. Curvature of a Foint-Roulette. To investigate the curvature of any point P fixed relatively to the rolling curve, let / be the point of contact, and let /' be a consecutive point of contact, P' the corre- sponding position of P. Since the displacement of the point of the rolling curve which comes to I' is of the second order of small quantities, the angle through which the figure has turned is Se = ^lPI'P' (1), ultimately. Let the normals to the path of P, viz. PI and 164-165] CUEVATUEE. 441 PT, be produced to meet in G. If h^ be the inclination of these normals, we have 8^ = ^101' = ^-^ CI (2). if ^ be the angle which IP makes with the normal at I. Fig. 137. Also, from the figure Sx^ = z PIT' - Z TPF 5v , Ss cos d) = ^0 — PT- 1 cos ~pr by Art. 164 (1), if R and R' be the radii of curvature of the fixed and rolling curves. Equating (2) and (3), we find =Kj R'^ R' ) (3), cos 'f'[m+ip)=R+R' (^>- This gives the limiting position of G, i.e. the centre of curvature of the path of P. The radius of curvature (p) is then found from p = GP=GI + IP (5). 442 INFINITESIMAL CALCULUS. [oh. We have taken as our standard case that in which the two curves are convex to one another, as in Figs. 135, 137. Any other case may be included by giving proper signs to R and R'. Ex. 1. In the cycloid, if a be the radius of the generating circle, we have iJ = oo, R' = a, /P = 2acos^ (6). Substituting in (4), we find CI = 2a cos = IP (7), andtherefore p = 2IP (8). Ux. 2. In the epicycloid (Art. 137) we have P^a, E' = h, IP = 2b cos - 2ab , a whence CI= 2{a + h) IP ...(9), (10), P = - .IP (11). a+26 "We note that if 6 = - Ja, we have p = oo ; cf. Art. 138, Ex. 2. The result contained in (4) and (5) may be put in a simple geometrical form as follows. On the normal at / mark off a length IH such that IH R^R" ■(12), and describe a circle on IH as dia- meter. Let IP meet this circle in Q. We have then IQ^ IH cos ^ " [r'^R') ^^"^ ^' and the relation (4) takes the form '+7i=:^-(i3). Eig. 138. CI ' IP' This shews that if P coincide with Q, GI is infinite; i.e. any point of the moving figure which lies on the circle just defined is at a point of inflexion of its path. For this reason, the circle in question is called the ' circle of inflexions.' 165-166] CURVATURE. 443 From (13) and (5) we find IP.IQ ci=- QP IP' (14). The latter result shews that p changes sign with QP ; that is, the paths of the various points of the moving figure are concave or convex to I, according to the side of the circle of inflexions on which they lie. In the standard case repre- sented in the figures, the paths are concave or convex according as P is outside or inside the circle. An example is furnished by the trochoidal curves figured on p. 349. The circle of inflexions has in this case half the size of the rolling circle. The preceding theory has an important application to the problem of 'rocking stones' in Statics. When one rough body rests on another, with a single point of contact, its centre of gravity must be vertically above this point. And for stability of equilibrium it is necessary that the path of the centre of gravity, in any possible rolling displacement, should be concave upwards. 166. Curvature of a Line-Roulette. The curvature of a line-roulette, i.e. of the envelope of a straight line carried by the rolling curve, can be found still more simply. The per- pendiculars IZ, I'Z' let fall on two consecutive positions of the line, from the corre- sponding positions (in space) of the instantaneous centre, are normals to the envelope, and the angle which they make with one another at their in- tersection {(J) is equal to the angle of rotation SO. Hence if n. If n' The equation implies that y is to be a differentiate function of X, and that dyjdx is to be continuous. The mode of derivation of a differential equation of the first order from a primitive involving an arbitrary constant, explained in Art. 170, may suggest that the general solution of (1) will in all cases consist of a relation between x and y involving an arbitrary constant. With some qualification, due to the occurrence of ' singular ' solutions (Art. 180), this is in fact the case. The rigorous proof, however, is difficult, and may be passed over here without inconvenience, since in almost all cases for which practical methods of integration have been discovered the process itself contains the demon- stration that the solution is of the kind indicated. 171-172] BIFFEEENTIAL EQUATIONS OF FIRST ORDER. 463 In such problems as ordinarily arise, either the left-hand side of (1) is a rational integral algebraic function of dyjdx, or the equation can be transformed so that this shall be the case. The 'degree' of the equation is then fixed by that of the highest power of dyjdx which occurs in it. The general equation of the first degree may be written M+N^ = (2), or Mdx + My = (3), whore M, N are given functions of x and y. The form (2) is also equivalent to |-f=*('.^) <*)■ li , we have (7 = 0. In this case the velocity with which the particle arrives at a distance a from the centre of force is ^{2fji,ja), or ^(2ga), if g' = /ija^. This gives the velocity with which an unresisted particle, falling from rest at a great distance, would reach the Earth, provided a denote the Earth's radius, and g the value of gravity at the surface. Ex. 4. In a suspension bridge with uniform horizontal load the form of the chain is determined by the condition that any L. 30 y-r-^ = iX, 466 INFINITESIMAL CALCULUS. [OH. XI two tangents to the curve intersect on the vertical bisecting the chord of contact. If the lowest point be taken as origin of rectangular coordi- nates, and the corresponding tangent as axis of x, the subtangent of any other point must be equal to one-half the abscissa. Hence dx ^=2^ (11). y X ^ ' the integral of which is log y=2 log X + const., or y=a?/a (12), where a is arbitrary. That is, the curve formed by the chain must be a parabola with its axis vertical. 174 Exact Equations. The case of the preceding Art. comes under the head of ' exact equations.' An equation Mda> + Jlfdy = (1) is said to be ' exact ' when M and N are of the forms du/dx and dujdy respectively. The form £■^+1*-" <^' is equivalent to du = (3), and its integral is u=C (4), where G is the arbitrary constant*- It may be shewn that every equation of the type (1) is either exact, or can be rendered exact by a suitable ' inte- grating factor.' The number of such factors is unlimited ; for if we suppose the equation (1) to have been brought to the * The rule for ascertaining whether a proposed eijuation of the first degree is exact will be given in Art. 210. 173-174] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 467 form (3), it will still be exact when multiplied by /'(«), where /(m) may be any function of u. The integral of f'{u)du = (5) is /(m)=C (6), which is obviously equivalent to (4). Ex. 1. (pKe + hy + g)dx+{h!ic + hy+/)dy = (7). This is equivalent to d{aii?+'2k>yy + hf + 2gx+'ify) = (8), whence aai?+2hxy + hy^+'2,gx + %fy =C (9). Ex. 2. xdx + ydy = k{xdy-ydx) (10). This may be written d{x' + y^) = 2Tcx'd(^ (H), and so becomes exact on division by a;" + y\ thus d(x' + y') _ W /i2\ i>^+y' ,y ^ '' Hence, integrating, log(jc2 + 2/'') = 2Atan-'^ + C' (13). The equation (10) may also be solved as follows. Its form suggests the substitutions jc = r cos 0, y = rsva.d (14), which give xdx + ydy = rdr, oady-ydx = r'dO (15). The equation therefore reduces to — = kde (16), whence log r = A6 + C (17). This is obviously equivalent to (13). Ex. 3. To find the form of a solid of revolution such that the centre of mass of the volume cut off by any right section shall 30—2 468 INFINITESIMAL CALCULUS. [CH. XI be at a distance from this section equal to 1/nth of the length of the axis. If the axis of x be that of symmetry, and y be the ordinate of the generating curve, w& must have (Art. 121) or I a:y^dx = ''^-^i I y^dx (18), where f is the abscissa of the bounding section. Hence, if i; denote the radius of this section, we find, on difierentiating with respect to i according to the rule of Art. 90, n n Jo ^'>={n-l)j y^dx (19). or A second difierentiation gives ^^m-ir^-'^)v' (20), whence ^|f = (-l)f (21). Integrating, we find irf = .4^"-'. The generating curve must therefore be of the type y^ = Ax''-^ (22). Since we have differentiated twice with respect to $, the differential equation actually solved is somewhat more general than the original problem. In fact the same differential equation would have been obtained if, instead of zero, we had had other (and distinct) constants as the lower limits of the two definite integrals in (18). It is therefore necessary to examine d, posteriori whether the solution finally obtained satisfies the original equa- tion with the actual lower limits. This is easily verified to be the case if n > 2. We note that if n = 3, the solid is a paraboloid of revolution, and that if n = 4 it is a cone; cf. Art. 121, Exs. 1, 2. 174-175] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 469 175. Homogeneous Equation. Let us suppose that, in the equation M, N are homogeneous functions of x and y, of the same degree. In this case the fraction MjN is a function of yjx only, and we may write I-/© w- If we put y = aiv this becomes "Tx + '-^f^"') (2). The variables x, v are now separable, viz. we have dx^_dv_ X f(,v)-v ^'^^' whence logfl;= j:^^ — +G (4Y After the integration has been effected we must write V = yjx. ■E'^- {^-f)%.-i^^y=^ (5). ' ^^'^ £=-\ (6)' whence x^- + v-- or x-^ = ~A, „'. dx 1 - ■»»' doc 1 - u= „ c?a; l-«2 , /l 2« \ , Hence — = -y= dv = \~--^ Adv n\ Integrating, we have log a; = log w - log (1 + 1»») + const., 470 INFINITESIMAL CALCULUS. [CH. XI which is equivalent to a!(l+«») = C«, or a? + y'=Gy (8). In the geometrical interpretation, the general solution of a homogeneous differential equation mtlst represent a system of similar and similarly situated curves, the origin being a centre of similitude. For the equation (1) shews that where the curves cross any arbitrary straight line {ylx = m) through the origin, dyjdx has the same value for each, that is, the tangents are parallel. Thus, in the above Ex. the solution represents a system of circles touching the axis of x at the origin. If in (4) we put C=logc, yjx or V is determined as a function of xjc. In other words, the primitive is homogeneous in respect to x, y, and c, and is therefore of the type *e'f)=» w This is in accordance with the geometrical property above stated, since if x, y, and c be altered in any the same ratio, the equation (9) is unaltered. In other words, a change in the value of c merely alters the scale of the curve. EXAMPLES. ZLVIII. 1. Integrate dy_y dx x' \.y=Gx?^ 2. Integrate dy y dx a?-\' . a;+l . 3. Integrate d/u -^ = cota;coty. [sin a; cos y= G.\ 4. Integrate ^%^y='- ri^=l + Ce'/«.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 471 Solve m {y + h) dx + n {x + a) dy = 0. [(a! + a)"(2/ + 6)"=C.] 6. Solve dy _l+y'^ \'^,_«' + G dx \ +^' r x + Q -| 7. Solve (\+y^dx-xy{\+a?)dy = Q. \il+x^){\+f) = Ca?.-\ 8. Find the curves in which the angle between the tangent and the radius is one-half the vectorial angle {&). [The cardioids r = oi (1 — cos ff)!\ 9. Find the curves in which the perpendicular from the origin on the tangent is equal to the abscissa of the point of contact. [The circles r='2a cos d.'\ 10. Find the curves such that the portion of the tangent included between the coordinate axes is bisected at the point of contact. [The hyperbolas xy = C] 11. Find the curves in which the subtangent varies as the abscissa. [?/ = Ca;™.] 12. Prove that if the subnormal bears a constant ratio to the abscissa the curve is a conic. 13. Find the curves in which the perpendicular from the foot of the ordinate to the tangent has a constant length a, [The catenaries y= a cosh [x- a)/a.] 14. Find the curve in which the polar subtangent is constant (=a). [r = a/(e-a).] 15. Find the curve in which the polar subnormal is constant (=a). \r = a(e-a).] 16. Find the curves such that the area included between any two ordinates is proportional to the intercepted arc. [The catenaries y = a cosh {x — a)/a. ] 17. Find the curves such that the area included between any ordinate, the axis of x, and the curve may be 1/jith of the rectangle contained by the ordinate and the correspondin" abscissa. [y = Gx'^~\'\ 472 INFINITESIMAL CALCULUS. [CH. XI 18. Find the form of a solid of revolution in order that the volume cut oflF by any right section may be 1/wth of the product of the area of this section into the length of the axis. [The equation of the generating curve must be i^ = Ax''~\'] 19. In a suspension-rod of uniform strength the area of the cross-section ( a constant, we have « = Ce~''+ r (15)- This might have been obtained more simply by writing the differential equation in the form i(.-|)..(.-|)=o ^u), whence M-|=Ce-*' (17). As t increases, v tends asymptotically to the constant value g/k. Ex. 2. If an electric current of strength x be flowing in a circuit of self-induction L and resistance H, and if £ be the extraneous electromotive force in the circuit, we have the equation L^+Iix=I! (18). If J? be a constant, the solution of this is «==| + C«^ (19), where C is arbitrary. The current therefore tends to the constant value EjE. If, for example, we suppose that the circuit is completed at time * = 0, we have to determine G so that a; = for t = ; this gives -l-l"-^' (20)- The second term represents the ' extra current at make.' 476 INFINITESIMAL CALCULUS. [CH. XI Again, if E = E„ cos (j>t + e) (21), we have -y (ae^ ) = -y e^ cos {^pt + e), whence, integrating, and dividing by e^ , we find Rf E ^t ( -5( x = Ce '' + ^e '' je ^ cos{pt + e)dt = G^~^' + ^J^AR ''OS (pt + e) +pL sin (pt + c)}... {22); see Art. 79 (14). Hence as t increases, the current settles down into the steady oscillation «'=;/(^T^^'=°'(^* "■'-'") (^^)' pZt where ei = tan~'^ (24). The effect of the self-induction (L) is therefore to diminish the amplitude of the current in the ratio HIJiE' + p'L^ and to retard its phase by Cj. 177. G-eneral Linear Equation of the First Order. We return to the general linear equation of the first order, l + ^2/ = Q (1). If Q = 0, we have li^^'" (^). whence log y +JPdx = A, or yefP^ = G (3). This shews that e-^^*" is an integrating factor of (1), viz. we have ^^''"(S+^^)=l^^^^""> 176-177] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 477 Hence (1) may be written ^(ye!P^) = Qe!P^ (4,). Integrating, we find yeSP^=SQe!Pd''dx+G (5). JUx. 1. -^ + ycoix = 2cosx (6). Here P — cotx, jPdx = \ogs>inx, e!P^ = &m.x. Hence, multiplying by sin x, -J- {y sin x) = 2 sin x cos x (7), y sin X = sin" x + G, G y = smx + - (S). ^«^-2. {l-x>)^-xy=l (9). Dividing by 1 -a?, we have dy X 1 ^~r^=^"ri^2 (10). Here Multiplying (10) by the integrating factor, we get ..^ . dy X 1 ^Mi-«^)y} = ;7(il:^) (11). Hence, integrating, ;^(1 -x')y = sin-' x + C, sin~^ X G y=7(X^)^7iP^) (^2>- 478 INFINITESIMAL CALCULUS. [CH. XI The integrating factor will often suggest itself on inspec- tion of the equation, without recourse to the above rule. Ex. 3. ^ + ny-==^ (13). (too OS The steps are *" ^ '*' '^"~ V = "''"^''' " m+n+l ™m+l 2'=;;rz;rTi + ^^~" - 178. Orthogonal Trajectories. Suppose that we have a singly-infinite family of curves 4>(x,y,G) = (1), where is a variable parameter, and that it is required to determine the curves which cut these everywhere at right angles. We first form the differential equation of the family, by differentiation of (1) with respect to ic, and elimination of G. See Art. 170. If two curves cut at right angles, and if yfr, -yfr' be the angles which the tangents at the intersection make with the axis of X, we have •^ — •^' = + ^tt, and therefore tan -^ = — cot i/r'. Hence the differential equation of one family is obtained from that of the other by writing -l/^for^. / ax ax Otherwise : if dx, dy be the projections of an element of one of the curves (1) we have S<^H-|*.0 (2,. 177-178] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 479 Hence, if dx, dy be the projections of an element of the orthogonal curve through the point {x, y), we have dx dy , , 3^=3^ ^^^- dx dy The differential equation of the trajectories is then obtained by elimination of G between (1) and (3). Ex. 1. To find the orthogonal trajectories of the rectangular hyperbolas a;y=(7 W- Differentiating, we find xdy + ydx = (5), and therefore, for the trajectories, xdx-ydy = Q (6), whence a?-y' = G' (7). This represents a system of rectangular hyperbolas whose axes coincide in direction with the asymptotes of the former system. Ex. 2. To find the curves orthogonal to the circles a? + f + 2ii.y-¥ = Q (8), where /a is the variable parameter. Difierentiating, we have xdx + {y + ij) dy = 0, and therefore, for the trajectory, xdy - {y + It) dx = Q. EUminating /x, between this and (8), we find 2xy^ + {a?-y'-l^) = (9), or x^^-y'^-o^ + k'' (10). This is linear, with y^ as the independent variable. The in- tegrating factor, as found by the rule of Art. 177, or by inspection, is l/ar*. Introducing this we have dx\xj ^s>?' 480 INFINITESIMAL CALCULUS. [CH. XI whence or ^ = -x + 2X, X X (11). X being arbitrary. The original equation represents a system of coaxial circles, cutting the axis of x in the points {±k, 0). The trajectories (11) consist of a second system of coaxial circles having these points as 'limiting points'; viz. if we put X = ±k we get the point- circles {x + kf + f = (12); see Fig. 148, Fig. 148. If the equation of the given family of curves be in polar coordinates, thus f(r,e,G) = (13), 178] DIFFERENTIAL EQUATIONS Of FIRST ORDER. 481 and if ^, ^' denote the angles which the tangents to the original curve and to the trajectory make with the radius vector, we have in like manner tan p + y){yp-x) = (i (7); and the solutions of xp + y = 0, yp-x=0, are, respectively, xy = C, o^ — y^ = G (8). The product of the two values of p given by (6) is - 1. This shews h, priori that the two branches of the primitive curves which pass through any point {x, y) will be at right angles to one another. Cf. Art. 178, Ex. 1. 180. Clairaut'B form. When the equation (1) of Art. 179 cannot be conveniently resolved into its linear factors, we may in certain cases have recourse to other methods. These are for the most part of somewhat limited utility, and are accordingly passed over here; but an exception may be made in favour of Clairaut's form, which is very simple in theory, and more- over often presents itself in questions where a curve is defined by some property of the tangent. If we write p for dyjdx, the form in question is y = xp+f{p) (1). It was proved in Art. 53 that the intercepts (a, /3) made by the tangent to a curve on the axes of x and y are given by a = (xp-y)/p, P = y-xp (2), respectively. Hence any equation of the form (1) expresses a relation between either intercept and the direction of the tangent, or (again) between the two intercepts*. Now it is * Yiz. the equation is equivalent to P=/(-j3/a), or (a,p)=0. INFINITKSIMAL CALCULUS. [CH. Xr evident that this relation is satisfied by &ny straight line whose intercepts have the given relation. Along any such straight line we have P^G (3), and we thus get the solution y = Cx+f(C) (4), involving an arbitrary constant G. But the equation will also be satisfied by the curve which has the family (4) of straight lines as its tangents ; in other words, by the envelope of this family. This envelope is found by expressing that (4), considered as an equation in G, has a pair of equal roots, i.e. by eliminating G between (4) and ic+f'{G) = (5); see Art. 156. JEx. To find the curve whose pedal with respect to the point (a, 0) as pole is the straight line x = 0. The expression of this property is where ^ is the intercept on the axis of y, or 2/ = acP+| (6)- This is satisfied by any one of the family of straight lines y = Gx + ^ (7), and also by their envelope y' = 4:ax (8); see Art. 157, Ex. 2. The usual method of deducing the above solutions is to differentiate (1) with respect to x ; thus whence {«,+/' (^)) g = (9). 180] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 487 This requires, either that l = « <^«)' or that x+f'{p) = Q (11). The former result makes p = G, and y=Ox+f(G) (12). The alternative result (11), combined with (1), leads, on elimination of p, to a particular relation between x and y. Since the result of eliminating p between (1) and (11) must be the same as that of eliminating G between (4) and (.5), we identify this second solution with the envelope aforesaid. The solution (12), involving an arbitrary constant G, is called the 'complete primitive.' The second, or envelope- solution, is not included in the complete primitive, i.e. it cannot be derived from it by giving a particular value to G, It is therefore called a ' singular solution*.' EXAMPLES. L. 1. Solve (|y-(a.^)|.„^ = 0. [2/ = ax + C, y = Px + C] 2. Solve ©■-'-• [y = G± cos X.] 3. Solve (l)'-v. [2/ = Ce±'"»^.J 4. Solve '•(I)'=- \if = G±iax.] 5. Solve '-(ST- [y=G±2J{ax).-\ * The general theory of singular solutions of equations of degree higher than the first must be sought for in books specially devoted to the subject of Differential Equations. It is closely related to, but not altogether co- exteusive with, the theory of envelopes. 488 INFINITESIMAL CALCULUS. [CH. XI 6. Solve <'-)(l)'='- [?/ = + sin-i X.] 7. Solve [y^ix'^+C, y=l-x+ Ce-".] 8. Solve y^\-x + Ge-'.] 9. Solve Kl)'-'»l— »■ [x'='iGy + G\] 10. Solve 11. Find the curve such that the product of the intercepts made by the tangent on the coordinate axes is constant (= W). [The hyperbola 4cxy = k\'\ 12. Find the curve such that the perpendicular from the origin on any tangent is equal to a. [The circle x^ + y'^ = a?.^ 13. Solve y = xp + J{b' + aY)- r x^ y^ ~\ Singular solution : — + j-^ = 1. 14. Find the curve such that the product of the perpendiculars from the points (+ c, 0) on any tangent is equal to V. The conies ^-,^+1=1, ^-^^-f^ = l.^ 15. Find the curve such that the tangent intercepts on the perpendiculars to the axis of x at the points (+ a, 0) lengths whose product is b'. The conies ^+^^1.1 16. Solve y = xp + ap(l —p). [Singular solution : (x + a)' = 4ay.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 489 17. Solve {x-a)p^ + {x-y)p-y = 0. [Singular solution : (x + yf = 4ay.] 18. Find the curve such that the sum of the intercepts made by the tangent on the coordinate axes is equal to a. [The parabola {m - yf - 2a (a; + y) + a" = 0.] 19. Shew that any differential equation of the type represents a system of parallel curves. 20. Shew that any differential equation of the type represents two systems of orthogonal curves. CHAPTEE XII. DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 181. Equations of the Type (Py/dai' =f{pc). This chapter is devoted principally to differential equa- tions of the second order, and especially to such tjrpes as are of most frequent occurrence in the geometrical and physical applications of the Calculus. Occasionally, the methods will admit of extension to equations of higher order. We begin by the consideration of a few special types, and afterwards proceed to the study of the linear equation, and in particular of the linear equation with constant coefficients. We take, first, the type da? /{"=) (1). This requires merely two ordinary integrations with respect to X ; thus %=\md.+A, y=={W{.!c)daa]dx + Ax + B (2), where the constants A, B are arbitrary. 181] DIFFERENTIAL EQUATIONS OP SECOND ORDER. 491 Ex. 1. The dynamical equation S=/(') (3)' which determines the motion of a particle in a straight line under a force which is a given function of the time, is of the above type, with merely a difference of notation. In the case of a particle subject to a constant acceleration g we have S=^ W' whence -j- =gt + A^ Cbt x = ^gf + At + B (5). Again, if -£^=fsinnt (6), the force varying as a simple-harmonic function of the time, we have dx f ^ . -rr = — -ao%nt + A, at n f aj = -— „sinM< + ^< + 5 (7). The constants A, B which occur in these problems may be adjusted so that at any chosen instant the particle shall be in a given position and have a given velocity. Ex. 2. To solve the equation ^2-'^(^-"')=° («)' subject to the conditions that y = Q and dyjdx = for a; = 0. This is the problem of determining the flexure of a bar which is clamped in a horizontal position at one end (x = 0) and supports a given weight ( W) at the other end {x = I). Two successive integrations of (8) give ^i=W{lx-ix') + A, ^'t/ = W{^h?-ia?) + Ax+B (9), 492 INFINITESIMAL CALCULUS. [CH. XII where A, B are arbitrary. The terminal conditions require that A = 0, 3 = 0, whence y = i^^{i-^) (10). 182. Equations of the Type d^yjdci? =fiy). If the equation be of the type %-ny) (1)' a first integral may be obtained in two ways. In one of these we multiply both sides by dyjdvs, and then integrate with respect to x; thus dxda?~-'^yUx' ^&J=h^TJ'+^=h^^y+'' (2). The second method is to introduce a special symbol (p) for dyjdx. Since this makes d^^dp^^^^ dp ,g, dx^ dx dy dx ^ dy ^ we have, in place of (1), p%=ny) w> which may be regarded as an equation of the first order, with p as dependent, and y as independent, variable. Integrating (3) with respect to y, we have hP'=Sf{y)dy + A (5), which is equivalent to (2). To complete the solution, we write (2) in the form dy V{2//(2/)c«y+24} = ±dx (6). 181-182] DIFFERENTIAL EQUATIONS OF SECOND ORDER, 498 The variables are here separated (Art. 173) ; but on account mainly of the occurrence of the radical the further integration is often impracticable, even with comparatively simple forms of the function f{y). A very important case is where f{y) is a linear function of y, so that the equation takes the shape d?y + ay = b (7). By a change of dependent variable, writing y^ + b/a for y, and afterwards omitting the suflBx, this is reduced to the somewhat simpler form g+»*^'> w d The first integral of this is {th^y'='- («)• If a be positive, we may write m = sla, G=m^(k^ (10), it being evident that, if we are concerned solely with real quantities, G must be positive. Thus V(^)=±'"'^ (">' whence cos~^ - = + (mm + e), a or 2/ = a cos {mx + e) (12). This is the complete solution of (8), and involves the two arbitrary constants a, e. If we put 4 = acoS6, 5 = -asin6 (13), we obtain the equivalent form y=-A cos TUX + B sin mx • (14). These results are exceedingly important, and should be remembered. 494 INFINITESIMAL CALCULUS. {VU. All The case where a is negative, = — m", say, can be treated in a similar manner, and we should find, as the complete solution y = A cosh mx + B sinh mx (15), where m — \/(- o)- -A. simpler method of treating this case will however be given later. The type (1) is of very great importance in Dynamics. Thus, the equation of rectilinear motion of a particle subject to a force which is a given function of the distance from the origin is of the form 5=/(«') (16), which is identical with (1), if regard be had to the difference of notation. The first method of integration consists in multiplying both sides by dxjdt, thus dx (Px_ .. .dx and integrating both sides with respect to t. In this way we obtain ^ (§)' = /^(^^ S "^^ + ^ = /^(^^ '^'^ -^ ^ (^^)' which is the ' equation of energy.' The second method consists in writing v for dxjdt, and therefore vdv/dx for d^x/dt" ; cf. Art. 39. Thus dv Hence, integrating with respect to x, we have W = J/{x)dx+G (18), in agreement with (17). Hx. 1. If a particle be attracted to the origin with a force varying as the distance, the equation of motion is rf? = -/^ (19)- This is of the special type (8), and the solution is x = a cos ( JfU + €) (20). 182] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 495 This represents a ' siuiple-harmonic ' motion. The values of x and dxidt both recur whenever J fit increases by 27r ; the period of oscillation is therefore in/^fi. The arbitrary constants a and c are in this problem known as the ' amplitude ' and the ' epoch,' respectively. The equation of motion of any 'conservative' dynamical system having one degree of freedom, when slightly disturbed from a position of stable equilibrium, is also of the type (19). For example, the accurate equation of motion of a pendulum is l^ = -gBme (21), where g is the acceleration of gravity, and Z is a certain length depending on the structure of the pendulum. In the case of a ' simple ' pendulum I is the length of the string. If the extreme angular deviation from the equilibrium position be small, we may write 6 for sin 6, thus ^-=-7^ (22). The solution of this equation is e = acoaf /^^.t+e^ (23), and the period is therefore 2-n-J(l/g). The accurate equation (21) can be integrated once by the method above explained; we thus find hl(fJ = 9<^os0+C (24), but the second integration cannot be effected (except in the particular case of (7 = 0) without the introduction of elliptic functions. Jix. 2. If a particle move in a straight line under an attraction varying as the inverse square of the distance from the origin, we have whence, as in Art. 173, Ex. 3, /dx\^ --S (25). fdx\^ _ IJ. ^ \dt) ~ X + G (26). 496 INFINITESIMAL CALCULUS. [CH. XII If the particle start from rest at the distance a, we have G=—iJt.la, !=-(¥)'•("-?)' •■••■■•■; <^"' the minus sign being taken since the velocity is towards the origin. The second integration is facilitated by the substitution a; = acos=e (28). Separating the variables, we find {I + cos 26) de = (^ydt (29), e + lsinie=(^\ t + G (30). As X diminishes from a to 0, 6 increases from to ^tt. Hence the time (^) of falling from rest at the distance a into the centre of force is given by -"JTSS <*"• The time (<„) in which a particle would describe a circular orbit of radius a about the same centre of force is '""^ <'^)- ^""■^ rj72=-'" <»>■ We infer that if the orbital motion of a planet (or of a planet's satellite) were suddenly arrested, the planet (or the satellite) would fall into the sun (or into the primary) in about ■\'n of its period of revolution. 183. Equations involving only the First and Second Derivatives. If the equation be of the type HS'I)=« (■)■ i.e. the variables x, y do not appear (explicitly), then, writing p for dyjdx, we have HS-")-" <^). M 182-183] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 497 which is an equation of the first order with p as dependent variable. Ex. 1. To find the curves whose radius of curvature is constant (=a, say). By Art. 152 we have (1)}'""" ' dp dx ... or rz — {-—-=+ — (4). (1 +p')i -a ^ ' Integrating this we have (Art. 76 (13)) {\+p')i--^r ^^'' where a is an arbitrary constant. This gives dy _ _ X — a . d^~P~- J{a^-{x-af} --^1' whence y- ^ = ± ^l{a'-{x-af] (7), if p be the arbitrary constant introduced by this last integration. The result may be written {x-af + {y-pf=a' (8), and so represents a family of circles of radius a. This investigation is given merely as an example of the general method;' the problem itself can be solved more easily in other ways. Ex. 2. To determine the rectilinear motion of a particle subject to a force which is a given function of the velocity. The equation of motion is of the form S = A cos kr + B sin hr, A cos kr + B sin kr „ _, or <^ = (15). E..2. (l-=)S-| + 2/ = (16)- A particular solution is obviously y = x. We therefore put 2/ = a!« (17), whichleadsto x{l-x^)^ + (2 -3x^)^=0 (18). Separating the variables, we have da? 2 X . ,,., ■^^x-T^^=^ (1^)' dx '"^^'''^ ^=^-J^aF) (2^)' z = -A^^^Kb (21). X ^ ' The complete solution of (16) is therefore y = A^{:i-^)+Bx (22). Ex. 3. (l+«?)g + 3a: J + y = (23). This happens to be an ' exact equation,' i.e. the left-hand side is the exact differential coefficient of a function of x, y, and dyjdx, for it may be written The integral is {}. + a?) -^ + xy= A (24). 185] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 505 This is linear, of the first order, and the integrating factor is seen to be l//y/(l + a"). We thus find ^{1 + x') .y = Asmh.-^x + B (25). EXAMPLES. LI. 'S-- \]j=x log X + Ax + 5.] s=- [y = {x-2)e?' + Ax + B.] -s- y = a log - + Ax + B. 3. 4. The difierential equation for the deflection of a horizontal beam subject only to its weight and to the pressures of its supports is where w is the weight per unit length. Integrate this, on the supposition that w is constant, and determine the constants so that 2/ = 0, d?yjdoi? — both for a; = and for x=l. (This is the case of a uniform beam of length I supported at its ends.) [352/ = ^wx {I - x) {V + lx- a?).] 5. Solve the same equation subject to the conditions that ^ = 0, dyjdx^O ior x=0 and x = l. (This is the case of a beam clamped at both ends.) [^y = -^wa? (I - xf.] 6. Solve the equation of Ex. 4, subject to the conditions that y = 0, dyjdx = for a; = 0, and d^yjda? = 0, d^yjda? = for x=l. (This is the case of a beam clamped at one end and free at the other.) [% = -^wx" {6P - ilx + x^.] 7. Solve the equation d'x . and interpret the result. [a; =/Iij. + a cos {Jfd, + e).] 506 INFINITESIMAL CALCULUS. [CH. XII 8. Shew that the solution of the equation of motion of a particle moving in a straight line under a force of repulsion varying as the distance, viz. is of one or other of the types : x=a cosh ( ^fd + e), x = a sinh {ijid + e), x =ae**''''+' ; and interpret these results. 9. A particle moves from rest at a distance a towards a centre of force whose accelerative effect is /* x (dist.)"'. Prove that the time of falling to the centre is a^jijft.. 10. Obtain a first integral of the general differential equation of central orbits, viz. dSt P where P is a given function of u. f-jj: j + m" = 2 / .-y-^ du + C. 11. Solve the equation d'r _ fi. d^~?' and shew that the solution is equivalent to 1^=1 + 231 + 0^, where A, B, C are connected by the relation AC-£' = ^,. «S = ^(l)T- [.-/^-eosh^«.] DIFFERENTIAL EQUATIONS OF SECOND OKDEB. 507 ^^' ds?'*"^'"^' [y = A. + Bx + Ccosx + I)Bin x.] 29. 30. ^{(^-/^'^Ill^^- [» = ^+^tanh-V.J 508 INFINITESIMAL CALCULUS. [CH. Xll 31. ^Ul-tJ,^)p\+2u=0. [M = ^/* + £(l-/itanh-».] 32. Find the curves in which the radius of curvature is equal to the normal, on the same side of the curve. [The circles (x - af + y' = ^".J 33. Find the curves in which the radius of curvature is h, this is equivalent to {D-X,){D-\,)y = (8), where Xj, Xj are the roots of X'H-aX + 6 = (9), \ = -\a±^{la?-h) (10). VIZ. Xa If we write (D-\)y = z (11), the equation (8) becomes {D-\)z = (12), a linear equation of the first order. The solution of this is, by Art. 176, z = Ae>^^'> (13). Substituting in (11), we have (D-X,)2/ = ^e^=^ (14), whence, by Art. 176, 1°, y = G^e^^«-\-G^ (15), 186] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 511 if (7i = A/(Xi — \). Since A is arbitrary, the constants C^, Cj are both arbitrary; and the process shews that (15) is the most general solution of the proposed equation (6). If Ja" = b, the roots of the equation (9) in \ are coincident, and the equation (14) becomes (D-X,)2/ = ^e^'» (16). The general solution of this, as found in Art. 176, 2°, is y ^ (Aid + B) e'^^^ (17). If ia^< b, the values of Xi, Xj are imaginary, but we can still obtain, by the foregoing process, a symbolical solution involving »J{— 1). Into the question as to what meaning can be attributed to such a result it is not necessary to enter here, as the difficulty can be evaded in the following manner. If we write y = e~i'"'z (18), we find By = e-*"* {B - ^a) z, By = e-*"* (D^ -aB + ia=) z, so that the equation (6) becomes £ + (&-K)^ = o (19). The solution of this, when b > |a^ has been shewn in Art. 182 to be z = A cos /3a; + jB sin /3a; (20), where ;8 = V(6 - ia') (21). Hence the solution of (6), in the present case, is y = e-i'^ (A cos ^x + B sin ^x) (22). This is also equivalent to y = Ce-i'"' cos i^cc+e) (23), where the constants G, e are arbitrary. To summarize our results : (a) If la^ > b, the solution of (6) is y = C^e*'^ + G^e^" , 512 INFINITESIMAL CALCULUS. [CH. All where \i, A-j are the roots of X" + a\ + 6 = ; (6) If la" = b, the solution is y = (Aoo + B) e-i"" ; (c) If Ja" < b, the solution is qj = e-i'"'(A cos ^x + B sin ySa;), if ;8" = 6 - ia^. ^->- . . %*%-'y-o (")■ The equation in A. is X" + X - 6 = 0, whence X = 2, or - 3. Hence y^Ae^ + Be-"^ (25). The equation in X, viz. (X + 2)^ = 0, has the double root — 2. Hence y = (,Ax + B)e-'^ (26). .£"0!. 3. The free oscillations of a pendulum in a medium whose resistance varies as the velocity are determined by an equation of the form d?x ^ dx ^ .-_^ .7? + '^rf*-^'^^ = « (2^)' where A is a coefficient of friction. The same equation also serves to represent the motion of a galvanometer-needle as affected by the viscosity of the air and by the electro-magnetic action of the currents induced by its motion in neighbouring masses of metal. When regard is had to the difference of notation, the solution of (27), when the friction falls below a certain limit, is a; =Ce-5" cos («!« + £) (28), where »Ji=V(/*-i^) (29). The motion represented by (28) may be described as a simple- harmonic vibration of period 2^/%, whose amplitude diminishes asymptotically to zero according to the law e"***. 186-187] DIFFERENTIAL EQUATIONS OF SECOND OEDER. 513 The solution (28) assumes that k^K^ii.. When liii, the proper form is a! = ^eM + 5eM (30), where \, X^ are the roots of k^ + kk + ^=0 (31). By hypothesis, these roots are real. Since their product (ft) is positive they have the same sign ; and since their sum (— k) is negative, the sign is minus. Hence the displacement x sinks asymptotically to zero after passing once (at most) through this value. This case is realized in a ' dead-beat ' galvanometer, or in a pendulum swinging in a very viscous fluid. In the critical case oiT^ — ifi, we have x={A+£t)e-i'^ (32). The first factor increases (in absolute value) indefinitely with t, •whilst the second diminishes. The decrease of the second factor prevails however over the increase of the first, and the limiting value of the product, for t= , is zero. Cf. Art. 28, Ex. 3. 187. Determination of Particular Integrals. We have next to consider the problem of finding a particular integral of the linear equation of the second order with constant coefficients, when this equation has a right- hand member, thus : (D' + aB + b)y=r (1), where T^ is a given function of «;. As already remarked, any particular integral, however obtaiiied, will serve the purpose. Thus, we may omit from the particular integral any terms which occur in the complementary function, since these will contribute nothing to the left-hand side of (1). Conversely, if for any purpose it is convenient to do so, we may add to the particular integral any groups of terms taken from the complementary function. Again, if V be composed of a series of terms, the problem consists in finding values of y which, when substituted on the left-hand side of (1) will reproduce the several terms, and adding the results. It will be sufficient here to notice the most useful cases. h. 33 514 INFINITKSIMAL CALCULUS. [CH. XII 1°. If y contains a term Ee^ (2) the corresponding term of the particular integral is w= , ^ J ^ (3). For if we perform the operation B^ + aD + b on the right- hand side of this, we reproduce (2). This rule fails if a" + aa + 6 = 0, i.e. if e"* be one of the terms which occur in the complementary function. Using the notation of the preceding Art., we will suppose that a = \i, so that the equation to be solved is (D-\){D-\,)y = He'''' (4). If we write, for a moment, (D-\,)y = z (5), this takes the form (D--\i)^ = ^e*'* (6). It was found in Art. 176, 2°, that a particular solution of (6) is z^Hx^^" (7). It remains only to solve {D--K)y = Hxe^'^ (8). The integrating factor is e""''*; thus J)(yg-M) = ^^.e(x,-A,)a: (9) Integrating the right-hand member by parts, and omitting a term already included in the complementary function, we find Xi — Xj JT or 2/ = - —xe^'" (10). Ai — Aj A further modification is necessary if a be a double root of the equation D' + aB + 6 = 0. The equation to be solved has now the form {B-Xyy^Hf" (11). 187] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 515 The first step is as before, but in place of (8) we have {D-\)y = Hx^^ (12). We found in Art. 176, 3°, that a particular integi-al of this is y = lHa?^ (13). The forms of these results being once established, the student will probably find it the easiest and safest course to assume y = Ge°^, y=Cxe'^, ovy = G%H'^ (14), as the case may be, and to determine the value of G by actual substitution in the equation {D'' + aD + h)y = He'^ (15). The work is facilitated by formulse to be given in Art. 188. ^'"•^- %^%-^y=^^'~"° (^^>- As in Art. 186, Ex. 1, the complementary function is y = Ae^ + Be-^. If we assume y = G#^, we find on substitution that the first term on the right-hand side of (16) is reproduced provided C = ^. The second term comes under one of the exceptional cases above discussed, since - 3 is a root of X^ + A. — 6 = 0. If we assume y = Gxe~^, we find that the term in question is reproduced, provided G = — ^. The complete solution of (16) is therefore y^Ae^ + Be-'^ + le^-^xe-^ (17). ^^•2- 2+^J + *2/ = «-+«-=" (18). The complementary function has been found in Art. 186, Ex. 2, to be y = {Ax + B) e-^. To reproduce the first term on the right hand, we assume y = Ce^, and find G = Jy. The second term corresponds to a double root in the equation for \; assuming, therefore, y = G3?e~^, we find C = J. Hence the complete solution of (18) is y = {Ax + B)e-'^+^-^e^ + ^x^e-'^ (19). 33—2 516 INFINITESIMAL CALCULUS. [CH. XII 2°. If Y contains terms of the form H Gos ax + K sin ouc (20), we may assume y = A cosow; + jBsin ax (21). Substituting in (1), \ve obtain, on the left-hand side, (— o?A + aaB + hA) cos ax + {-a^B — aaA + hB) sin ax. Hence the terms (20) are reproduced, provided {-a^Jrh)A+aaB = H, -aaA + {-a^ + h)B = K...(22). Except in the particular case where a = 0, a? = h, which will be considered presently, these equations determine A and B; thus (,-a? + h)H-aaK aaE+(-a' + b)K (a»-6)» + aV ' {a'-by+aV •••^^**^- JSx. 3. -To find a particular integral of -^ + k -^ + fix =/ cos (pt+e) (24). This is the equation of motion of a pendulum subject to a resistance varying as the velocity, and acted on by a force which is a simple-harmonic function of the time. We assume x = A cos (pf + e) + B sin (pt + e) (25), and find, on substitution, - p^A + hpB + ixA =/, - p'B - kpA + fiB = } (26), mi nr}" JcO If we put A=Rcosei, £ = Bsmei (28), the solution (25) takes the form X = IiGOS{pt+ £ — «i) (29), where -^=-777 4^ — 72~ai ' ei = tan-'— ^ (30). We have thus determined the 'forced oscillations' due to the given periodic force. The ' free oscillations,' which are in general 187] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 517 superposed on these, are given by the complementary function (Art. 186, Ex. 3). Unless k=0 they gradually die out as t increases. The foregoing results simplify when the coefficient a in the differential equation is zero; a particular integral of -tA+ by = H cos aso + K sin CUV (31) being obviously V = 5 iCosaa! + , ^„sina« (32). A singularity arises, however, when o^ = h. To find the proper form in this case we may assume y — u cos ax + v sin a.x (33). This makes -A + a.^y = {2-^ + . . . + An-.D + A^. . .(3), then ■^{D)e^=y]r{\)e>^ (4). For D^'e'^ = >S^, and thus the several terms of ■<^ (D) give rise to the several terms of ■^^ (\) as factors of e^. * Usually, in the physical applications, the equation (37) is approximate only, being obtained by the neglect of powers of x higher than the first. Hence when the amplitude increases beyond a certain limit, the equation ceases to apply, even approximately, to the subsequent motion. 187-189] DIFFEKENTUL EQUATIONS OF SECOND ORDER. 519 2°. With the same meaning of ■>^ (D), if u be any function of X, then f{D).^''u = e'^.ir{D + X)u (5). For we find in succession JD .e'^u = e'"'{D + \)u, ==e^{D + Xfu, and so on ; the general result being IX.^u = e^(D + \yu. Hence the several terms of the operator ylr (D) give rise to the several terms of the result given in (5). 3°. If ylr (D) contain only even powers of B, it may be denoted by {D).^{D) (2), the equation (1) is obviously satisfied by any solution of t(^)2/ = (3). And since the factors are commutative (Art. 186), it is also satisfied by any solution of HD)y = o (4). Hence (1 ) is satisfied by the sum of any solutions of (3) and (4). Continuing the resolution we see that if f{D) = ,{D).cj,,(D).,{D)y = 0, <^,(D)2/ = 0, s(D)y = 0, (6). By a theorem of Algebra, already referred to in Art. 84, the function f(I)) can be resolved (if its coefficients be real) into real factors of the first and second degrees, the sum of the degrees of the several factors being equal to the degree (n, say) of the function. Moreover the factors of the first degree are of the forms Z»-\i, I)-\^, D-Xs, ... where Xi, X^, X3, ... are the real roots of /W = (7). If X be a simple root of (7), one of the equations (6) is of the form (D-\)y = (8), the integral of which is known to be y=Ge>^ (9). And if the roots of (7) be all real, say they are Xi, X2, ... X„, a solution of (1) involving n arbitrary constants is y = de'^'* + C^e''^ + ... + O^e^"* (10) ; of. Art. 186 (15). If the equation (7) has a multiple root, two or more of the terms on the right hand of (10) coalesce, and the number of distinct solutions thus obtained is less than n. To supply the deficiency, we remark that, if X be an r-fold root of (7), /(D) contains a factor (D — X)"". To solve (B-\yy = (11), we assume y = e^ z (12), which makes {B -\y y = (I) -Xf .e^ z = &^ !>■' z, by Art. 188, 2°, and the solution oiiy z = Q'\% obviously z = B,^-B^x-\- B^^ + . . . + Br-X~\ whence y = (i?„ + 5, + jB^a;^ + . . . + Br-X'^) e^ (13). 189] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 621 We have here r arbitrary constants, corresponding to the ir-fold factor of /(D). Cf. Art. 186 (17). If /(-D) contains, once only, an irreducible quadratic factor D^ + aB + 6, where \a? < h, then part of the solution of (1) consists of the solution of (D' + ai) + 6)y = (14). If we put y = e-i'^2, 0' = h-la? (15). we have, by Art. 188 (5), {D' + aD + h)y ={{!) + ^af + /S"} e-i«» 3 And the solution of (D" + ^)z = 0, is z = E cos ^ OB + F single. We thus find y = e-i<^ (E cos ^so + F sin ^x) (16), as in Art. 186 (22). Hence for every distinct quadratic factor of /(D) we obtain a solution involving two arbitrary constants. Finally, if /(D) contains an irreducible quadratic factor which occurs r times, we have to solve (B' + aB + byy=0 (17), or, making the substitutions (1-5), as before, (B' + fi'yz = (18). To solve this, we assume 0= M cos /3a; + ?; sin /8« (19). Now, by actual differentiation, we find (B' + ^'').ucos^x = 2^.Bu.cos(l3x + i-!r) + ..., {B^ + ^J . u cos ^x = (2;S)= . ICu . cos (jSa; + tt) + . . ., and, generally, (D'' + ^y. u cos ^x = (2/3)'- . IKm. cos (/8a! + J r-Tr) + (20), where only the terms of lowest order in the derivatives of u are expressed. Similarly (D' + ^y. V sin ^x = (2/8)^ B^v . sin {^x + ^ttt) + (21). 522 INFINITESIMAL CALCULUS. [CH. XII Hence the equation (18) is satisfied, provided l>u = 0, D'-D = (22), ie. u = Eo + E^x + E^^ + . . . + ^,.-i a'*""'! , (23). v = F^+F^x+F^^ + ...+Fr-iar--'] The complete solution of (17) is therefore y = (E„ + EiX + E^ +... + ^,.-1 «'^') e- J«^ cos ^x + {F^ + F^x-\-F^+...+ i?;_ia'->)e-i«* sin ^x. . .(24), involving 2r arbitrary constants. s=s w This may be written Z*^ (Z) — 1) y = 0, and the complete solution is accordingly made up of the solutions of Cfy = 0, (Z) - 1) y = 0, thus y^A+Bx + Ge?' (26). ^a'-a. % = '"''y (27). This may be written {D -m)(D + m) (D^ + m?)y = 0. Adding together the solutions of {l)-m)y=0, (D + m)y=0, {B' + m')y = 0, we obtain y = Ae^ + Be-'"" + B cos mx + F sin mx (28). ^-^- S-^S^^ = « (29)- This is equivalent to {D^+D+l){D'-D+l)y=0. Hence y = e-i^r^cos^a; + ^sin^a;) + ei^^4'cos^a; + ^'sin^a;) (30). £;x.i. g + 2^^g + mV = (31). Writing this in the form we find y=(Bf, + Ejps) cos mx + {Fo + F^x) sin mx (32). 189-190] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 523 190. Particular Integrals. We proceed to the determination of a particular integral of the equation f{D)y=y • (1). in the two most important cases. 1°. If F contain a term H^ (2), the corresponding term in the particular integral is »=/-|r <'>• 77 for this makes /(i))2/ = jT-.f{D)0-^ = Ife^ by Art. 188 (4). The rule fails when a is a root of f{D) = (4). If it be a simple root, we may write f{D)^i{B){B-a) (5), where ^(D) does not contain the factor D — a. The equation {D){B-a)y = He- (6), is satisfied provided {D — a)y= , e"*, and we have seen, in Art. 176, 2°, that a particular integral of , ■ • H this IS V = -rr^a'e°* (7). If a be an r-fold root of (4), we may write f{,D)^4>{D){D-ar (8), where (D) does not contain Z) — a as a factor. The equation <\>(D){B - af y = He^ (9), is satisfied, provided Now if we put y = e^z. 524 INFINITESIMAL CALCULUS. [CU. XII we have, by Art. 188 (5), (I)-ayy = e'''D'z, whence D^z = , , . . <^(a) This is evidently satisfied by r ! ^ (a) A particular integral of (9) is therefore y= nfe,'^ (10). ^ r!^(a) ^ ' 2°. Let V contain terms of the type jEf cos aa; + ^ sin OB (11). Since the operation /(D), performed on y = A cos oue + B sin ate (12), must result in an expression of the same form with altered coefficients, a particular integral can in general be found by substituting this value of ?/ in the equation f(D)y = H cos aa; + -B^sin ax (13), and determioing A and B by equating coefficients of cos ax and sin ax. In one very frequent case, the values of A and B can be written down at once, viz. when the equation is of the type ^ (Z)") y = H cos ax-\- K sin ax (14), i.e. f{D) contains only even powers of I). We have, then, by Art. 188, 3° 2/ = ^(l^)«°««^ + 0(3^)«i°«'^ (15)- This result fails if ^(—a^) = 0, i.e. if {B") contain D" + o" as a factor, in which case terms of the type (11) occur in the complementary function. If the factor jy + a' occur once only, we write 0(D'Os;^(i)^)(i)'^+a») (16). Now the equation X {D-) {B^ + a") y = H cos ax + K sia ax (17), 190] DIFFEEENTIAL EQUATIONS OF SECOND ORDER. 525 will be satisfied if H K (If + «")« = — 7 s^ COS ax H — 7 — ^ sin ax. . .(18). The problem is thus reduced to one already solved under Art. 187, 2°, viz. we have the particular integral y= a — 7 ^ajsinoa;— ^5 7 rra;cosaa; (19). ^ 2ax(-a0 2ax(-a'') If the factor D" + a" occur r times in ^ (D^), we write ,^(I»OsxW(^" + «')'■ (20), and the problem is reduced to finding a particular integral of H K (D^ + «')'■ y = — 7 rr cos ojx H 7 jT sin ax . . .(21). X(-a) XC-") If we assume 2/ = M cos (aa? ^ ^tt) + ^^ sin (aa; — ^r-ir) (22)*, we find {B' + ay y = {2ay. iTu. cos ax ^{^ay.iyv. sin ax +... by Art. 189 (20), (21). Hence (21) is satisfied provided Ex^ Kx'- . . °'" " rl{2ayx(-«-y '"~rl(2ayx(.-a')'"^ ^' A particular integral is therefore ^ " r\(2ayx(-«'') ^^ ^^ ^'^ ~ ^"^"^^ ^ ^ ^'" ^°"^ " ^'^^^ (25). In the general case, where /(If) contains both odd and even powers of D, the assumption (12) fails in like manner if, and only if,/(i)) contains the factor D' + a". Writing f{D) = x{B){D^ + ay (26), * The assumption y=ucoaax + vsriiax would serve equally well, but the form in the text enables us to write the final result in a more compact manner. 526 INFINITESIMAL CALCULUS. [CH. XII where ;)^ (D) does not contain the factor Z)* + a', we first obtain a particular integral of the equation X(.^)y= -ffcos ax+ KsinaoB (27), in the form y = Si cos ax + K^ sin ax (28). It then remains only to solve (iy-\^ay y=H^ cos ax + K^sm ax (29). This has been treated above. Another case where a particular integral can be obtained is that of f{D)y = X (30), where X is a rational integral function, of (say) degree r. We put y = x^v, where m is the lowest index of D which occurs in /(D), and » is a rational integral function of x, of degree r. The coefficients in v are then determined by substitution. 191. Hoino§reneous Linear Equation. An equation of the type + A^,x£+A„ij = V. (1), is sometimes called a 'homogeneous linear equation.' The complementary function in this case consists in general of a series of terms of the form Gk™, where C is arbitrary, and the values of m are to be found by substitution on the left-hand side. Moreover, if V contains a term HxP, the corresponding term in the particular integral will in general be BxP, provided B be properly determined. To see the truth of these statements we may take the homogeneous linear equation of the second order. To solve -^S^-l+^^=« (2), we assume y = Gx™ (3). This will satisfy the equation provided {m{m-l) + am + b} 0x^=0 (4). 190-191] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 527 Equating to zero the expression in { }, we have a quadratic in m. If TWi, TTia be the roots of this, the solution is y = G^x^^+G^x'^ (5). Again, a particular integral of the equation a?P. + ax^ + hy = HxP (6), da? ax ^ will be ' y = BxP (7), provided {p{p-V)-\- ap + h]B = H (8). f;^^s=« » If we multiply by r^ this comes under the form (1) ; thus rPV iJV ^'J-^^t, 2/ = ^ie-^0 ^^ . and x = A^e~'^*, y = B^-''J') where the relation between A^ and B^, or A^ and B^, is given by either of equations (10) with -X, or —A, written for X. On account of their linear character, the equations (8), with E — F=Q, are satisfied by the sums of the above values of x and y, respec- tively. If E and F are not zero, but given constants, a particular integral of (8) is evidently E F '"=R' y=s' and the complete solution is E I '\ (13), y=-n+ Bje-f^i* + Bffi-'^it 34-2 532 INFINITESIMAL CALCULUS. [CH. XII where the relations between A-^ and B^, and between A^ and B^ are as above indicated. The first terms in these values of x and y represent the steady currents due to the given electro-motive forces ; the remaining terms represent the efiects of induction. Since we have, virtually, two arbitrary constants, these can be determined so as to make the actual currents have any given initial values. Another important case is where ^ is a simple-harmonic function of the time, and F is zero. Putting, then, S=S,cospt, F=0 (14), a particular integral of the equations (8) may be found by assuming x = A cos pt + A' sin pt,' y = B cospt + B' sinpt On substitution we find, equating separately the coefficients of cospt and sinpt, pLA'+pMB' + RA = M„,' -pLA-pMB + RA' = 0, pMA'+pNB' + SB = 0, -pMA-pNB + SB' = (i These formulae give A, A', B, B', and so determine the elec- trical oscillations in the two circuits due to the given periodic electro-motive force. The free currents are given by terms of the same form as in (12) Their values depend on the initial circumstances, and in any case they die out as t increases. .(15). .(16). Fx. 5. As a final example, we take the equations dt^ . CPX rrCpy , „ ■(17), which determine the motion of any ' conservative ' dynamical system, having two degrees of freedom, in the neighbourhood of a position of equilibrium. To find the free motion, we put X = 0, 7=0, and assume x = Fe^, y=Ge>-* (18). 192]- DIFFERENTIAL EQUATIONS OF SECOND ORDER. 533 We thus get {A\^ + a) F+ {HX? + h)O = 0,\ .^g> {H\^ + h)F+(£X^ + b)G = o\ '' Eliminating the ratio F : G, we have {AX^ + a){BX? + b}-(HX?+hY = (20), or (AB - H') X* + {Ah + Ba- IHh) X^ + {ah - A") = . . .(21). This is a quadratic in \^ The expressions iHt)*"'i'i*K%'\ <^^>' and l{aa?+2hxy + hy^) (23), represent the kinetic and potential energies of the system. The former is essentially positive; hence A, B are positive and AB^H"". It follows that the left-hand side of (20) or (21) will be positive both for A.^ = + oo and for X^ = — oo , whilst for X^ = the sign is that of ah — h^. Also, from (20), it appears that the left-hand member is negative for X^ = — ajA and for X" = - hjB. Hence if the expression (23) be essentially negative, so that a, h are negative, whilst ab — W is positive, the equation (21) is satisfied by two positive values of X^, one of which is greater, and the other less, than either of the quantities —ajA, —Bjh. Denoting these roots by Xj", X/, we have the solutions X = F^e>^^t + Fj'e-^'f' + F^e>^t + FJ[e-'"f,\ , (24). Of the eight coefficients, only four are arbitrary. The ratio F^ : Gi, which is the same as Fj' : G-l, is determined by (19), with Xj" written for X''. Similarly, the ratio F^ : G^ or F^ : G^ is determined by the same equations, with X/ written for X^. The four arbitrary constants which remain may be utilized to give any prescribed initial values to x, y, dxjdt, dyjdt. It appears that X and y will increase indefinitely with t, unless the initial circumstances be specially adjusted to make F-^ and F^ vanish. Hence if the potential energy in the equilibrium position be greater than in any neighbouring position, an arbitrarily started disturbance will in general increase indefinitely; so that the equilibrium position is unstable. If, on the other hand, the expression (23) be essentially positive, so that a, h, ah— h^ are positive, the roots of the quadratic in X^ will both be negative, viz. one will lie between 534 INFINITESIMAL CALCULUS. [CH. XII and the numerically smaller of the two quantities — a/ A, — hjB, and the other will lie between the numerically greater of these quantities and — oo. This indicates that in place of (18) the proper assumption now is x = Fcospt + F' sin pt, y = Gcospt+ G' sin pt... (25). This leads to equations of the forms (19) and (21), with —p^ written for A.'. It follows that the two roots of the quadratic in p^ will be real and positive ; denoting them by pi^ and p^, we get the solutions x = Fi cosp^t + F-^ sin jPi* + F^ cos ^3* + F^ sinp^t,') [■■■(26), y = Gi cosp^t + G^ sin^i< + G^ cosp^ + G^' cosp^t J. where the ratios FJG^, F-^jG-l, F^G^, F^'/G^' are determined in the same manner as before. Since Fi'/Gi'= F-JG^, and F^jG^ =FJGi, the results may also be written X = F^C09{pJt+€^) + FsC0s{p^ + e^, ) y = (?i cos (pi« + £,) + G'a cos (pa* + Ej) J '' where F-yjG-i and FJG^ are determinate. Hence when the potential energy in the equilibrium position is less than in any neighbouring position, a slight disturbance will merely cause the system to oscillate about the equilibrium position, which is therefore stable. The two roots of the quadratic in X" (or in p') have been assumed to be distinct. It may be proved that they cannot be equal unless ajA — hjB = hjH ; and that if these conditions be fulfilled the solution is of one or other of the two types : x=:Fe>^ + P'e~>^, y = (?e« + G!'e-« (28), a: = -f cos^*+ F' sin^i, y= G cos pt+ G' cos pt... (2^), where the four constants are in each case independent. Finally, we have the case where the expression (23) for the potential energy may be sometimes positive and sometimes negative. In this case ah - h^ is negative, and one root of the quadratic in A." is positive, the other negative. The complete solution is now of the type x=Fe^ + F'e->^t + F" cospt + F" sm.pt,\ y = Ge» + G'e->'i+G" cos pt + G"'mnpt] ^ '' It is clear that an arbitrary disturbance will in general increase indefinitely, so that the equilibrium position must be reckoned as unstable. 192] DIFFERENTIAL EQUATIONS OF SECOND ORDEE. 535 A slightly different method of treating the question is to assume y=t^ (31)- The equations to be solved now take the form {A + fiH) -j-^ + {a + fih)x = Q, .(32). These are both satisfied by x = Fe>^t (33), provided — -S-=-i r = -T3 v-^*)- Hence /x is determined by the quadratic {Eb- Bh) ^^ + {Ah - Ba) p.+ {Ah- Ha) = (35). If /ii, /Ha be the roots of this, the corresponding values of )sirima!.] [2/ = e*"* (A cos noo + B sin. na;).] 8. ^-4^ + 13y = 0. [2/ = e^(/icos3a; + ^sm3a;).] 9. S"^^^"^^^"*^' [2/ = «"°'(^cos2a! + 5sm24] 10. ^-6^^ + 13y = 0. [2/ = e«'"(^cos2a! + ^8m24] 11. ^ + mV = 0. r / , , mx _ . , 7nx\ mx ly = (^ ''"^'^ 72 ^ ^ "'"'^ 72 j '°' j2 / ., . mx „, . , mx\ . nix "I DIFFERENTIAL EQUATIONS OF SECOND ORDER. 537 [2/ = ul cos (ma: + a) + JS cos (ma; + )8).] 18. Shew that the solution of the equation cPx , dx . dt- dt '^ is of the form x = Ae~°-t + Be^*, where a, ^ are both positive (if k and fi are positive) and a > /S. 19. V^ - 2m -r- + inv = sin nx. dx- dx [(m? — n') sin nx + 2mn cos nx . "1 2/ = ^ (m' + ny +H 20. 21. -5-^ + -^ = 1 + cosh a:. \y = x + \e?' — ^.re~* + ikc.J dx^ dx 23. -r4 - ni^'l = cos te + cosh kx. dx* ^ y ~ ta — 4 (""'^ ^^ "*" ^^^^ ^^) + "^c. „>. a , dW \dV . 24. Solve -n + -^5- = ar' r dr as a homogeneous linear equation. \Y=A log r + .S.] 30. 31. Prove that 5.38 INFINITESIMAL CALCULUS. [CH. XII 25. '^^~^S=^- [y=^+^^og«' + C!»^] 26. x'^-^y^x. [y^AaP + ^-ix^j 29. x'^-3x^ + iy = a?. [y = {A + B log x) a? + x>.] at 32. Prove that /(« ^) a;™ log a; - x" {/(m) log a; +/' (m)}. 33. ~ + 7x-y = 0, ^+2x + 5y = 0. [x = J {(4 + ^) sin t + (A-B) cos «} e"", y = (Aco3t+£ sin «) e"".] _- dx „ dw [x = (A + Bt)e''', y^{A-B + Bt)e^:\ 35. ^+5£c + y = e', ^ + Zy-x = e^. at "at [x = {A+Bt)e-» + ^^-^^, y = -{A + B + Bt) e-« + ^ef + ^^.] 36. g-,3a,-4y+3 = 0, g+a,-y + 5 = 0. [a; = (.4 + £<) cos t + (A' + Ft) sin « - 17, y = ^(A+B' + Bt)co&t + ^{A' -B + B't) sin « - 12.] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 539 37. Determine the constants in the solution of the simul- taneous equations cPx so that, for < = 0, df=f^' -dT-^f^y . dx ^ d/u „ \x=a cosh ^fLt, y = V/Jijl . sinh ^^/t*.] 38. The equations of motion of electricity in a circuit of self- induction L, and resistance B, which is interrupted by a condenser of capacity G, are r dx - a da ^dt-'^''=-G' jr''' where x is the current, and q the charge of the condenser. Find the condition that the discharge should be oscillatory. [Z > ^R^C] and shew that the solution represents a conic symmetrical with respect to the axis of x. and prove that the curves represented by the solution include a family of hyperbolas. ., „ , ) = 8n(a>) + Bn(^) (5). The quantity jR„ (x) is called the ' remainder after n terms ' ; it is of course the sum of the series 4„a!» + ^„+,a;»+i+ (6). By hypothesis, the sequence 8,{x), S,(x), 8,(x), (7) has (Art. 6) the limiting value S(x). It follows that the sequence B,(x), B,(x), R,{x), (8) must have the limiting value zero. Ex. In the case of the geometric series l+x + 3?+ ...+x^+ (9) we have ^»(^)=T^"' '^(^)=r^' ^»(^)=i?s (10), provided |a!| \K+i{a:)\l,and u„{x) = Sj,(x) j ^^v- £;X.l. If S,{X)=:^^, (15) we have S{x) = limS^{x) = (16), n=oo for all values of x without exception. The series is however not uniformly convergent over any range which includes x = 0. For li„(x) = S{x)-S,{x) = -^^, (17), and for any given value of n this attains the values + 1 for x = ± 1/n. * Thus a power-series which is convergent for x = a, where (for defi- niteness) we suppose a positive, but not for a;>a, has been proved to be uniformly convergent for values of x ranging from to any fixed value short of o. As a matter of fact it will be uniformly convergent up to a;=a, inclusively, but this cannot be established by the above method. + By this notation it is meant that x may range from a to 6 inclusively. L. 35 546 INFINITESIMAL CALCULUS. [CH. Xlll The question may be illustrated graphically* by drawing the curve y = S (x), and also the ' approximation-curves ' y = S„ (a;), for ?i= 1, 2, 3,...t. If the series be uniformly convergent, then, however small the value of o-, from some finite value of n onwards the approximation-curves will all lie within the strip of the plane xy bounded by the cui-ves y = S(x)±(r (18). In the above Ex. the approximation-curve y = Sn (x) is derived from Fig. 17, p. 31, by contracting all the abscissae in the ratio 1/ji The curves tend ultimately to lie between the limits y = +') - Rn {(c). . .(1), and the condition of continuity is that, o- being any assigned magnitude, however small, a value of hx shall exist such that \8{af)-8{x)\)i d^, (11). J The radical can be expanded by the Binomial Theorem (see Art. 200) in the series l_le=sin»^-^6*sin^0-i^««sin» 0. Fig. 151. 197. Derivation of the Iiogarithmic Series, and of Gregory's Series. The following are important applications of the theorem just proved : 1°. If I a; I < 1, we have 1 1+0! = l—x + x' — a^ + . ■in 197] INFINITE SERIES. 553 Hence, integrating between the limits and as, log (1 + «) = «- la!" + ia!'-ia''+ (2), which is the 'logarithmic series*.' The proof applies only for | a; | < 1 ; and we cannot assert without examination that the result is valid for a; = + 1. For x=\, the terms on the right-hand of (2) are alternately positive and negative, and diminish continually, and without limit, in absolute value. Hence, by a known theorem (Art. 6), the sum is finite, and the remainder after n terms is in absolute value less than l/(n + l). Hence this remainder can be made less than any assigned magni- tude by taking n sufficiently great. The same result holds d, fortiori when x is positive and < 1. Hence the series on the right-hand of (2) is uniformly convergent up to the point a!=l inclusive; it follows by the method of Art. 195 that the sum is continuous for values of x ranging up to unity, inclusive. The two sides of (2) are therefore finite and continuous up to x= 1, and since they have been proved equal for values of x which may be taken as nearly equal to unity as we please, it follows that they are rigorously equal for a;=l. Hence log2 = l-i-t-i-H (3). This formula, though exact, is not suited for actual calcu- lation, on account of the very slow convergence of the series. It maybe shewn that about 10" terms would have to be included in order to obtain a result accurate to n places of decimals. If we subtract from (2) the corresponding series obtained by reversing the sign of x, we obtain \og\^^=2{x + \a? + l!^+...) (4), which might also have been obtained directly by integration from the identity m + l^ = ra=2(l+*^ + ^^+-) (^)- * Apparently first obtained by N. Mercator in 1G68. 554 INFINITESIMAL CALCULUS. [CH. XIIl If in (4) we put x= 1/(2to + 1), we obtain log (m + 1) - log m =log This series is very convergent, even for to = 1. Putting m = 1, 2, 3, ..., we obtain the values of log 2, log 3-log 2, log 4-log 3,... in succession, and thence the values of the logarithms (to base e) of the natural numbers 2, 3, ... . When log 10 has been found, its reciprocal gives the modulus /x by which logarithms to base e must be multiplied in order to convert them into logarithms to base 10*. 2°. Ifa!=< 1 we have ^ = l-a? + a--a^+ ...(7). Hence, by integration between the limits and tv, tan-ii» = a;-^a» + ^a=-fa!'+ (8). This is known as ' Gregory's Seriest.' The function tan~' x here appears as the equivalent of dss Jo I and must therefore be taken to denote that value of the ' multiform ' function tan~' x which lies between + ^tt. For reasons similar to those given at length in connection lyith the formula (2), the result (8) holds up to both the limits +1 of «, inclusively. Putting a; = 1, we find i'^=l-* + i-f+ (9). * The most rapid way of determining /n. is by means of tlie identity log 10=3 log 2+log|. The two logarithms on the right hand are found by putting m = 1, m = 4, in (6). t After the discoverer James Gregory (1671). 197] INFINITE SERIES. 555 The series (9) converges very slowly, and lias been superseded for the calculation of ir by others. Euler used the identity ^ = tan-'^ + tan-iJ (10), which gives 1 1 \ /I 1 1 * \2 3.2« 5.2= 7 V3 3.. 3»"^"5.3» .)...(11). Machin had previously employed the formula i,r = 4 tan-'^- tan-Vis (12), which, like (10), is proved in most elementary books on Trip;ono- metry. This leads to ^'' " * (5 " 3T5» "^ 57^' " ■ ■ 7 V239 ~ 37239= "^ 57239= ~ ■ • 7 (13). On account of the importance of the matter, it is worth while to give the details of the calculation of ir from Machin's formula. To calculate tan~' J, we first draw up the following table : n 1 5» n.5» 1 ■200 000 000 + -200 000 000 3 8 000 000 - 2 666 666 7 5 320 000 + 64 000 7 12 800 1 828 6 9 512 + 56 9 11 20 5 1 9 13 8 + 1 Tlie sum of the positive terms in the last column is + -200 064 057 0, and that of the negative terms is -•002 668 497 2. Hence tan->i.= -197 395 559 8. Again to calculate tan^'^-j^ we have the table : 556 INFINITESIMAL CALCULUS. [CH. XIII n 1 239" *n.239'' 1 3 •004 184 100 4 73 2 + -004 184 100 4 24 4 This makes tan-i^= -004 184 076 0. Hence iir = 4 tan~*-|- - tan-'-^-^ = +■789 582 239 2 -•004 184 076 = ■785 398 163 2, ir = 3-141 592 652 8. The last figures are of course liable to error. To estimate the possible error of the final result, we remark that in the calcula- tion of tan~'|- there were five errors, each not exceeding half a unit in the last place, and that there are two such errors in the computation of tan~'-!j|-5-. The errors, therefore, in the inferred value of TT, even if cumulative, cannot exceed 4 X (4 X 5 X ^ + 2 X J), =44, times the unit of the last place. The first seven decimal places cannot therefore be aflfected, and we can assert that the last three must lie within the limits 484 and 572. As a matter of fact the errors are not all in the same direction, and the correct value of IT to ten places is jr = 3-141 592 653 6. 3°. If I a; I < 1, we have, by the Binomial Theorem (see Art. 200), v(T^=^+r^+2-7l^+2-:ltl-'+ (1^)- Hence, integrating term by term between the limits and «, la? 1.3af> 1.3.5X' sin-^a; = a; + H7r + a-^-F + a . ^ 1;-+ (15); 23 a series due to Newton 2.4 5 2.4.6 7 197-198] INFINITE SEBIES. 557 If we put « = i, we get [2 ' 2 . 3 . 2» ' 2 . 4 . 5 . 2' from whicli TT can be calculated without much trouble. 198. Differentiation of a Power-Series. If the series A, + A^o) -{- A^'' + ... + Ansif' + (1) be convergent for any one value (a) of cc, the series ^i + 2J.2a; + 3A«'+ ... + n^„«'^'+ (2), composed of the derivatives of the successive terms of (1;, will be essentially convergent for all values of x such that For the hypothesis requires that 1 4„a" | shall tend with increasing n to the limiting value 0. Hence if M denote the greatest of the quantities \A,a\. |^a»|,...|4„a»|, (3), the terms of (2) will be in absolute value respectively less than the corresponding terms of the series ^{l+2t + Bt'+...+nP>-^+...) (4), where t = | x/a \ . The ratio of successive terms in (4) is of the form n + 1 -t or i^-l> and the limiting value of this, for increasing n, is t, which is by hypothesis < 1. Hence after a finite number of terms the series (4) will converge more rapidly than a geometric progression whose common ratio is ti, where ^ may be any quantity between t and 1. The series (4), and d fortiori the series (2), is therefore essentially convergent. 558 INFINITESIMAL CALCULUS. [CH. XIII We can now shew that if S{a;) denote the sum of the series (1), and/(a;) that of the series (2), then /(^)=«'(«^) (5), i.e. the sum of (2) is the derivative of the sum of (1), for all values of x within the range of convergence of (1). Since f{x) = A.,-\-2A^^-^A^^ + ...+nA„x^^-^ (6), we have, by the theorem of Art. 196, ^J{x) = A,^ + 4,p + ^3^ + . . . + ^„p + . . . = 'S(^)-A (7). provided f lie within the range of convergence of (6). Hence, differentiating both sides with respect to f, and afterwards writing X for f, we obtain the result (6). We have thus ascertained that, for values of x within the range of convergence, the sum of a power-series is a differen- tiable, as well as a continuous, function of x, and that its derivative is obtained by differentiating the series term by term *. A more general theorem is that if the series S{x) = u^{x) + rii{x)+u^{x)+ ...+Un{x)+ (8) be convergent for values of x extending over a certain range, and if the series M„' (a;) + V(a;) +1*2' (»;) + ...+ «„'(«)+ (9), composed of the derived functions of the several terms of (8), be uniformly convergent over this range, the sum of this latter series will be ^' (a;). Ex. 1. If I a; I < 1, we know that .r^ = \+x + x' + a?+ (10). 1 —X ^ ' * The plan of deducing this theorem as a corollary from that of Art. 196 is attributed to Darboux. It has considerable advantage in point of simplicity over the inverse order more usually followed. 198-199] INFINITE SERIES. 559 3, we obtain = l+2x + 3x^ + 4:0? + (11). Differentiating both sides, we obtain 1 (1-a;)^ A second differentiation leads to 1 {l-xf |(1.2 + 2.3a;+3.4a;^ + 4.5a^+...) ...(12). Sx. 2. On the other hand, consider the series (considered already in Art. 194) in which ^"(^)=r!^ (1^)- This makes S(x) = 0, and therefore S' (x) = for all values of x; but, if we differentiate term by term, the sum of the resulting series is lim^„'(a!) = lim ,{.,.., (U). 71=00 U=QO \± -T fi Jj J This vanishes if a;=t=0, but it becomes infinite for a;= 0; the series of derivatives is in fact divergent for aj = 0. 199. Integration of Differential Equations by Series. Given a linear differential equation, with coefficients which are rational integral functions of the independent variable (x), it is often possible to obtain a solution in the form of an ascending power-series, thus y = Ao + AiX + A^+ ...+Anx'^ f (1). If we assume, provisionally, that this series is convergent for a certain range of x, it can be differentiated once, twice, . . . with respect to x, by the theorem of Art. 198. Substi- tuting in the differential equation, we find that this can be satisfied if certain relations between the coefficients Ao, Ai, A^,... are fulfilled. In this way we obtain a series involving one or more arbitrary constants ; and if this series proves to be in fact convergent, we have obtained a solution of the proposed differential equation. Whether it is the complete solution, or how far it may require to be supplemented, are of course distinct questions, which remain to be discussed independently. 560 INFINITESIMAL CALCULUS. [CH. XIU One or two simple examples will make the method clear. 1°. Let the equation be dx = 2/ (2). The solution of this is of course known otherwise ; but if we were ignorant of it we might obtain a series, as follows. Assuming the form (1), and substituting in (2), we get (^1 - A,) + {2A, - ^,)a; + (3A - A,)x^+ ... + M„-^^0«="~' + -=o (3), which will be satisfied identically provided ■0-1 = -4,,, Ai=\AT^ = i^A,i, Ai = ^Ai=-^ — q -^oj and, generally, ■4» = ~ -^»-i = • • • ~ ^1 -^1" ('^)' We thus obtain y=AoE(x) (5), where, as in Art. 17, E(a:)=l+a>+^^+... + ^+ (6), and A a is arbitrary. If we were previously ignorant of the existence and properties of the exponential function, it would present itself naturally in Mathematics as involved in the solution of the problem: To determine a function whose rate of increase shall be proportional to the function itself. That (5) constitutes the complete solution of (2) may easily be shewn. For, writing y = vE(x) (7), where v is to be determined, and substituting in (2), we find dx E{x) + v[E'{x)-E(x)}=0 (6), 199] INFINITE SERIES. 561 or, since E{x) satisfies (2), 1 = 0'--^ (9> 2°. Let the equation be 2+^=« (^«)- Assuming the form (1), and substituting, we find (1. 2^2 + ^) + (2.3^3 + A)« + (3.4A + ^>'+. . • + {(n-l)mJ„ + ^^2]a;"-2+... = 0...(ll), which is satisfied identically, provided and, generally, 79:;nrTr9:7 -^«i-2 = (-)" ^;;;ri -^o' ...(12). ^^" ~ (2n - 1) 2n ^™-= ~ ^"^" 2»i ! ^»' ^'^+'~ 2«(2n + l)~^ ^"(2m+l)!'*" We thus obtain the solution 2/ = A(l-2i + 4-,-...j+A(^-3-;+^5^-. ..)... (13). The series in brackets are easily seen to be convergent, and their sums therefore continuous, for all values of x. It has been shewn in Art. 182 that the complete solution of (10) is 2/ = J. cos a; + iJ sin a; (14). Hence, given Ao and A^, it must be possible to determine A and B so that the expressions (13) and (14) shall be identical. ^- 36 562 INFINITESIMAL CALCULUS. [CH. XIII For example, putting A^ = 1, A^ = 0, we must have 1— ^, + —.— ■■■ = A cosx + Bsinx, 2 ! 4! and, changing the sign of x, 1 — — - 4- -7-7 — . . . = J. COS a; — 5 sin «. 2 ! 4! Hence we must have jB = 0, and putting a; = 0, we find A = \. We thus obtain the formula cosa;=l-2-, + ^- (15). In the same way, if we put A^ = 0, A^ = 1, we find ^ = 0, B=\, and therefore sm« = a;-g-, 4-g-j- (lb). The foregoing method is, for various reasons, not always practicable. It may also lead to a solution which is in- complete ; thus in the case of the linear equation of the second order, the method may yield only one series, with one arbitrary constant. This occurs not infrequently in the physical applications of the subject. The solution may, in this case, be completed, at least symbolically, by the method of Art. 185, 3°. 200. Expansions by means of Differential Equa- tions. The method of the preceding Art. may sometimes be utilized to obtain the expansion of a given function in a power-series, provided we can form a linear differential equation, with rational integral coefiBcients, which the function satisfies *- For example, let y = {l+xy»' (1), where m may be integral or fractional, positive or negative. Taking logarithms of both sides, and then differentiating, we have \dy _ m ydx 1 -I- a; ' * This method was first employed by Newton, to whom the series for cosx and sxnx are also due. The manner of obtaming these series was, however, different. 200] INFINITE SERIES. 563 or (l+a,)^_m,y = (2). Assuming y = Aa + AiX + A^^-\- ...+ J.„a;"+ (3), and substituting, we have (1 + x){Ai + 1A^+... + w4„a!"-'+ ...) - TO (4„ + ^1* + ^a®" +... + Aa?" +...) = 0- or + {n4„-(m-n + l)J.„_i}a!'^'+... = (4), which is satisfied identically provided A -'^ A . m—X A. = _ w(m-l) -^1 j--2 — "' 2 _m-2 ^ _ m(m-l)(OT-2 ) ^ » --3-^2 17273 "' and, generally, . m — w + 1 . m(m — l)...(?n — n.+ 1) . .^j, ^»= - ^„_a- i.2.3...n ^o.---W- We thus obtain y=AAl+-x- m m(m-l) »n(m— 1)...(to — m + 1) ^ /r"+...j ...(6), as a solution of (2); and it is easily verified that the series is convergent so long as | a; | < 1. Now if we retrace the steps by which the differential equation (2) was formed, we see that its complete solution is y^Oa+xy (7), 36—2 564 INFINITESIMAL CALCULUS. [CH. XIII where G is arbitrary. Hence (6) must be equivalent to (7), and putting a; = in both, we see that G=Aa. Hence (l+a,r=l+^. + '?^^>«.+ ... ^ m(m-l)..^(m-n + l) ^„^ ^^^^ for all values of x such that | a; | < 1. This is the well-known ' Binomial Expansion*.' As a further example, we take the function sin~' X ^=7(1^ ^'>- Multiplying up by ^^(1 —a?), and then differentiating, we find v(i-^)S X 1 dx J{l-x')^~^{l-x')' (l-.=)l-.,=l (10). Assuming y — A„ + AiX + Affie'+ ... + A^x" + (11), we find (1 -ar*) (A, + 2A^ + 3A,x^+ ... + w^„x"-'+ ...) ~ x{Aa + A^x + A^ + ... + A^x" + ...) = 1 ...(12), or (^,-l) + (2^-^)a: + (3^-2^)a^+... + M„-(ra-l)^.5}a;«-i+...=0...(13), which is satisfied identically, provided A = l, A-'^A 4 2 2 .3,1.3, 2.4 A 5. 1-3.5, « ~ ft -^4 = (^ — i — ;; -^n } (14)- "*" 5"*' 3.5' "''~6'''~270"'" * Newton (1676). The oases of «= ± 1 would require special iuveBtigation. 200] INFINITK SERIES. 566 * We thus obtain the solution 2 . 2.4 . 2.4.6 . , /, 1 „ 1.3 , 1.3.5 , \ ,,., Now if we retrace the steps by which the linear equation (10) was formed, we see that its general solution is ^(1 - a;") . 2^ = sin-^jc + .4 (16)*, sin-'a; A ^„, 2^=,/(T^)+7(T^^) ^ ^' and, by the nature of the case, (15) must be included in this form. If we put a; = in (15) and (17), we see that A = A^ The identity of the two expressions for y then requires that sin"'a! 2 , 2.4 , ,,„, 7cr:^)='^ + 3'^^3T5^+ (^^)' These series are both of them convergent for | a; | < 1. The result (19) is a mere reproduction of the binomial ex- pansion of (1 - a?)~i. If we put X = sin 6, the former series may be written e = sinecose(^l+|sin''e + |^sin*e+...^ ... (20). Again, if we put tan =z, we obtain the form This series has been made the basis of several ingenious methods of calculating ir. It may be shewn, for example, that ^7r = 5tan '^+2 tan '/y, whence 28 " = T0 {^+1(4) +0(4)'+-} 30336 f 2 / 144 \ 2^4 / Jf^V 1 ■^ 100000 13 viooooo/ "^3.5 \iooooo/ "^ ■■■/ ■■■^ ^" * See also Ait. 177, Ex. 2. 566 INFINITESIMAL CALCULUS. [CH. XIII These series are rapidly convergent, and are otherwise very con- venient for computation, owing to the powers of 10 in the denominators*. Another remarkable series follows by integration from (18), viz. HBin-.)»=f'-.|^+|^'|\ (23). EXAMPLES. Lm. 1. Prove by repeated diflFerentiation of the identity t; =l+a;-l-a;'' + a!°+..., 1 —X where | a; | < 1, that, if ?» be a positive integer (l-a,)-»=l + ma,+!^^^= + !?^(^?^±il^±^a=»+ ... 2. If I a; I < 1, prove that tauh~' x = x + ^k' + ia;" + . . . . 3. If I a; I < 1, prove that iT2" 273 +3:4~--=<^+"')^°S (!+«')-«=• Does the result hold f or a; = + 1 ? 4. If I a; I < 1, prove that Tr2~ O "^ gTg- •■• =a'taa-a;- Jlog (1 + x'). Hence shew that l-i-J + i + T-- = -43882.... 5. Prove that /. ^sina; , a? a? dx = x — s — ST + 1 'o X 3.3! 5.5! •" ' /■■'' sinh X , a? x' Jo X 3 . o ! 5 . ! * For the history of these series, see Glaisher, Mess, of Math., t. ii., p. 119 (1873). INFINITE SERIES. 567 Prove that f»e* , ,_ 6 „ , b^'-a' 6' - a' 3.3! 7. Obtain the following results by calculation from the series (6) of Art. 197: log 2 = : -693 147 181 log 7 = = 1-945 910 149 log 3 = : 1-098 612 289 log 8 = = 2-079 441 542 log 4 = : 1-386 294 361 log 9 = = 2-197 224 577 log 5 = : 1-609 437 912 log 10 = = 2302 585 093 log 6 = : 1-791 759 469 A* = = -434 294 482. 8. Prove that log 2 = 7a-26 + 3c, log 3 = lla-36 + 5 c, log 5 = 16a -46 + 7c, and thence log 10 = 23a-66 + : lOc, where a -- 1 = 10-^ 1 2 . lO'' "^ 3 1 .10'+-- = -1053605157, h-- 4 4« ~ 100 ' 2 . 100" "^ 4' 3 . 100' "*" ■ .. = -0408219945, 0-- 1 "80 1 2.80""*" 3 1 .80' ■••" •0124225200. Apply this to find log 10. (Adams.) 9. Prove that log 2 = 7P+ 5Q+ ZR, log3 = llJ'+ 8^+ 5R, log5 = 16P+12e-f 75, and thence log 10 = 23P + 17§ + 105, where 5 = 2 (ij + 3-^+ ^+ ...) = -0645385211, «=Ki-^3:i9-' + 5:i9B-^-)='°*«^^i^^^^' ^=KT^"3n6P+5n6T= + -) = -°12*22^2«0- Apply this to find log 10. (Glaisher.) 568 INFINITESIMAL CALCULUS. [CH. XIII 10. Shew graphically that the series in which M„ (x) = (1 - k) a;" is not uniformly convergent in the neighbourhood of a; = 1. 11. In the series for which Un (a;) = 91+1 n+2 we have S (x) = x for values of x ranging from to 1 inclusively, whilst the series of derivatives of the several terms has the sum 1 for 1 > a!> 0, but the sum for a; = 1. Account for this. 12. Examine the character of the convergency of the series for which S^ {x) has the forms = 5- , a;=''-i , nx^e-'^, sm" x, , respectively. 13. If Sn {x) = - log cosh iix, prove that »S' (a;) = | a; | for all values of x. 14. If the series flSo + Oi + flSa + ... + a„ + . . . be essentially convergent, the series ttj + »! cos x + a^ cos 2x + ... +a,n cos nx+ ... will be uniformly convergent for all values of x. 15. If «„, «!, a^, ... a„, ... be a descending sequence of positive qualities whose lower limit is 0, prove that the series Oi, + «! cos X + a^ cos 2x+ ... +a„ cos nx+ ... will be uniformly convergent over any range of x which does not include any of the points x = 2sTr, where s = 0, + 1, + 2, + 3,... 16. Prove that / •1 dx 11 1^ 1 1.3.5 1 oV(l-a;*)~ "^2-5'^2.4'9^2TT76"T3'^-- INFINITE SERIES. 569 17. From the formula ■ ds 1-6^ - = a-, for the radius of curvature of an ellipse, shew that the perimeter is equal to o /•. »\/i 3 , 3^5 , 3^5^7 . \ and verify that this is equivalent to the result of Art. 196 (13). 18. The time of a complete oscillation of a simple pendulum of length I, oscillating through an angle a (< ir) on each side of the vertical, is 'I fi" d s/ gj sl{\.— sin^ |a sin"* <^) ' prove that this is equal to / 1 / 1^ P 3" I'' 3" 5^ \ 19. The perimeter of an ellipse of small eccentricity e exceeds that of a circle of the same area in the ratio 1 + ^jfi*, approximately. 20. The surface of an ellipsoid of revolution (prolate or oblate) of small eccentricity e exceeds that of a sphere of equal volume by the fraction y^^e* of itself. 21. Prove that - = ^3(1-3^-3+ 5^.- 7^+..-). 22. Assuming the series for sin x, prove Huyghens' rule for calculating approximately the length of a circular arc, viz. : From eight times the chord of half the arc subtract the chord of the whole arc, and divide the result by three. Prove that in an arc of 45° the proportional error is less than 1 in 20000. 570 INFINITESIMAL CALCULUS. [OH. XIII 23. Obtain a particular solution of the equation in the form y^-^y P "^ F72^~ P72^rp"'' '")' 24. Obtain a particular solution of the equation dr^ r dr . ^^ r. . ./, kV k'r^ in the form 25. Integrate (1 -a,^)g-a,g = 0, by series, and deduce the expansion of sin"' a; (Art. 197 (15)}. 26. Prove that y = 8inh~'a; satisfies the differential equation Hence shew that, for | a; | < 1, log{a;+^(l + «0} = «'-||' + yf--- 27. Obtain a solution of the equation in the form \ ^ • 4 ! / , n(, {n-V){n + 2) { n-Z){n-\){n + 2){n + i:) , \ INFINITE SERIES. 671 28. Obtain a solution of the equation in the form \ 1-7 1.2.7(y+l) a(a + l)(a + 2)^(^+l)(^ + 2) X 1.2.3.y(y+l)(y+2) ="+-)■ bion of the equation rdr \ r V 29. Obtain a solution of the equation in the form *~ *■ 1/ 2"('2ji+T)"*' 2.4:(2w+2)(2j» + 4)~ ■■■/■ 30. Obtain the solution of the equation ar r a/r in the form „ ■ A kV ¥r* \ V 2(2»i+3)'^ 2.4(2»i + 3)(2»i + 5) ■■"/ " 2 (1 - 2?i) "^ 2 . 4 (1 - 27i) (3 - tn) +5r--ri ^r-+ ^r \ V 2 (1 - 2ji) 2 . 4 (1 - 27i) (3 - 2ji) 7' 31. If y = sin(msin ^a;), prove that (1 — ar*) -r^ — x-~ + rr?y = 0. Hence shew that cos«.e= l-^;sin=fl + ^(^!f^sin*e- .... 2 ! 4 ! 32. Obtain a solution, by a series, of and give the symbolical expression of the complete solution. CHAPTER XIV. Taylor's theorem. 201. Form of the Expansion. Let /(«) be any function of x which admits of expansion in a convergent power-series for all values of x within certain limits + a. It has been proved, in Art. 198, that the derived function /' {x) will be given by a similar series, obtained by differentiating the original series term by term, for all values of X between + o. By a second application of the theorem cited, the value of /" («) will be obtained, for values of x be- tween the above limits, by differentiating the series for/' (x) term by term. And so on. Hence, writing fix)^A, + A^x + A^^ + ...+ A^a!" -l- (1), we have /'(«)= A,-l-24ja! + ...-|-w4„a;»-i -I-.. f"(x)= 2.\A^ + ...+n{n-l)Anaf^ + .. .(2). /(")(«)= w(re-l)...2.1A„+.. Putting a; = in these equations, we find A=/(0), A,=/'(0). 4, = l/"(0),... A„ = l/(») (0) ...(3), where the symbols /(O), /' (0), /" (0), . . . are used to express that X is put = after the differentiations have been per- formed. 201] Taylor's theorem. 573 The original expansion may now be written /(«')=/(0) + ^/'(0) + ^/"(0) + ...+|^/<»)(0)+... This investigation was given by Maclaurin*. It will be noticed that the proof depends entirely on the initial assumption that the function f{x) admits of being expanded in a convergent power-series. The question as to when, and under what limitations, such an expansion is possible will be discussed later (Arts. 203, 204). If we write (a + x) =f{iB) (5), we can deduce the form of the expansion of ^(a + x) in a power-series, when such expansion is possible. For if we write, for a moment, u = a + x, we have /(w) = (/> (u), /"(.) = ,4f(.)=|^fW.|=f'(.). and so on (Art. 39, 1°). Hence, putting « = 0, u= a, we find /(O) = <^ (a), /' (0) = f (a), f" (0) = f (a), . . ./<») (0) =<^<») (a) (6); so that (4) takes the form (a+x) = cl>ia) + w4>'{a) + ^cj>"(a) + ...+^^-^{a)+... ' (7). This is known as Taylor's Theorem-)-. We have deduced it from Maclauriil's Theorem, but the two theorems are only slightly different expressions of the same result. Thus assuming (7), we deduce Maclaurin's expansion if we put a = 0§. * Treatise on Fluxions (1742). The theorem had been previously noticed by Stirling. + Given (under a slightly different form) as a corollary from a theorem in Finite Differences, Methodus Incrementorum (1716). § The virtual identity of (4) with Taylor's Theorem was clearly recog- nized by Maclaurin. 574 INFINITESIMAL CALCULUS. [CH. XIV 202. Particular Casea. Before proceeding to a more fundamental treatment of the problem suggested in the preceding Art., the student will do well to make himself familiar with the mode of formation of the series. In the following examples the possi- bility of the expansion is assumed to begin with ; and the results obtained are therefore not to be considered as esta- blished, at all events by this method. 1°. If " (o) = m (m - 1) a"^\ . . . ^(»)(a) = m(m-l)...(m-n + l)a"^, (2). Taylor's formula then gives (a+ xy = a*" + ma!^-^x + ^ ' a™-W + . . . , m(m-V)...{m-n + l) ,„ + 1.2...m " " ^ ^^^' which is the well-known Binomial Expansion. That Taylor's Theorem cannot hold in all cases, without qualification, is shewn by the fact that the series on the right-hand is divergent if | a; | > | a |. For | a; | < | a | the series is convergent, but it is not legitimate to affirm on the basis of the investigation of Art. 201 that its sum is then equal to (a + a;)™*. A valid proof of the equality has been given in Art. 200. 2°. If f{«>) = ^ (4), we have /'"' («) = eF, and therefore /(0) = 1, /<")(0)=1. Maclaurin's expansion is therefore a? a? a;" ,_. This is in any case a mere verification, as the series on the right-hand was adopted in Art. 17 as the definition of e*. * There are in fact cases where Taylor's expansion is convergent, whilst the sum is not ec[ual to (a + :£). 202] Taylor's theorem. 575 3". Let /(a;) = cosa! (6). It was shewn in Art. 63 that this makes / W (x) = cos {x + ^tt), so that /(0) = 1, /'")(0) = cos|%7r (7). Hence /'"' (0) vanishes when n is odd, and is equal to + 1 when n is even, according as \n is even or odd. Substituting in Maclaurin's formula, we get cos.= l-fj+ J, -...+(-)'. 2^,+ (8). 4°. Let /(a;)=sina; (9). This makes /<"' («) = sin {x + ^ir), so that /(O) = 0, /<"> (0) = sin \ntr = sin {^ (w - 1) tt + Jtt} . . .(10). Hence /'"' (0) vanishes when n is even, and is equal to + 1 when n is odd, according as \(n — \) is even or odd. Mac- laurin's formula then gives sm. = ^-- + - -...+(-)» ^-2^-^^+ (11). The results (8) and (11) have been established rigorously in Art. 199, 2°. 5°. Let /(a;) = Iog(l+a;) (12). This makes 1 i—'T-^ Tw — IV •^'(^)=rTS' ^°'^' *'°^' ^>i'/'"'(^)= (i+V • Hence /(O) = 0, /' (0) = 1, and, for n > 1, /(«)(0) = (-)»-i(n-l)! (13). Substituting in Maclaurin's formula, we get log(l+«j) = ^-| + |-... + (-)5-J+ (14). Of Art. 197, 1°. When the general formula for the wth derivative of the given function is not known, the only plan is to calculate the 576 INFINITESIMAL CALCULUS. L^H. XIV derivatives in succession as far as may be considered necessary. The later stages of the work may sometimes be contracted by omitting terms which will contribute nothing to the final result, so far as it is proposed to carry it. Ex. To expand tan x as far as a?. Putting / {x) = tan x, we find in succession /' (x) = sec^ flj = 1 + tan' x, f" (.x) = 2 tan a; sec" a; = 2 tan a; + 2 tan' a;, /'" (x) = (2 + 6 tan" x) sec" a; = 2 + 8 tan" a; + 6 tan^ x, f" (a;) = (16 tan a; + 24 tan' x) sec' x = 16 tan 03 + 40 tan' a; + 24 tan° x, ,p {x) = (16 + 120 tan" a; + 120 tan* a;) sec" x = 16 + 136 tan= x + 240 tan^a; + 120 tan'a;, f^ {x) = 272 tan x sec" x + dtc, /"'(a:) = 272sec*a; + &c., where, in the last two lines, terms have been omitted which will contribute nothing to the value of /'" (0). Hence / (0) = 0, /' (0)=1, /"(0) = 0, /"'(0) = 2, /''(0) = 0, r (0)=16, /''(0) = 0, /-«(0) = 272, and the expansion is 2a? 16a!» 272a!' tana; = a! + -3j+-gy + ^rj- + ... = x + \a? + ^-^<^ + ^,a?+ (15). That odd powers, only, of x would appear in this expansion might have been anticipated from the fact that tan x changes sign with x. 203. Proof of Maclaurin's and Taylor's Theorems. Let f{x) be a function of x which, together with its first m — 1 derivatives, is continuous for values of x ranging from to h, inclusively ; and let us write f{x) = ^„{x) + Rn{x) (1), 202-203] TAYLOR'S THEOREM. 577 where (2); i.e. 4>„(a;) is the sum of the first n terms of Maclaurin's expansion, and i2„ {x) is at present merely a symbol for the difference, whatever it is, between f{x) and $»(«). The object aimed at, in any rigorous investigation of Maclaurin's Theorem, is to find (if possible) limits to the value of Rn (ao) ; in other words, to find limits to the error committed when f{os) is replaced by the sum of the first n terms of Maclaurin's formula. If we can, in any particular case, shew that, by taking n great enough, a point can be reached after which the values of R^ («) will all be less than any assigned magnitude, however small, then Maclaurin's series is neces- sarily convergent, and its sum to infinity is /(«). It is evident that the argument cannot be pushed to this con- clusion if /(«) or any of its derivatives be discontinuous for any value of x belonging to the range considered. The notion of representing a function f(x) approximately by a rational integral function of assigned degree, say A„ + AiX + A^^+...+A„_iX^-'^ (3), has already been utilized in Art. 112. The plan there adopted was to determine the n coefficients A,,, A^, A^, ... A^-i so that the function (3) should be equal to /(x) for n assigned values of x, which were distributed at equal intervals over a certain range. In the present case, the n values of x are taken to be ultimately coincident with 0; in other words, the coefficients are chosen so as to make the function (3) and its first n — 1 derivatives coincide respectively with f{x) and its first n—l derivatives for the par- ticular value x = 0. The result of this determination is, by Art. 201, the function $„ (as). In the graphical representation, the parabolic curve y = *„ (a;) is determined so as to have contact of the (w-l)th order (see Art. 206) with a given curve y=f{x) at the point a5= ; and the problem is, to find limits to the possible deviation of one curve from the other, as measured by the difierence of the ordinates, for values of x lying within a certain range. This is illustrated by Fig. 149, which shews the curve y = log(l +a;) (4\ L. 37 578 INFINITESIMAL CALCULUS. [CH. XIV and (by thinner lines) the curves y = x, y = x-^a?, y = x-^aP + ^a?, (5), obtained by taking 1, 2, 3, ... terms of the 'logarithmic series' (Art. 202 (14)). The dotted lines correspond to a! = ± 1, and so mark out the range of convergence of the latter series*. s, 3 Pig. 152. It appears, from the conditions which 0„(a;) haa been made to satisfy, that B„ («) and its first n — 1 derivatives * This very instructive example is due to Prof. Felix Klein. 203] taylok's theorem. 579 will be continuous from a; = to x — h, and will all vanish for x = 0. Now we can shew that any function which satisfies these conditions, and has a finite j?th derivative, must lie between A — , and B — , , where A and B are the lower and upper limits to the values which the wth derivative assumes in the interval from to h. For, let F{x) be such a function. By hypothesis, we have F(Q) = Q, F'(0) = 0, ii"'(0) = 0, ... J?'<«-')(0) = 0...(6), and A»(«) is a rational integral function of degree m — 1, its wth derivative is zero (Art. 63); and the wth derivative of i?»(a;) is therefore, by (1), equal to/'"' («), if this latter derivative exist. We infer, then, that Rnia>) = Gf^ (13), where G is some quantity intermediate to the greatest and least values which/'"' {x) assumes in the interval from to A. And if, as we will suppose, this latter derivative is continuous from x = Q to x = h, there will be some value of x, between and h, for which /'"' {x) is equal to G. Denoting this value by dh, we have J?«(«') = 5/""(^/0 (14). where all we know as to the value of 6 is that it lies between and 1. The formula (14) holds from x = to x = h, inclusively. Putting x = h, and substituting in (1), we obtain 5/'»'(eA)...(15). fih) =/(0) + hf (0) + ^/" (0) + ... + 7;f^,/"-" (h) In this form Maclaurin's Theorem is exact, subject to the hypothesis that f(x) and its derivatives up to the order n, inclusively, are continuous over the range from to h. The conditions, however, that /'"' (x) is to exist, and to be continuous over the above range, include the rest. If we write /{x) = > 0. This is an accurate form of Taylor's Theorem. It holds on the assumption that (^'"' (x) exists and is continuous from x = a to x=a + h, inclusively. The last terms in (15) and (17) are known as Lagrange's forms of the ' remainder' in the respective theorems. The formula (17) is a generalization of some results obtained in the course of this treatise. For example, putting n—1, we get {a + h) = '(a+eh) (18); and, putting n=2, <-.(.).. (20), since the integrated term vanishes at both limits. Performing this process »i — 1 times, we obtain Since the function under the / sign is continuous, we infer, by Art. 89, 3°, that ^^^^ =("^"l)!(^ - ^r'^"" W (22), where 1> 6> 0. 582 INFINITESIMAL CALCULUS. [CH. XIV It appears, then, that the last term in (15) may be replaced by (ir^T)-!^!-^)""^''"'^^^) (23), and the last term in (17) by --^(l-0r^.l,m(a + eh) (24). These forms of remainder are due to Cauchy. 204. Another Proof. The proof of Taylor's (or Maclaurin's) theorem which is most frequently given follows the lines of Art. 66, 2°. Considering any given curve 2/=/W (1). we compare with it the curve y = Ao+ A^x + A^x^ + . . . + ^,wa;»-' + ^„a;" . . .(2), in which the n + 1 coefficients are determined so as to make the two curves intersect at a; = and x = h, and, further, so as to make the values of dy d?y d^-'y dx' dx^' '" d«"-i respectively the same in the two curves at the point a; = 0. These conditions give ^=/(0), A,=f'iO), 4, = l/"(0), ... ^- = 0^/'"-MO) ...(3), as before, and f(h) = A^ + AJi + AJi" +...+ iL^i/i"-' + Anh"" . . .(4), this latter equation determining An. Denoting by F(a;) the difference of the ordinates of the two curves, it appears that F (0) = 0, F' (0) = 0, F" (0) = 0, . . . J?'(«-') (0) = . . .(5), and F(h) = (6). Since F(x) vanishes for a; = and a; = A, it follows, under the usual conditions, that F' (x) vanishes for some value of x 203-205] Taylor's theorem. 583 between and h, say for x = OJi, where 1 > ^i > 0. Again, since F' {x) vanishes for a; = and x = OJi, F" {x) will vanish for some value of x between and 6Ji, say for x = 6Ji, where ^1 > ^2 > 0. Proceeding in this way we find that J^'i"-') (x) vanishes for a; = and x = O^^h, where 1 > On-^ > 0, and hence that Fm(eh)=0 (7), where 1 > 0>O. Now, on reference to (1) and (2), we see that ^w («)=/(») (x)-nl A„ (8). It follows from (7) that -^« = ^/'"'W (9). Hence, substituting from (3) and (9) in (4) we obtain f{h) =/(0) + hf (0) + |/"(0) + ... +^-^/(»-i) (0) + ^/'"'W (10), as before. The conditions of validity are as stated in Art. 203, after equation (15)*. 205. Derivation of Certain Expansions. We proceed to consider the value of the remainder for various forms of f{x), or ^ {x) ; and in particular to examine under what circumstances it tends, with increasing n, to the limit 0. In this way we are enabled to demonstrate several very important expansions ; but it is right to warn the student that the method has a somewhat restricted application, since tlie general form of the wth derivative of a given function can be ascertained in only a few cases. Moreover, even when the method is successful, it is often far from being the most instructive way of arriving at the final result. 1°. If f{x) = ^ (1), nr ^ n\ we have =l^/w (t^a;) = - e"^ (2) * The foregoing proof is substantially that given by Homersham Cox, Camb. and Dub. Math. Journ., 1861. 584 INFINITESIMAL CALCULaS. [CH. XIV Now, whatever the value of x, we have lim^ = (3), «=»«! since the successive fractions XXX X V 2' 3''" n'"' diminish indefinitely in absolute value. Hence the expansion (5) of Art. 202 holds for all values of x. As already pointed out, this merely amounts to a verification. 2°. If /(a;) = cosa; , (4), we have — /("'(^a;) = — jcos (fe + i«7r) (5). The limiting value of the fraction x'^/n] is zero, and the cosine lies always between + 1. Hence the expansion (8) of Art. 202 holds for all values of x. The same reasoning applies in the case of sin x. S°. If f(x) = (l+x)'^ (6), we find E! /•« (f)^\ _ m (m-l)...(m-w + l) x^ . ny ^ > 1.2...W (i + ex)^'^'-^'- „,y V— /- 1.2...W {\ + exY This may be regarded as the product of (1 + Ox)™ into n factors of the type m — r + 1 X /, m + lN x or r ' l + 6x (-l+^)Tfffi-W If 1 > « > 0, the fraction xl(l + Ox) lies between and x, and since the first factor in (8) tends with increasing r to the limiting value — Ij it appears that by taking n great enough the value of the expression (7) can be made less than any assignable magnitude. Hence, for 1 > a; > 0, we may write (! + «;)'»= 1 + ma; + "*^~-^a,«+ (9) ad infinitum. 205-206] TAYLOR'S THEOREM. 585 We cannot make the same inference when x is negative, even if I a; I < 1. For if we put x = — Xi, the fraction Xij(l — tej) is less than 1 only if ^ < (I — iXi)/xi . And ii Xi>^, we have no warrant for assuming that lies below this value. Oauchy's form of remainder (Art. 203 (23)) is now of service. We have, in place of (7), m{m-l)...{m-n + l) {I -$)"'-'■ a^ , , 1.2...(n-l) (1 + te)"-™ ^ '' This is equal to mx (1 + ^a;)™"' multiplied into n—l factors of the type / . m\w-6x . (-^+7JrTfe ^''^' and, if |a;| < 1, the fraction (x— Ox)l(l + Ox) is easily seen to be less in absolute value than x, whether x b,e positive or negative. Hence the limiting value of the remainder (10) is 0, so long as 1 >x> — 1. we find ]f-'m=^^{^J (13). 4°. If /(a;) = log(l+a;) (12). The limiting value of the first factor is 0, and, if x be positive and :^ 1, x/{l + 6x) if 1. Hence the limiting value of (13), for re = 00 , is zero, and the expansion (14) of Art. 202 is valid from a; =0 to « = 1, inclusively. Of. Fig 149. The above form of remainder does not enable us to determine the case of x negative, even when | aj | < 1. In Cauchy's form, we have, in place of (13) /_i).-i_^_/'^^r' (14). ^ > i + ex\\ + dx) ^ '• If I a; I < 1, the limiting value of this is 0, whether x be positive or negative. 206. Applications of Taylor's Theorem. Order of Contact of Curves. If two curves intersect in two points, and if, by continuous modification of one curve, these two points be made to coalesce into a single point P, then the two curves are said ultimately to have contact 'of the first order' at P. An 586 INFINITESIMAL CALCULUS. [CH. XIV instance is the contact of a curve with its tangent line. And whenever two curves have contact of the first order, they have a common tangent line. Again, if two curves intersect in three points, and if by continuous modification of one curve these three points are made to coalesce into a single point P, then the two curves are said ultimately to have contact ' of the second order ' at P. An instance is the contact of a curve with its osculating circle (Art. 154). Let us suppose that the two curves y = ^(x), y = f{x) (1) intersect at the points for which x = a;„, x^, x^, respectively. The function F(x) = <^{x)-^(x) (2), which represents the difference of the ordinates of the two curves, will vanish for *• = «„, ajj, x^. Hence, on the usual assumptions as to the continuity of F {x) and F' (x), the derived function F' («) will, by the theorem of Art. 48, vanish for some value of x between x,, and x^, say for x=Xo', and again for some value of x between Xi and x^, say for x = x^. Hence, by another application of the theorem referred to, if F" (x) be continuous in the interval from x^ to x^, it will vanish for some value of x between x^ and «/, say for x = x,,". Hence if, by continuous modification of one of the curves (1), the three points x = Xq, x^, x^he made to coalesce into the one point x = x„, the values of F{x^, F' (x^), F" (x^ will all be zero; i.e. we shall have ^{x)=ir {x), <^' {X) = i|r' {x), cj>" (x) = Vr" (x) . . .(3), simultaneously, for x = x,,. In other words, if two curves have contact of the second order at any point, the values of ^' dx' dx^ will at that point be respectively identical for the two curves. 206] Taylor's theorem. 587 Ex. To determine the circle having contact of the second order with the curve y = 4'{'>^) (4) at a given point. The equation of a circle with centre (|, 17) and radius p is {x-tf + iy-vf-p' (5)- If we differentiate this twice with respect to x, we find ^-^+(2/-'?)J = (C), '*(I)*(»-')ll-» w In these results, y is regarded as a function of x determined by the equation (5). But if the circle have contact of the second order with the curve (4) at the point {x, y), the values of y, dyjdx, and d^yjda? will be the same for the circle as for the curve. We may therefore suppose that in (5), (6), (7) the values of X, y, dyjdx, d^yjda? refer to the curve (4). The equations then determine the circle uniquely, viz. we find that the coordinates of the centre are t \dx) ) dx \dxj dx' dx' {,.(*Yl' \dx) J and that the radius is p= '' ,g (9); d^ cf. Art. 152. The above considerations may be extended, and we may say that if two curves intersect in m + 1 consecutive points, or have contact ' of the wth order,' the values of dy d?y d'^-y ^' 5«' d^' '" dof" must be respectively identical for the two curves at the point in question. The investigations of Arts. 203, 204 give a measure of the degree of closeness of two curves in the neighbourhood 588 INFINITESIMAL CALCULUS. [CH. XIV of a contact of the nth order. By hypothesis we have at the point x — a (say) <^(a) = V^(a), f(a) = ./n'(a),... ./,'») (a) = ,|r(») (a)...(10), and therefore, with F(x) defined by (2), F{a) = ^, F'{a) = 0, ^"(a) = 0,... i?'<") (a) = 0...(ll). It follows that, under the usual conditions, F(a + h)=j^^^Fi-+^^{a+eh) (12), where 1 > ^ > 0. Hence, if h he infinitely small, the difference of the ordinates is in general a small quantity of the order n + 1. Moreover, it will or will not change sign with h, according as n is even or odd. For example, the deviation of a curve from a tangent line, in the neighbourhood of the point of contact, is in general a small quantity of the second order, and the curve does not in general cross the tangent at the point. Again, the deviation of a curve from the osculating circle is a small quantity of the third order, and the curve in general crosses the circle. See Pig. 123, p. 422. But if the contact with the circle be of the fourth order, as at the vertex of a conic, the curve does not cross the circle. The same thing is further illustrated in Fig. 149, p. 572, where the curves numbered 1, 3, 5 do not cross the curve y = log(l + a!) at the origin, whilst the curves numbered 2, 4, 6 do cross it. 207. Maxima and Minima. If ^ (x) be a function of x which with its first and second derivatives is finite and continuous for all values of the variable considered, we have {a + h)-4>{a) = h'{a)+^^4,"{a + 0h) ...(1), where 1> 6>0. By taking h sufficiently small, the second term on the right-hand can in general be made smaller in absolute value than the first, and (f>{a + h) — ^ (a) will then have the same sign as hcf)' (a), and will therefore change sign with h. 206-207] TAYLOR'S THEOREM. 589 Now if (f> (a) is a maximum or a minimum value of (x), the difiference ^{a + h) — (a) must have the same sign for suflBciently small values of h, whether h be positive or negative. Hence we cannot, under the present conditions, have a maximum or a minimum unless ^' (a) = 0. Let us now suppose that (f>' (a) = 0, so that (1) reduces to cji(a + h)-{a)=~" (a). Hence if this be positive we shall have (j)(a + h)>(i) (a), whether h be positive or negative ; i.e. ^ (a) is a minimum. Similarly, if ' (a), it is necessary to continue the expansion in (1) further. To take at once the general case, if we have (x) = cosh a; + cos a; (6). This makes tf,' (x) = sinh X — sin x, " (x) = cosh x — cos x, (0) is a minimum value of (x). 208. Infinitesimal Geometry of Plane Curves. Let the tangent and normal at any point of a plane curve be taken as axes of coordinates; it is required to express the coordinates of a neighbouring point P of the curve in terms of the arc OP, =s, say. If, for brevity, we use accents to denote differentiations with respect to s, we have, as in Art. 109, a!' = cosi/r, 2/' = sin'\/r and thence (c" = — sin\]r. i^', su'" = — cos yfr . yfr'^ - sin i^ . yfr",. . . ' y" = cos \}r . -^p-', y'" = — sin i|r . ■>|r'2 + cos ^ . yfr",. . . and so on. Now, by Maclaurin's Theorem, ■■(1). ...(2), « = «o + j-a!o' + j~2 *o" + ■ y = yo+iyo' + Y^yo"+ ] ■(3), where the suffix is used to mark the values which the respective quantities assume for s = 0. But, putting ■>fr = in (1) and (2), we have sCf, = 1, Xq =0, Xq = — ■„,... P yo'=0, 2/o" = where Ijp has been written for dyfr/ds. Hence '^ = ^-67^+' s' s' dp (4), (5). 2p 6p» where p and dp/ds refer to the origin. These formulae are useful in various questions of ' infini- tesimal geometry.' 207-208] Taylor's theorem. 591 £x. 1. Thus the second formula in (5) shews that the devia- tion of the curve from the osculating circle at is ultimately since dpjds = for the circle. And, generally, for all purposes where s' can be neglected, the curve may be replaced by its osculating circle. Ex. 2. Again, the normal at P meets the normal at in a point whose distance from is y + x cot i/f. If we neglect terms of the second order in s, this s . s' dp p'^p'Js i'*iii)- Hence the distance of the intersection from the centre of curva- ture at s is ultimately i4s (^)- When p is a maximum or minimum we have (in general) dpjds = 0, and the distance is of a higher order, the evolute having then a cusp at the point corresponding to 0. EXAMPLES. LIV. , 2V 2*3? X. cosh X cos X= I -r-r + -5-Y — ... ... , 2V 2W» sinh a; sin as = a?" - -^-r + -rprr - D ! lU ! ^ . 2a? 2V 2. cosh X sin x = x+ -77-, — =-- - . . . , o ! 5 ! 2a;' 2V sinh a; cos a; = a; --„-: — =rr+ .... 6 I ! 2a? 2V 2V 2V 3. e'coBx=l+x-j^-^-~+^+.. , 2a? 2V 2V 2V e-smx = x + a? + jj-^-^-^ + . 592 INFINITESIMAL CALCULUS. \_Cn. XIV i. seca:=l + 2-, + -^ + -g-j-+.... a? sH^ a^ 5. logseca;= 2-+y2 + 45+ — • 6. ta,nhx = x-^si? + -^af-^^x^+ — 7. cos" a; = 1 - ar + -jj — ^ + — « 1 w , n(3w — 2) , 8. cos''a;=l--=-;a^+~i-i — «?-■■■■ 2 4 ! 14. /sin ojN" 9. f^i^^ =i_|_^+!L|!^)^^. in a; , a^ 14a^ 10. ^ — = 1 + 5-: + -^-p + sin a; a ! o ! X , !»? 14a!* = 1- s-: + sinh X 3 ! 6 ! ■n a! 1 1 1 a'' 1 "'^ , 12. tan(j7r + a;) = l + 2a!+2a;'i + |a? + ^a:*+... 13. logtan(Jir + a:) = 2a; + ^a;' + |a!'+ .... ,,^ Isin^e 1.3siii*e 1.3.5sin»e 15. If D^djdx, prove that Tjngaioosa cos (a; sin a) = e*<'osacos(a!sin a + raa). Hence shew that a^ ™j8 ga;oosacos(a;sina)= 1 + a; cos a + jr- cos 2a + 5- cos 3a+ ... . 16. Draw graphs of the functions as* of of respectively, and compare them with the graph of sin x. TAYLOR'S THEOBEM. 593 17. Draw graphs of the functions /y>2 /yi2 /m4 respectively, and compare them with the graph of cos x. 18. If ^(a) = 0, i/f(ffl) = 0, prove that in general 4.{x) ^4>'{a) -"fix) "ij,' (a)' 19. Prove that, in the formula (17) of Art. 203, the limiting value of 6, when h is indefinitely diminished, is in general ll(n + l). 20. Prove that when h is sufficiently small the error in Simpson's formula (Art. 112 (8)) of approximate integration is equal to nearly. 21. Prove that the mean value of a function tji (x) over the range extending from x = a-h to x = a + h is (x,y) (2), we can form the second derivatives A _9^/'9m\ J./SwA ^/M ■)' dy\d(cj' dx\dy)' dy\dy)' (3). or, as they are usually written, da?' dydx' dxdy' dy^ It will be noticed that there is (primarily) a distinction of meaning between the second and third of these symbols, the operations indicated being performed in inverse orders in the two cases. It will be shewn, however, in Art. 210 that under certain conditions, which are generally satisfied in practice, the results are identical. The first derivatives of (x, y) are sometimes denoted by x{so,y), y{x,y) (4), 38—2 596 INFINITESIMAL CALCULUa |.CH. XIV and the second derivatives (3) by ^xx{i!o,y), yx(a!,y), ^xy{(B,y), ^yy{x,y)...{5). These are often abbreviated into «. <^» (6), and ^xxt ^yx, 4>ii>y, vy 0)' respectively. Ex.\. If u^Ax'^y'" (8), we have —=mAx^~^y'', -- = M4a;™«""* (9), 9a! " dy * ' -^ = OT (m - 1 ) Ax'^-^y, 5^ = w (w - 1 ) Aary^~\ oar oy 3^w , „_,-. , 3°M ,,„, ^-— = mnAx'" y-i = — — (10). dydx axay ' Ex.2. If is = atan-i^ (11), dz ay dz ax .^ . dx x' + y' dy x^ + y' ^z _ 2axy 9z _ a(f-Qg) _ 8'g ff'z _ - 2aa;y ^~(^+YP' ^9a!~ {a? + y''f ~ dxdy' df~ {a? + yy'^ '' 210. Proof of the Commutative Property. Let u = (a!, y) (1), and let us suppose that the functions du du dhi dhi . ' dx' By' dydx' dxdy are continuous over a finite range of the variables, including the values considered. We proceed to shew that, under these conditions, _3!« ^ a^ .gx dydx dxdy To this end, we consider the fraction , j.^ {x + h, y + k)-(x + h, y)- (x, y + Ic) + (x, y) ^^ ' ^ hk (4). 210] TAYLOR'S THEOREM. 597 in which x, y are regarded as fixed, whilst h, k will (finally) be made infinitely small. Let us write, for a moment, F(a;) = {a;, y + k)- ,j>{x, y) (5). By the mean-value theorem of Art. 56 (8), we have F(x + h)-F{(B) = hF'(x + eji) (6), or, in full, {4>{x-irh, y + k)-(x, y + k)-{x, y)] = h{<^^{x + 0Ji, y + k)-(j)^(x + 0A y)} ...(7), where 1 > ^j > 0, the value of y not being varied in this process. Hence ,^^ ;a ^ i>^ (^ + ^1^. y + k)-^^(x+ eji, y) ^ iG If we now write f(y)=M^+^A y) (9). we have, by a second application of the theorem referred to, f(y + k) -f{y) = kf{y + ejc) (10), or x(x+6Ji, y + k)-i; + dA y) = k + 0Ji, y+djc) (11). Hence xi^' k) = (^y^ix + QJi, y + Ojc) (12), where ^„ 6^ lie between and 1. By a similar process we could shew that xQi, k) = ^(x + e,'h, y + 0,'k) (13), where 0i, 0^ also lie between and 1. These results are exact, provided x + h, y + k lie within the range of the variables for which the conditions above postulated hold. . If we now diminish h and k indefinitely, it follows from the comparison of (12) and (13), and from the continuity of the derivatives, that 4>yx(«', y) = ^«y{«!, y) (14), as was to be proved*. * This proof appears to be due to Oasian Bonnet. An alternative proof is indicated in Art. 211. 598 INFINITESIMAL CALCULUS. [CH. XIV It appears from (4) that 1 ^(x,y + k)-4,^(x,y) - k ^^^''■ Hence lim^j^o lim*=o X (''.*) = "#>!«(«'. 2/) (16). Similarly, we find liinft=o limj=(,x(^. ^='i>Tu{«, V) (17). If, then, we could assume that the limiting value of the fraction (4), when h and k are indefinitely diminished, is unique, and independent of the order in which these quantities are made to vanish, the theorem (3) would follow at once. A simple example shews, however, that the assumption is not legitimate without further examination. If we have linij.^, limft„„/(7i, A) = - 1, lim4_„ limj„o/(A, ^) = + 1. Ex. The condition that Mdx + Ndy (18) should be an exact differential (Art. 174) ia f=f- <-)■ For if the expression (18) be equal to du, we have ^=£' ^=a? (^°)> and therefore each of the partial derivatives in (19) is equal to cfu/dydx or d'^ujdxdy. Conversely, we can shew that, if the condition (19) hold, (18) will be an exact differential. Let v denote the function jMdx, obtained by integrating as if y were constant. We have, then, S=' <^')' 210-211] taylob's theorem:. 599 , ,, „ 3iV ciM 9v and thereiore -r- = -r— = ■:r-:r i ox oy oxoy 5('-|) = «-- W This shews (Art. 56) that the function N - dv/di/ is constant so far as x is concerned. Denoting its value by /' (y), we have ^-^y^fiy) (23). Hence, if we write u = v+f(^/) (24)i we have, by (21) and (23), 3^ = ^' ry = ^ •••(2^)' and therefore Mdx + Ndy = du (26)*. It follows from the above theorem that in the case of a function of any number of independent variables x,y,z,.., the operations 1 1. 1 dx' dy' dz''" or, as we may denote them for shortness, JJx, -Dy, JJz, ••• are in general commutative, i.e. the result of any number of them is independent of the order in which they are performed. For example, = DJDJ)yU = etc. 211. Extension of Taylor's Theorem. Let

{a,h + k) + h^x(a,b + k) + ^h':,:,(a,b + k)+ (2). Again, by the same theorem, {a,b + k) = xx{a,,b + k) = ^ (a, b) + 2hk4>y^ (a, b) + k'yy (a, 6)} + (4). If we regard the forms of the several ' remainders ' (Art. 20-3) in the preliminary expansions, it appears that the re- mainder in (4) will be of the form ^-ARho + SSh^k + STht+Uli^} (5), where R, 8, T, U are functions of a, 6, h, k which remain finite when h, k are indefinitely diminished. The remainder is therefore of the third order in h, k. The conditions for the validity of the foregoing result are that {x, y) and its derivatives up to the third order should be continuous for all values of the variable considered. It may be remarked that in the piool of (4) it was not necessary to assume that ^yx{ci',V) = <^{a,V) (6). If we had begun by expanding (1) in powers of k (instead of h) we should have arrived at a result similar to (4), but with <^a^ {a, h) in place of <^j„ (a, h). From a comparison of the two forms we can obtain an independent proof of the theorem of Art. 210. 211] TAYLOR'S THEOREM. 601 With a slight change of notation we may write (4) in the form -*e^'iJ-^^^al.-^4yt)- (^)' where, on the right-hand, stands for {x, y). A more compact form is (j>(x + h,y + k) = ^ (x, y) + (h^^ + k^y) + i {h^4>xx + 2M(/>^^ + ¥4>yy) + (8). Again, if ii be any function of the independent variables X, y, and if, as in Art. 60, Sit denote the increment of u due to given increments 8*, 8y of these variables, the formula is equivalent to (9)'. An independent investigation, giving the general term of the expansion (7), is as follows. We write h = at, k = ^t, and F(t) = ^(x + h, y-\-k) = ^{x + at, y + ^t) ...(10). Regarded as a function of t, this can be expanded by Maclaurin's theorem, and the general term is ^,^'"'(0) (11). Now if we put for a moment x+ at =u, y + ^t = v, we have d^ _ddu _d(j) d(j) _d^dv _d!J) , , dx dudx du' dy dvdy dv where ^ is written for ^ (u, v). Hence = (4+^|)^(«'^) ^^•^> * The extension of the investigations of this Art. to cases where there are three or more independent variables will be obvious. 602 INFINITESIMAL CALCULUS. [CH. XIV The result is evidently a function of u and v ; hence, by a repetition of the argument, 'By) and, generally, i^'"'(0 = («| + /39|)"'^K^') (15), where the operator admits of expansion by the Binomial Theorem, in virtue of the commutative property of the operators d/dao and d/dy. Since t only occurs in the com- binations x+at, y + ^t, it is evidently immaterial in (15) whether we put t = before or after the differentiations indicated on the right-hand side. The general term of our expansion is therefore nl\ daf dai^'dy 1.2 dx^-'dy'' J (16), where is now written for {x, y). Ex. To prove tliat if a, + yi>y = m (17). ar>^ + y^4>yy = m{m-l)(t> (18). The general definition of a homogeneous function of degree m is that if X and y be altered in any ratio /i, the function is altered in the ratio /a™, or ^{^x,,xy) = ^^{x,y) (19). In this equality, let us put fi=l + f. Since (x + xt,y + yt) = (x,y) + t {x^^ + yxfy^) + ¥'' i«^xx + 2a!!/<^^ + 2/V„) + • • • by (8), and by the Binomial Theorem, the results (18) and (19) will follow, 211-212] TAYLOR'S THEOREM. 603 on. equating coefficients of t and ^. More generally, equating coefficients of t\ and making use of (16), we find „3"^ n-i 3""^ n{n-\) „„, 3»d. dx"" ^aa;''-% 1.2 y day^-^df ■ = m{m-l}(m-2) ... {m-n+l)+|"* ; » Unless 9u/9« and dujdy both vanish, the sign of (1) is reversed by reversing the signs of hx and hy. Hence for some varia- tions Sm will be positive, and for others negative. In other words, u cannot be a maximum or minimum unless we have s-"' |=° «• simultaneously. Let us now suppose the conditions (2) to be fulfilled. We have, then, a. = i{g(S.)» + 2^^8.Sy + g(Sy)j + (3).^ When Bx and By are sufficiently small, the sign of hu will be that of the terms written. Now it is known from Algebra that the sign of a homogeneous quadratic function A^^ + 2H^7]+Brf (4) is invariable, if (and only if) AB>H^ (5), * It ia assumed in the investigation of Art. 211 that these derivatives are continuous and therefore finite. That is, we exclude ah initio the two- dimensional analogues of the cases considered in Art. 51. 604 INFINITESIMAL CALCULUS. [CH. XIV and that the sign is then that of A (or B). We infer that when the conditions (2) are satisfied Su will have the same sign for all values of | Sa; | and \Sy \ not exceeding certain limits, provided a=M a»M / dhi, Y , 9a^ dy'^Kdwdy) ^ ^' and that the sign will then be that of d^u/da? and dhi/dy'. And u will be a maximum or minimum according as this sign is negative or positive. Tf d^d^ ( S'u y ,,_. dx^dy^'^Kdcody) ^ '' then for some values of the ratio Sy/Sx the increment of u will be positive, for others negative, and the value of u, thoxigh ' stationary ' (cf. Art. 50) is neither a maximum nor a minimum. If dhidhi^/ dhi Y daf" dy' \dxdy) ^ ^ the terms which appear on the right-hand side of (3) are equal to + the square of a linear function of Sx and By, and therefore vanish for a particular value of the ratio Sy/Sx. Since Su is then of the third order it appears that there is in general neither a maximum nor a minimum, but the question cannot be absolutely decided without continuing the expan- sion further. The same remark applies when the second derivatives dhi/dx^ d'u/dxdy, d^ujdy^ all vanish. The preceding investigation has an interesting geometrical interpretation. If, as in Art. 45, z be the vertical ordinate of a surface, and x, y rectangular coordinates in a horizontal plane, the first condition for a point of maximum or minimum altitude is that 1 = 0. ?^ = (9) simultaneously. Since these equations ensure that hz shall be of the second order in Sjc, Sy, it follows that at the point (P, say,) in question the tangent line to every vertical section through P will be horizontal ; in other words, we have a horizontal tcmgent plane. 212] TAYLOR'S THEOREM. 605 We have next to examine whether the surface cuts the tangent plane at P. Along the line of intersection (if any), we shall have 8is = 0, and therefore from (3), if we put hy = mSx, and finally make Sa; vanish, the directions of the tangent lines at F to the curve of intersection are determined by 8^ + 29-^- + g^m»=0 (10). This quadratic in m will have imaginary roots if dx> dy^^\dxdy) '^ '' the surface then, in the immediate neighbourhood of F, will lie wholly on one side of the tangent plane, and the contour-line at P reduces to a point. Hence P will be a point of maximum or minimum altitude according as ^zjda? and cFzjdy^ are negative or positive, i.e. (Art. 68) according as the vertical sections parallel to the planes zx and zy are convex or concave upwards. If we imagine the axes of x, y to be rotated in their own plane, we can infer that every vertical section through P is in this case convex upwards, or concave upwards, respectively. S 5" (&)"■■ :<-'■ the roots of (10) are real and distinct. The contour-line has a node at P, the two branches separating the parts of the surface which lie above the tangent plane .from those which lie below. ^* 3l^=(&J/) ^^^)' the roots of (10) are real and coincident. The contour-line has in general a cusp at P, and the question as to whether the altitude at P is a maximum or minimum cannot be determined without further investigation. Ex.\. Let z = a?-'Saa?-iay'+G (14). This makes -^ = 5x{x—'ia), ~ = — 8ay (15), — = = 6(a! — a), —-— = 0, ;r-5 = — ba (16). da? ^ " dxdy dy^ ^ ' The conditions (9) are satisfied by x = 0, y = 0, and also by x = a, y = 0. The former solution satisfies the inequality (11), 606 INFINITESIMAL CALCULUS. [CH. XIV and since ^z/dy^ is negative, a is a maximum. The latter solution comes under (12); « is then neither a maximum nor a minimum. The contour-lines for this case are shewn in Fig. 74, p. 335. Ex.2. Let z = {x' + i/'y-2a'{x>-y')+0 (17). We find ^=4:x(os' + y^-a'), ^^= iy (a^ + y^ + a^) (18), g = 4(3«? + 2^-«^), 35^=8«^.g = 4(.'H-32^ + a«) ...(19). The real solutions of (9) are, in this case, x = 0, y=0, and x=±a, y = 0. The former values fulfil the relation (12), so that z is neither a maximum nor a minimum. The solutions x = + a, y = satisfy (11), and, since they make ^zjda? positive, z is then a mvnimu7n.. The contour-lines of the surface (17) are shewn in Pig. 113, p. 389. The surface has two symmetrical hollows with the 'bar' between them. If we reverse the sign of the right- hand side of (17), we get two peaks, with the 'pass' between them. 213. Applications of Partial Differentiation. Numerous problems of partial differentiation present themselves in geometrical and physical questions. As a rule, they are best dealt with as they arise; but we give one or two simple cases which may serve to elucidate the chief points to be attended to. 1°. Let u = 4>(v) (1), where u is a function of the independent variables x, y ; and let it be required to form the successive partial derivatives of M with respect to these variables. By Art. 39 we have du ,,y^dv du , . dv ,„, aS = '^(^>9i' d-y='f'^'^dy <2>- Again, dhi d ,, , . dv , .,, ^ dhi ,„ , .dvdv , cfv d^y^d-y'^^'^-d'x^'^^'^d^y^^ ^'^ dxdx^ '^ ^'^ d^y W, 213] Taylor's theorem. 607 and so on. -^a;. 1. Let 8 = <^(a!-c«) + x(a; + c<) (6), where the variables x and < are independent. Putting x — ct=-u, x + ct=v, for shortness, we find ^£ = " M + X" W. H = c^<^" (u) + oY {u) . . .(8). ^^""^ 3? = «3^ W. 2°. Let u = 4>{x,y) (10), where «, y are given functions of the independent variable t ; and let it be required to calculate the derivatives of u with respect to t. We have, by Art. 62, 1° du _d(f) dx d^ dy dt~dxdidydt ^ ^" Differentiating again, we find dhi^d^d^ Md^ d /9^\ dx d^ (d^\ dy dt' dx dt^ dy dt^ ^ dt\bx) dt "^ dt [dyj ~dt"'^ '^' Now, by the theorem referred to, d /S^\ ^ d_ /d^\ dx d_ /a^\ dy dt \dxj dx \dx) dt dy \dx) dt ' dt \dyj dx \dyl dt dy \dy) at Substituting in (12), and recalling the commutative property established in Art. 210, we have d?u_d^d^x d^d?y^ d^ /dxV d^ dx dy dFif) (dy dt" ~ dx'dP dy dt" dx" \dt) "^ dxdy dtdi'^dyAdi (13). 608 INFINITESIMAL CALCULUS. [CH. XIV The process might be continued, but it is seldom necessary to proceed beyond this stage. This process is sometimes required when we transform the coordinates in a dynamical problem. Thus, to change from rectangular to polar coordinates in two dimensions, we have x = r cos 0, y = r sin 6, and the above method enables us to express d^x/dt^ and d'^y/dt' in terms of the differential coefficients of r and with respect to t. 3°. Let y be a function of «, defined ' implicitly ' by the equation {^,7,) (18), where f, 17 are given functions of x and y, and let it be required to calculate the second partial derivatives of u with respect to so and y. We have 9m _dud^ du drj du doe 9f 9a! drj dx ' dy Hence 9^ _ 9w 9^ /9^ 9| dx^ ~ 9? dx^ V9f dx du 9^ du dr) ' 9^ dy dv dy' 9^ 9| _9^ drA 9| .(19) ^^ 9« 9^917 9«/ 9a; 9m9^ /^u^d^ d^drAdji drj dx' {.d^drj dx dif dx) dx d^ [dx) + 2 d'u d^dr, d^drj dx dx ^ dry dPu/dvY '^d^AdxJ 9m9^ 9m9^ '^d^da^'^dridx^ In like manner we should find _9^_9^9|9| ^ /d^dv d^dr) dxdy ~" 9p 9a! dy d^dr} \dx dy dy dx. (20). + + 9m 9*^ du d'rj d^ dxdy drj dxdy d'u d^dv . dhi 9*^ drj drj drj' dx dy ■ (21), d^' \dyj d^drj dy dy drj' \dy, dij\' dy' iW du d'^ du d'v d^df drjdy' (22). Ex. 2. To change from rectangular to polar coordinates in the expression Putting we find du dr' (hi 8^ (23). (24). du dx du dy „ 3m 5- 5::+ 3- /=oose — + sm dxdr dy or dx , du du dudx dudy I . ^du . du\ de = d^d-0^d^i = '[-''^^rx^''°'%), ..(25), 39 610 whence INFINITESIMAL CALCULUS. [CH. XIV Hence -=cose--sm0-^, _ = sm«-+cos^-^...(26). rdS' du . . du\ cose;--sme-^j, ^=(cose|:-sine^^)(, — -5- = ( sin S — + cos e — t: I sin 5 — + cos S —7; ) ...(27). It is not necessary to perform all the operations indicated, as several of the terms will obviously cancel when we form the sum (23). The remaining terms give ^ ^ _ 3^ Idu Id'u .(28). 1. If verify the relations 2. If EXAMPLES. LV. x + y' du du ,^u d3? d'u ffu y prove that 3. If prove that z = a? tan"' ^-ifi tan~' X " ^z 3? — y' dxdy a? + y^' z=F{x)+f(y), -0. 3'g _ dxdy Conversely shew that, if cfz/dxdy = 0, z must have the above form. 4. Prove that the equation dr^ r'dr r^de^ is satisfied by ^=(.ilr" + — j cos n{d-a). TAYLOR'S THEOREM. 611 5. Prove that any differential equation of the type F {a? + if) {xdx + ydy)+f (pi {xdy - ydx) = becomes exact on division by a? + y". 6. Prove that sinh ydx — sin xdy cosh y — cos X is an exact differential of a function u • and find u. dx" ^ Sy "' prove that a function ifr exists such that d dij/ d dij/ dx dy' dy dx ' prove that a function ij/ exists such that d(ji 8i/f d dxj/ ^^~rd6' rde^ Br' and that i^ satisfies the same partial differential equation as <^. 9. If ?^+^l+l^A = o da? dy^ y dy ' prove that a function i/r exists such that a^_ 13^ 9^_ia^ dx y dy' dy y dx' andthat £f-?-t-^? = 0. Bar dy y dy 10. Prove* that in the surface ast = x^- y^, the ordinate (z) is stationary, but not a maximum or minimum, when x = 0, y = 0. Sketch the contour-lines of the surface. 39-2 612 INFINITESIMAL CALCULUS. [CH. XIV 11. Prove by means of the rule of Art. 212 that the parallelepiped of least surface for a given volume is a cube. 12. If A, B, G be the vertices of a triangle, and P a variable point, the sura PA' + PB^ + PG^ is a minimum when P coincides with the mass-centre of three equal particles at A, B, G. 13. With the same notation, the sum my.PA^ + m^.PB^ + mi.PC^ is a minimum when P coincides with the mass-centre of three particles nij, m^, m,g situate at A, B, G. 14. Find for what values of x, y the ordinate of the surface z = a? + '^ — Zaxy is stationary. [The values are a, a, and 0, 0. The latter do not make z a maximum or minimum.] 15. Prove that the ordinate of the surface e^a = a^ — a? is stationary, but not a maximum or minimum, when jb = 0, y = 0. Sketch the contour lines. 16. Find for what values of x, y the function SB* -I- y* - 2 (a; - 2/)" is stationary. [There is a stationary value when a; = 0, y = 0, and two minima when x = ± ^% y- + ^'i.'\ 17. Prove that the function {a? + y^)e^-^ has a minimum value when x = 0, y = Q, and a stationary value which is neither a maximum nor a minimum when a! = 0, y = + \. 18. Prove that the ordinate of the surface z=f{aa?+2hxy + hy'^) is in general stationary when a; = 0, y = 0, and examine whether it is a maximum or minimum. Sketch the contour-lines in the several cases. TAYLOR'S THEOREM. 613 19. Find the stationary points of the function (a? - ay + {a? - a") {y" - ¥) + {y'' - by, and examine their nature. Sketch the contour-lines of the function. 20. If ■<&■ prove that a; — + « — = 0, ^ dx " dy ' 21. If z=f{oi? + y% prove that 5 + 5 = 4 {?? + f)/" (^ + f) + 4/' (^' + y'). 22. If u=/{r), where r = ^{a? + y"), prove that a^ + ^.=/ '('•) + ;:/('•)■ 23. If Vi^ stand for the operator S'jdx' + ^7^2/^ prove that Vi'logr=0, where r = J{{x - af + (y - pf}. 24. If u=f{x'' + y'' + z'), prove that S + 5 ■" S = * ('^^ + 2^' + *')-^" (^ -^ 2/^ + *') + ^•^' (^' + 2^' + ^'>- 25. If i*^/^, where r = ^^(a;^ + y'' + z^), prove that 26. If V^ stand for the operator 3^/3ar' + o^jdy'^ + 3Y3«', prove that where r= J{{x- af + {y- /Sf + {z- yf}. 614 INFINITESIMAL CALCULUS. [OH. XIV 27. With the same meaning of v'j prove that if , du dv dw ^ and 5~ + 5~ + a~ = "» 8a! oy dz then V° {"'^ + yv + zw) = 0. 28. If u, V be two functions of x, y, z satisfying the equations v'jA = 0, ^'v = 0, and if ii be a function of u, then v must be of the form Au + B. 29. If a; = r cos 0, y='r sin 6, where r, are functions of t, prove that ^cos0 + ^sine=-^-^-r(^^j. r; sm e + -r-r cos ^ = - -j- r= -3- . df dl' r dt\ dtj 30. If w = -(/.(c<-r)+-x(c'+'')> u - «-* e-a^'M"*, 3m _ /3^ 2 3m\ 3«~''\.9»^ ^rdr)' u = x^4>(^^, du du prove that 31. If prove that 32. If prove that 33, verify that X — ; + 2a;v -—.;- + w"— 5 = ra(w — l) w. 3a;= " dxdy " ^ ^ ' 34. If y-nz =/ {x - mz), . dz dz ^ prove that m -- + »i _- = 1. ^ dx cy TAYLOJS'S THEOREM. 615 35. « i-^-^\-rfV-P^ -=<^-")/(H). prove that (-c- a)^+ (2/-/3)^ = «-y. :.^ = ic^(cosh2^-cos2,)(g.^). 36. If x = c cosh ^ cos tj, y = c sinh ^ sin ij, prove that 37. If ^, 7j be current coordinates, the equations of the tangent and normal at any point (x, y) on the curve are (^ - a:) <^„ + (r, - 2/) ^^2ci>^£+^^(£) =0. Hence shew that the point is a node, a cusp, or an isolated point, according as {^xyTz'I'xx'I'mi- 616 APPENDIX. NUMERICAL TABLES. A. Table of Squares of Numbers from 10 to 100. 1 2 3 4 5 6 7 8 9 1 100 121 144 169 196 225 256 289 324 361 2 400 441 484 529 576 625 676 729 784 841 3 900 961 1024 1089 1156 1225 1296 1369 1444 1521 4 1600 1681 1764 1849 1936 2025 2116 2209 2304 2401 5 2500 2601 2704 2809 2916 3025 3136 3249 3364 3481 6 3600 3V21 3844 3969 4096 4225 4356 4489 4624 4761 7 4900 5041 5184 5329 5476 5625 5776 5929 6084 6241 8 6400 6561 6724 6889 7056 7225 7396 7569 7744 7921 9 8100 8281 8464 6649 8836 9025 9216 9409 9604 9801 B. 1. Table of Square-Roots of Numbers from to 10, at Intervals of -1. ■0 -1 -2 -3 •4 -5 6 -7 •8 -9 •316 -447 -548 -632 -707 •775 -837 -894 -949 1 1-000 1-049 1-095 1-140 1-183 1-225 1-265 1-304 1-342 1-378 2 1-414 1-449 1-483 1-517 1-549 1-581 1-612 1-643 1-673 1-703 3 1-732 1-761 1-789 1-817 1-844 1-871 1-897 1-924 1^949 1-975 4 2-000 2-025 2-049 2-074 2-098 2-121 2-145 2-168 2-191 2-214 5 2-236 2-258 2-280 2-302 2-324 2-345 2-366 2-387 2-408 2-429 6 2-449 2-470 2-490 2-510 2-530 2-550 2-569 2-588 2-608 2-627 7 2-646 2-665 2-683 2-702 2-720 2-739 2-757 2-775 2-793 2-811 8 2-828 2-846 2-864 2-881 2-898 2-915 2-933 2-950 2-966 2-983 9 3-000 3-017 3-033 3-050 3066 3-082 3-098 3-114 3-130 3-146 10 3162 APPENDIX, 617 B. 2. Table of Square-Roots of Numbers from 10 to 100, at Intervals of 1. 1 2 3 4 5 6 7 8 9 1 3-162 3-317 3-464 3-G06 3-742 3-873 4-000 4-123 4-243 4-359 2 4-472 4-583 4-690 4-796 4-899 5-000 5 099 5-196 5-292 5-385 3 5-477 5-568 5-657 5-745 5-831 5-916 6-000 6083 6-164 6-245 4 6-325 6-403 6-481 6-557 6-633 6-708 6-782 6-856 6-928 7-000 5 7-071 7-141 7-211 7-280 7-348 7-416 7-483 7-550 7-616 7-681 6 7-746 7-810 7-874 7-937 8-000 8-062 8-124 8-185 8-246 8-307 7 8-367 8-426 8-485 8-544 8-602 8-660 8-718 8-775 8-832 8-888 8 8-944 9-000 9-055 9-110 9-165 9-220 9-274 9-327 9-381 9-434 9 9-487 9-539 9-592 9-644 9-695 9-747 9-798 9-849 9-899 9-950 C. Table of Reciprocals of Numbers from 1 to 10, at Intervals of -1. -0 -1 -2 -3 -4 -5 -6 •7 •8 -9 1 1-000 -909 -833 ■769 •714 -667 -625 ■588 ■556 ■526 2 •500 -476 •455 •435 -417 •400 -385 ■370 ■357 -345 3 -333 -323 •313 ■303 -294 -286 ■278 ■270 ■263 •256 4 •250 -244 -238 ■233 -227 -222 •217 ■213 ■208 -204 5 •200 •196 •192 -189 -185 -182 ■179 •175 •172 •169 6 -167 •164 •161 ■159 -156 ■154 ■152 ■149 •147 ■145 7 •143 •141 -139 •137 -135 ■133 ■132 ■130 •128 ■127 8 •125 •123 ■122 -120 •119 •118 •116 ■115 ■114 ■112 9 •111 -110 -109 •108 •106 •105 ■104 •103 ■102 ■101 D. Table of the Circular Functions at Intervals of One-Twentieth of the Quadrant. eiiT sin coseo 6 tan $ cot e seo cos 00 00 1-000 1^000 100 ■05 •078 12^745 ■079 12^706 1-003 •997 -95 •10 •156 6-392 ■158 6^314 1-012 ■988 -90 •15 •233 4-284 ■240 4^165 1028 •972 -85 •20 •309 3-236 ■325 3-078 1^051 •951 ■80 •25 •383 2613 ■414 2-414 1-082 •924 -75 •30 ■454 2^203 ■510 1-963 1-122 ■891 -70 •35 •522 1^914 •613 1-632 1-173 ■853 -65 •40 ■588 1^701 •727 1^376 1^23e -809 •60 •45 ■649 1^540 ■854 1171 1-315 •760 -55 ■50 •707 1^414 1^000 1-000 1-414 •707 •50 COB e seo $ cot 8 tane cosec sine eiiT 618 APPENDIX. E. Table of the Exponential and Hyperbolic Func- tions of Numbers from O to 2-5, at Intervals of -1. X ex e-x cosh X ginh X tanh X 1-000 1-000 1-000 •1 1-105 -905 1-005 •100 •100 •2 1-221 •819 1020 ■201 •197 •3 1-350 •741 1-045 -305 •291 ■4 1-492 -670 1-081 •411 -380 •5 1-649 -607 1-128 -521 -462 •6 1-822 -549 1-185 -637 -537 •7 2-014 •497 1-255 •759 -604 •8 2-226 •449 1-337 ■888 -664 •9 2-460 •407 1-433 1^027 -716 10 2-718 •368 1-543 1^175 •762 11 3-004 •333 1-669 1-336 •801 1-2 3-320 ■301 1-811 1-509 •834 1-3 3-669 •273 1-971 1-698 -862 1-4 4-055 •247 2-151 1-904 -885 1-5 4-482 •223 2-352 2-129 -905 1-6 4-953 •202 2-577 2-376 -922 1-7 5-474 •183 2-828 2-646 -935 1-8 6-050 •165 3-107 2-942 -947 1-9 6-686 •150 3-418 3-268 -956 20 7-389 •135 3-762 3-627 ■964 21 8-166 •122 4-144 4-022 •970 2-2 9-025 •111 4-568 4-457 ■976 2-3 9-974 •100 5-037 4-937 ■980 2-4 11-023 -091 5-557 5-466 ■984 2-5 12-182 -082 6-132 6-050 ■987 F. Table of Iiogarithms to Base e • -0 ■1 -2 -3 -4 -5 -6 ■7 8 •9 1 -095 •182 •262 -336 •405 ■470 ■531 •588 -642 2 ■693 -742 ■788 •833 -875 ■916 ■956 ■993 1^030 1-065 3 1-099 1-131 1^163 1^194 1-224 1-253 1^281 1^308 1^335 1-361 4 1-386 1-411 1^435 1^459 1-482 1-504 1-526 1^548 1-569 1-589 5 1-609 1-629 1-649 1-668 1-686 1-705 1-723 1-740 1^758 1-775 6 1-792 1-808 1-825 1-841 1-856 1-872 1-887 1-902 1-917 1-932 7 1-946 1-960 1-974 1-988 2-001 2^015 2028 2-041 2-054 2-067 8 2-079 2-092 2-104 2-116 2-128 2^140 2-152 2-163 2-175 2-186 9 2-197 2-208 2-219 2-230 2-241 2-251 2-262 2-272 2-282 2-293 log 10=2-303, log 102=4-605, log 103=6908. INDEX. [The numerals refer to the pagei."] Acceleration, 68 angular, 69 tangential and normal, 397 Accidental convergence, 10 Algebraic functions, continuity of, 26 Amsler'a planimetre, 250 Anchor-ring, surface of, 272 volume of, 259 Approximate integration, 275 Arbitrary constants, 166 Arc of a curve, formula for, 264, 266, 268 Archimedes, spiral of, 354, 367 Area, definition of, 241 sign of, 245 swept over by a moving line, 249 Areas of plane curves, formulffl for, 242, 247 mechanical measurement of, 250 of surfaces of revolution, 270 Astroid, 359, 438 Bernoulli, lemniscate of, 370, 389 Binomial theorem, 564, 574, 584 Bipolar coordinates, 886 Calculation of tt, 273, 554 Cardioid, 356, 358, 869, 370, 383 Cartesian ovals, 388 Caasini, ovals of, 389 Catenary, 342 arc of, 265 curvature of, 397, 400 parabolic, 465 Centre, instantaneous, 434, 439 of curvature, 395, 400 of mass, see 'Mass-Centre' of pressure, 296 of rotation, 433 Centrodes, 445 Change of variable in integration, 179 Chord of curvature, 395, 403 Circle, perimeter of, 5 area of, 241 involute of, 354, 428 of curvature, 395 osculating, 406 Circular arc, mass-centre of, 293 disk, radii of gyration of, 315, 320 Circular Functions, continuity of, 33 differentiation of, 71, 72, 78 graphs of, 34 Circular motions, superposition of, 359 Cissoid, 337 Clairaut's differential equation, 485 Complementary function, 474, 502, 509 Concavity and convexity, 157 Cone, right circular, mass-centre of, 304, 305 surface of, 270 volume of, 257 Continuity of functions defined, 15 Continuous functions, properties of, 17, 18, 23, 52, 54 Continuous variation, 1 620 INDEX. Conjugate point, 333 Contour lines, 98, 606 Conveigence of infinite series, 4 essential and accidental, 10 uniform, S43 Convergence of a definite integral, 211 of power-series, 643 Corrections, calculation of small, 133, 138 Cotes' method of approximate in- tegration, 276 Crossed parallelogram, 447 Cubic curves, 333 Curvature, 394, 402, 406 centre of, 395 chord of, 395, 403 circle of, 396 radius of, 395, 397, 400, 402, 406 Cusp, 336 Cusp-locus, 420 Cusps, circle of, 444 Cycloid, 347 arc of, 348 area of, 350 curvature of, 398, 442, 444 evolute of, 423 Definite integral, see ' Integral ' Degree of a differential equation, 463 Density, mean, 288 Derived Function, definition of, 64 geometrical meaning of, 66 properties of, 100, 103, 104, 106 DifEereutial coefficients, 64, 96, 144 Differential Equations, 456 exact, 466 homogeneous, 469 integration of, by series, 659 linear, 473, 476 of first order and first degree, 462 of first order and higher degree, 484 of second order, 490 simultaneous, 529 Differentials, 132, 138 Differentiation, 69, 71 of a sum, product, quotient, etc., 73, 74, 76 of a function of a function, 80, 139 of a definite integral, 219 of implicit functions, 98, 139, 607 Differentiation of inverse functions, 85 of power-series, 557 partial, see ' Partial differentia- tion' successive, 144 Discontinuity, 21 Displacement of a plane figure, 433 Distributed stresses, 322 Elasticity of volume, 69 Elimination of arbitrary constants, 456 Ellipse, area of, 243, 248 perimeter of, 267, 548 curvature of, 398, 402, 404 evolute of, 422 Ellipsoid, volume of, 260 radii of gyration of, 317, 821. 328 of revolution, surface of, 273 Elliptic disk, radii of gyration of, 316, 327 Elliptic segment, mass-centre of, 326 ElUptic-harmonic motion, 346 Elliptic integrals, 268 Envelopes, 413, 415 contact-property of, 418 Epioyolies, 359 as roulettes, 448 Epicycloid, 350 arc of, 353 curvature of, 398, 442, 444 evolute of, 425 double generation of, 450 Epitrochoid, 354 Equations, theory of, 104, 161 Equiangular spiral, 366 curvature of, 399 Even and odd functions, 41 Evolute, 421 arc of, 425 Exact differential, condition for, 698 Exact differential equations, 466 Expansion, coefficient of, 69 Expansions by means of differential equations, 562 by Maclaurin's theorem, 573, 583 Exponential function, 35, 660 graph of, 38 Flexure of beams, 299, 323 Function, definition of, 12 graphical representation of, 13 INDEX. 621 Functions, algebraic and transcen- dental, 26 implicit, 97 inverse, 44, 85 Qeometrical representation of mag- nitudes, 2 Goniometric functions, 45 differentiations of, 87 graphs of, 86, 88 Gradient of a curve, 67 Graph of a function, 18 Gregory's series, 652 Gnldin, see 'Pappus' Gyration, Badius of, 313 Hart's linkage, 381 Hemisphere, mass-centre of solid, 804 mass-centre of surface of, 301 Homogeneous differential equations, 469 Homogeneous functions, Euler's theorem, 608 Homogeneous strain, 324, 327 Hyperbola, area of sector of, 243 Hyperbolic functions, 41 differentiation of, 84 graphs of, 42, 43 inverse, 48 differentiation of, 93 Hypocycloid, see ' Epicycloid ' Hypotrochoid, see ' Bpitrochoid ' Implicit functions, 97 differentiation of, 98, 140, 608 Indicator diagram, 211, 247 Inertia, moment of, 318 Infinite series, 4, 9 addition and multiplication of, 11, 35 differentiation and integration of, 549, 656 Infinitesimals, 61 Inflexion, points of, 157 Instantaneous centre, 434, 439 Integrals, definite, 208, 214 convergence of, 211 properties of, 217, 232 rule for finding, 221 approximate calculation of, 275 Integrals, multiple, 282 Integration, 166 by parts, 187 Integration by substitution, 179, 184 of irrational functions, 176, 203 of power-series, 549 of rational fractions, 171, 193, 195, 197 of trigonometrical functions, 182, 229 Interpolation by proportional parts, 156 Intrinsic equation of a curve, 397 Inverse functions, 44 differentiation of, 85 Inversion, 378 mechanical, 380 Involutes, 427 Involute teeth, 452 Leibnitz' theorem, 146 Lemniscate of Bernoulli, 370, 383 Lima(fon, 357, 368, 383 Limit, upper and lower, of an as- semblage, 3, 61 Limiting values, 8, 54, 55, 68 Limits, upper and lower, of a defi- nite integral, 209 Linear differential equations of first order, 473, 476 of second order, 501 with constant coefficients, 509, 513 Line-density, 289 Line-roulette, 443 Lissajous' curves, 344 Logarithmic differentiation, 91 Logarithmic Function, 46 graph of, 47 differentiation of, 89 Logarithmic series, 652, 676, 585 Maclaurin's Theorem, 573, 576, 682 Magnetic curves, 389 Mass-centre, 291 of an are, 293 of an area, 298, 295 of a surface of revolution, 301 of a soHd, 303, 306 Maxima and minima, 106, 160, 688 by algebraic methods, 111, 136 of functions of several variables, 135, 603 Mean density, 288 pressure, 296 Mean values, 279 Mean-value theorems, 129, 217 622 INDEX. Modulus (in logarithms) 48 Moment of inertia, 313 Multiple integrals, 282 Multiple roots of equations, 161 Newton's treatment of curvature, 402 Node-locus, 420 Node, 338 Older of a differential equation, 456 Orthogonal projection, 324 Orthogonal trajectories, 478 Pappus, theorems of, 307 Parabola, arc of, 265 curvature of, 398, 402, 403 evolute of, 421 Parabolic segment, area of, 243 mass-centre of, 294, 295 Paraboloid, volume of, 259, 260 mass-centre of, 304 Parallel curves, 428 Parallel projection, 324 Partial derivatives, 95 of higher orders, 695 Partial differentiation, commuta- tive property of, 596, 600 Partial Fractions, 172, 194, 195, 197 Particular integral of a linear dif- ferential equation, 474, 602 PeauceUier's linkage, 380 Pedal curves, 382 Pedals, negative, 384 Pericycloid, 350 Perimeter of a circle, 5 Planimeter, 250 Point-roulette, 433, 440 Polars, reciprocal, 384 Positions, curve of, 68 Power-series, continuity of, 547 diiferentiation of, 557 integration of, 549 Pressure, centre of, 296 Primitive of a differential equation, 457, 459 Prismoid, volume of, 261 Proportional parts, 156 Quadrature, approximate, 275 Eadius of curvature, lee ' Curva- ture' Badius of gyration, 813 Batioual fractions, graphs o^ 28 integration of, 171, 193, 195, 197 national integral functions, 26 Beciprocal polars, 384 Beciprocal spiral, 868 Bectification of curves, 264, 266, 268 of evolutes, 425 Beduotion, formulee of, 189, 190, 229 Bemainder in Taylor's and Mac- laurin's Theorems, 581, 582 Bing, radius of gyration of a, 317 surface of a, 272, 308 volume of a, 259, 308 Bolle's theorem, 105 Boots of algebraic equations, sepa- ration of, 105 Boulettes, 433 curvature of, 440, 443 Second derivative, 144 geometrical meaning of, 151, 154 Sector of a circle, see ' Circle ' Semi-cubical parabola, 336 Separation of roots of an algebraic equation, 105 of variables in a differential equation, 464 Series, see ' Convergence ' and ' In- finite Series ' Sign of an area, 245 Similar curves, 376, 470 Simpson's rules, 260, 277 Simultaneous differential equations, 529 Sin X, expansion of, 562, 675, 605 Sin~> x, expansion of, 556 Singular solutions, 487 Sphere, radius of gyration of, 817, 320 surface of, 272 volume of, 259 Spherical sector, mass-centre of a, 305 Spherical segment, volume of, 258 Spherical shell, radius of gyration, 316 - Spherical surface, area of a, 272 mass-centre of a 301, 302 Spherical wedge, mass-centre of a, 306 INDEX. 623 Spiral, equiangular, 366 of Archimedes, 354, 367 reciprocal, 368 Stationary point, 396 Stationary tangent, 167, 396 Stationary values of functions, 110, 604 Subnormal, 122 Subtangent, 122 Successive differentiation, 144, 589 of a product, 146 Surface-density, 289 Surface of revolution, area of, 270 mass-centre of, 301 Tangent to a curve, 66 Tangential polar equation, 371 Taylor's Theorem, 154, 572, 573, 682 expansions by, 574, 583 extension of, 599 Teeth of wheels, 450 Tetrahedron, volume of a, 257 JTheory of equations, 104, 161 Tirtie-integral, 210 Total variation of a function, 136, 601 Tractrix, 344 Trajectories, orthogonal, 478 Transcendental functions, 26, 33 Triangle, mass-centre of a, 293, 294 Trigonometrical Functions, , see 'Circular Functions' Trochoid, 350 Uniform convergence of series, 543 Variable, dependent and indepen- dent, 12 Variable, change of, in integration, 179, 184 Variation, continuous, 1, 2 Velocities, component, 267, 270 Velocity, 68 angular, 69 Volumes of solids, 255, 328 of revolution, 258 'Witch' of Aguesi, 337 Work, 210 CAMBEIDOE : PRINTED BT J. & 0. F, CLAY, AT THE UNIVEBBITT PBESS. mm