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A Overdue charge is made on all books, University of Illinois Library AN ELEMENTARY COURSE OF INFINITESIMAL CALCULUS. London: C. J. CLAY anv SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. Glasgow: 50, WELLINGTON STRIET. Leipsig: F. A. BROCKHAUS. New Work: THE MACMILLAN COMPANY. Gombay anv Calcutta: MACMILLAN AND CO., Lrp. [All Rights reserved.] AN ELEMENTARY COURSE OF INFINITESIMAL CALCULUS BY HORACE LAMB, M.A. LL.D. F.RS, PROFESSOR OF MATHEMATICS IN THE OWENS COLLEGE, VICTORIA UNIVERSITY, MANCHESTER; FORMERLY FELLOW OF TRINITY COLLEGE, CAMBRIDGE. SECOND EDITION, REVISED. CAMBRIDGE: AT THE UNIVERSITY PRESS. 1902 ) 1tOCHE I’ &- | J * _ ~ A , 7) Y oe Chia i two AATHEMATICS \Y 02 UEPAR OER PREFACE TO THE SECOND EDITION. HIS book attempts to teach those portions of the Calculus which are of primary importance in the application to such subjects as Physics and Engineering. Purely analytical developments, and processes, however in- genious, which are seldom useful in practice, are for the most part omitted. Although the author has had in view the needs of a special class of students, he has not scrupled to discuss questions of theoretical interest when these appeared to him to be at once simple and relevant. He trusts therefore that the book may be acceptable also to mathematical students who desire to take a general survey -of the subject before involving themselves deeply in any of its more elaborate developments. The arrangement is substantially that which has been adopted for a long series of years in lectures at the Owens College. As far as possible, priority is given to the simpler and more generally useful parts of the subject; in particular, the Integral Calculus is introduced at an early period. An attempt is made, as each stage in the subject is reached, to bring in applications of the theory to questions of interest. Thus, after the rules for differentiation come the applications of the derived function to problems of Maxima and Minima, and to Geometry; the rules for integration are followed by hath “ss 3 72370 v1 PREFACE. the application to areas; volumes, &.; and so on. Moreover, in the exposition of the theory, dynamical and physical, as well as geometrical, illustrations have been freely employed. The demands made on the previous knowledge of the student are not extensive. He is of course assumed to be familiar with the elements of Geometry, Algebra, and Trigonometry, and with the fundamental ideas of Analyti- cal Geometry, but not necessarily with such things as the theory of Infinite Series, or the detailed theory of the Conic Sections: Imaginary quantities are nowhere em- ployed in the book. The introductory pe hae will supply what is required outside the above range*; but its main object is to establish carefully the findatedval notions of continuity, and of limiting values, on the basis of the geome- trical conception of magnitude. In this edition the book has been carefully revised, and a number of errors have been corrected, principally in the Examples. A few paragraphs in the latter portion of the book, relating to Infinite Series, have been amplified. The author is much indebted to various friends, and in particular to Professor W. F. Osgood, who have Sa favoured him with criticisms and corrections. 1 Except, to some extent, in the latter part of the book, where the curvature and related properties of special curves are investigated. * In particular, the hyperbolic functions, direct and inverse, whose sys- tematic employment in the Calculus Prof. Greenhill has done so much to promote, are here introduced and studied. September, 1902. > Ps] 5 lime SAS ae ck | aie ag CONTENTS. CHAPTER I CONTINUITY, Continuous Variation . : : ° Upper or Lower Limit of a neti oo : Application to Infinite Series. Series with oatttye Meva Perimeter of a Circle . Limiting Value in a Sequence . : i ‘ ‘ Application to Infinite Series General Definition of a Function : Geometrical Representation of Functions . Definition of a Continuous Function. Property of a Continuous Function . PF Graph of a Continuous Function . . Discontinuity : : ° Theorems relating to Santos Peeuine Algebraic Functions. Rational Integral Functions . Rational Fractions Examples I. A Transcendental Functions, The Circular Functions The Exponential Function . The number e ‘ The Hyperbolic manciione ; Inverse Functions in general The Inverse Circular Functions ‘The Logarithmic Function . The Inverse Hyperbolic Functions Examples II, WI. . 4°] > Q & _ noon Frt we 49, 50 Vill CONTENTS. ART. PAGE 24. Upper or Lower Limit of an Assemblage . : . : - 51 25. A Continuous Function has a Greatest and a Least Value . 52 26. Limiting Value of a Function . : : “ : . 654 27. General Theorems relating to Limiting Values " ese S BD 28. Illustrations Z : : : : . : ; 2 bG 29. Some Special Timing Values . : ; : ; Ba ees) 30. Infinitesimals ; - ; . : Z . : ; Sage «Hs Examples IV. ; : ° ° ° : ; sn Oe CHAPTER IL. DERIVED FUNCTIONS. 31. Definition, and Notation . ? r ¢ ; Rae 7 ek 32. Geometrical meaning of the Derived Function . . : OG 33. Physical Illustrations : ; F 08 34. Differentiations ab initio . E ; : : ‘ : too Examples V s ; 3-40 35. Differentiation of Standard Functions P Petey | 36. Rules for differentiating combinations of simple eindet Dif- ferentiation of a Sum . : i , : 5 : cele 37. Differentiation of a Product ; ; ; : : : . TA 38. Differentiation of a Quotient . : F ; : : Lek Examples VI . , ; : 3 : F : ees 39. Differentiation of a Function of a Function . ; : - 80 Examples VII : ; . ; : 2 Oe 40. Differentiation of the Hyperbolic Functions . : ; . 84 Examples VIII : i ye ; : : : - 85 41. Differentiation of Inverse Functions . 42. Differentiation of a Logarithm . : : : : ee) 43. Logarithmic Differentiation : ; : j ; : eer Examples IX . : : pearare a ; : z cee 44. Differentiation of the Inverse Hyperbolic Functions . 2 O18 E; Table of Logarithms to Base e : : ; ; : - 618 INDEX ° e ° ° e r ® ° e e ° e z e 619 CHAPTER 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 may be 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, &e. Algebraically, any such magnitude is represented by a letter, such as @ or 2, 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,’ 7.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, b, c,... of the alphabet are generally used to denote constant, and the later letters ...u, v, w, 2, 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. os fi 2 INFINITESIMAL CALCULUS. [OHI 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 ae, | sin x, | log x|. It is important to notice that, if a and b have the same sign, ja+b|=|a|+|5| Soon (1), whilst, if they have opposite signs, la+b|<|@|+ | 0] ccccpee meester (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 O, 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 O; in cases where there is a variety of sign, OM must be drawn to the right or left of O 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 M in the line XX. x’ ; M Xx Fig. 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. 1-2] CONTINUITY. 3 2. Upper or Lower Limit of a Sequence. Suppose we have an endless sequence of magnitudes of the same kind | digg With, fot ap hues Aine Gee pe), Mare Gry each greater than the preceding, so that the differences Hg — Hy, Xz— Xa, Uy — Uz, oe are all positive. Suppose, further, that the magnitudes (1) are all less than some finite quantity a. The sequence will in this case have an ‘upper limit,’ z.e. there will exist a certain quantity m, 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 Retain which is less than yp. In the geometrical representation the magnitudes (1) are represented by a sequence of points : iE GE SCORERS eh RAMA ES 2 a ates CA each to the right of the preceding, but all lying to the left of some fixed point d. Hence every point on the line XX, Oo My, Ms M3 M,A Xx Fig. 2. without exception, belongs to one or other of two mutually exclusive categories. Hither it has points of the sequence (2) to the nght of it, or it has not. Moreover, every point in the former category hes to the left of every point of the latter. Hence there must be some point J, 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 w= OM, wp 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 ch less than the preceding, so that ais Haren H, — Xo, Ly — V3, Dem ai sates 4 INFINITESIMAL CALCULUS. (CH. I are all positive, whilst the magnitudes all exceed some finite quantity b, 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 ». 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 Uy + Ug Ug oe FU ie ee es (1) whose terms are all positive, and let Ses hs: 8 = 1G Sy = Uh Uy +c ee (2). If the sequence Sin Semen es Say ios Rie (3) 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 loeg+tt+it. If in Fig. 2 we make 04/,=1, WEL: and bisect 17,4 in M,, M,A in i, 3, and so on, the points M@,, M,, M,, ... will represent the magnitudes 8,, 8, 83,.... And since these points all lie to the left of A, whilst J/,,A =1/2"-1 and can therefore be made as small as we please by taking nm large enough, it appears that the sequence has the upper limit OA, =2. ix. 2, The series oy a 1.2 230 See If we write this in the form Ge nn ET (con a we see that S,=1— 2-4] CONTINUITY. 5 sta nm+1’ which has evidently the upper limit 1. Ex, 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 1:4142138... 4 |] 4S 2 ] 3 or 1+79+I@+ jot Ioit Iort Tot Since s, is always less than 1:5, there is an upper limit. Again, if (1) be a convergent series of positive terms, and if Rireete Maret ates fede ig aia a BUR tac bse (4), be a series of positive terms which are respectively less than the corresponding terms in (1), @.@. wy’ T,, whilst IT,'< Il,’.. 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 W,, WL, Us,..., and a descending sequence Uy ths A eee Also, every member of the former sequence is less than every member of the second. Hence the former sequence will have an upper limit II, and the latter a lower limit II’, and we can assert that II > IL’. 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. 7 is ultimately less than any assignable magnitude, the limits II and II’ are identical. For, let PQ be a side of the Pp Fig. 4. inscribed polygon, PT’ and QT the tangents at P and Q. If PQ meets O7' in JN, we have (BENS SS SRE) Sy OY oa Now if = be a symbol of summation extending round the polygons, we have == (PQ), Wy => P+ LQ). Hence, by a known theorem, the ratio Tr, _ =(PQ) Tee CLP ero) will be intermediate in value between the greatest and least of the ratios ON 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 I, is identical with that of I,, or II = Il’. 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 Il, and in another the limit P. Regarding II as the limit of an inscribed, and P as that of a circumscribed polygon, it is plain that Il + P. In like manner, regarding P as the limit of an inscribed, and II as that of a circumscribed polygon, it appears that P > II. Hence IJ = P. ; The definite limit to which, as we have here proved, the perimeter of an inscribed (or circumscribed) polygon tends, when each side is ultimately less than any assignable magnitude, is adopted by definition as the ‘perimeter’ of the circle. The proof that the ratio (7) 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 Uy Uy, Uey-<0 6 canes Cry: 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’ p. 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 p. The point in the sequence after which the difference of each member from w 1s less (in absolute value) than o will in general vary with the value of o, but it is implied in the definition that, however small o be taken, such a point exists. 4-6] CONTINUITY. 9 Ex, Let the sequence be that obtained by putting n=0, 1, 2, 3, ... in the formula . n+l Fin ror a The sine is alternately equal to +1, whilst the first factor diminishes indefinitely. The limiting value is therefore zero. Te 6. Application to Infinite Series. If in the infinite series EMINENCE Un ares a ected eth ce. (1), whose terms are no longer restricted to be all of the same sign, we write ett 8 == Uy 1+ tla, s00. 0 Sos ty Fe Ueto wa dln, eee wens (2), and if the sequence Spon, El Geek Se cetah ets ee sgh ee nee es (3) has a limiting value S, the series is said to be ‘convergent,’ aud S is called its ‘sum.’ The most important theorem in this connection is that if the series bg habit yin ce oe | tig | Ent ween ecbes oe (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 @ 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 sm4,, 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 — gp. We have, then, (P — 9) — Snen = (P — ¥) — (Pm = Yn) = (P — Dm) — (4 — Yn) eeeeeeee (5). If m+n be sufficiently great, p—p » and g — gp will both be less than o, where o is any assigned magnitude, however small; and the difference of these positive quantities will be a fortiori less than o in absolute value. Hence s,,4,, has the limiting value p— g. ea 10 INFINITESIMAL CALCULUS. GOH RE When the series (4), composed of the absolute values of the several terms of (1), is convergent, the series (1) is said to be ‘absolutely,’ or ‘essentially, or ‘unconditionally’ con- vergent. It is possible, however, for a series to be convergent, whilst the series formed by taking the absolute values of the terms has no upper limit. In this case, the convergence of the given series is said to be ‘accidental,’ or ‘ conditional.’ The following very useful theorem holds whether the series considered be essentially or only accidentally con- vergent : If the terms of a series are alternately positive and negative, and continually diminish in absolute value, and tend ultimately to the limit zero, the series is convergent, and its sum is intermediate between the sum of any finite odd number of terms and that of any finite even number, counting in each case from the beginning. Let the series be { 1 — Ag+ yi Cie @eereserseeerees (6), where, by hypothesis, Maes Fee ac Meee! In the figure, let OM,=4, M,M,=a, M,M,=as, M,M,=4a,, oove Soa ase een meee ot ee So ; M, M4MgM M;Mg3 Ma x Fig. 5. It is plain that the points M,, M;, 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 Mey, M. on+1 = Aen4+1) and therefore is ultimately less than any assignable magni- tude, these two limiting points must coincide, say in 7. Then OM represents the sum of the given series (6). 6] | CONTINUITY. 11 Ex. The series 1-3+24-44... converges to a limit between 1 and 1 —- 3. This series belongs to the ‘accidentally’ convergent class. It is shewn in books on Algebra that the sum of m terms of the series 1+$4+ 54+4+... can be made as great as we please by taking m 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 Te Sey) a Sree ry) es Se nt Alea pee (7) and (Tie ch BG fk So etme Hoes] Rw i ie oe (8) be two convergent series whose sums are S and JS’, respec- tively, the series (ty + ty’) + (Uy $ te) +... + (Un tin) +... ...(9), composed of the sums, or the differences, of corresponding terms in (7) and (8), will converge to the sum S+8’. This is easily proved. If sy, sp’ 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 + 8’) — (Sn + Sn’) = (S — Sn) + (8’ — Sy’) ...(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—sa|<4o, and |S’ —s,'|<4o...... LLY 2 12 INFINITESIMAL CALCULUS. [CH. I and therefore |(S + 8S’) — (Gn + Sy ) | ee (12), which is the condition that s, +8,’ should have the limiting — value S + 8”. 7 General Definition of a Function. One variable quantity is said to be a ‘function’ of another when, other things remaining the same, if the value of the latter be fixed that of the former becomes determinate. The two quantities thus related are distinguished as the ‘dependent’ and the ‘independent’ variable respectively. The notion of a function of a variable quantity is one which presents itself in various branches of Mathematics. Thus, in Arithmetic, the number of permutations of ” objects is a function of 2; the number of balls in a square or a triangular pile of shot is a function of the number contained in each side of the base ; the sum (s,) of the first m terms of any given series is a function of the number 7; and soon. In some of these cases there are definite mathematical formule for the functions in question, but it is to be noticed that the idea of functionality does not neces- sarily require this; for example, the sum of the first » terms of the series PL ee ptptptpt is a definite function of n, although no exact mathematical expression exists for it. So, again, the number of primes not exceeding a given integer » is a definite function of n, although it cannot be represented by a formula. In these examples, the independent variable, from its nature, can only change by finite steps. The Infinitesimal Calculus, on the other hand, deals specially with cases where the independent variable is continuous, in the sense of Art. 1. For instance, in Geometry the area of a circle, or the volume of a sphere, is a function of the radius; in theoretical Physics the alti- tude, or the velocity, of a falling particle is regarded as a function of the time; the period of oscillation of a given pendulum as a function of the amplitude; the pressure of a given gas at a given temperature as a function of the density; the pressure of saturated steam as a function of the temperature; and so. on. Tere, again, the existence or 6-8] ; CONTINUITY. 13 non-existence of a mathematical expression for the function is not material; all that is necessary to establish a functional relation between two variables is that, when other things are unaltered, the value of one shall determine that of the other. In general investigations it is usual to denote the independent variable by x, and the dependent variable by y. The relation between them is often expressed in such a form as y=$(2), or y=f(x), ke, the symbol ¢(«), for instance, meaning ‘some particular function of a.’ | When a quantity varies from one value to another, the amount (positive or negative) by which the new value exceeds the former value is called the ‘increment’ of the quantity. This increment is often denoted by prefixing 6 or A (regarded as a symbol of operation) to the symbol which represents the variable magnitude. Thus we speak of the independent variable changing from w# to «+ 6a, and of the dependent variable consequently chaning from y to y + dy. Hence if ION COM Fe 5 aie ee tc es ene ely we must have SEP OM AD ALOR ote We iy 3 eva es (2), and therefore by = (4 + 62) —(#) «0.0.0... (3). At present there is no implication that da or éy is small; the increments may have any values subject to the relation (2). ) Exe. It y=’, then if x=100, d2=1, we have dy = (101)? — (100)? = 30301. 8. Geometrical Representation of Functions. We may construct a graphical representation of the rela- tion between two variables #, y, one of which is a function of the other, by taking rectangular coordinate axes OX, OY. If we measure OM along OX, to represent any particular value of the independent variable x, and ON along OY 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. = ae) oy ) 14 INFINITESIMAL CALCULUS, | [cH. 1 Since, by hypothesis, IM may occupy any position on OX, between (it may be) certain fixed termini, we obtain Y K 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. Fig. 7. 8-9] CONTINUITY. 15 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 « OM?, and the points P lie on a parabola. The same curve will represent the relation between the space (s) described by a falling body and the time (¢) from rest, since sa 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 « there shall be a definite value of y; but there is no necessary relation between the values of y corresponding to different values of w, 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, 2.e. N occupies once at least every position between H and Kk. Further, that if the range AB be continually contracted, the range HA 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 # and y be corresponding values of the independent variable and of the function. Let dv be any admissible increment of w, and dy the corresponding increment of y. Then if, o 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 6% which are less (in absolute value) than e the value of dy will be less in absolute 16 INFINITESIMAL CALCULUS. (CH. I value than oc, the function is said to be ‘continuous’ for the particular value w of the independent variable. _ Otherwise, if ¢ (7) be the function, the definition requires that it shall be possible to find a quantity e such that lb(a+h)—b(»)|< cee ee (1), for all admissible values of h such that |h|, 3,... having & for its limit, the function will be contin- uous at &. Ex. 1. To shew that the function b(t) = 0 ve (2) is continuous, according to the above definition, for all values of «x. We have , : (0+ h)? — a? = Qh + A (3). Now, « being fixed, and o being any assigned positive quantity, however small, we can always find a positive quantity e such that de |e] + =O... eee ee (4). This is in fact a quadratic equation to find ¢, and it is easily seen that one root is positive. And it is evident that if |h| be less than the value of « thus found, we shall have |p (a + h) — p (x)| ) 2 / ; 56 [ <2) (e) » <5 iS 4 > Pressures ee Oe One ee | li. ius ats a ass F an. to] 40 60 80 100 120 140 160 Temperatures 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. i 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 w 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 line) a certain breadth, and the same is true of the streak which represents the axis of 2; the distance bee 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 due 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 (x) 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 z. Functions exist (and can be mathematically defined) which are discontinuous for every value of « within a certain range. But ordinarily, in the applications of the Calculus, we have to deal with functions which are discontinuous Gf at all) only for certain isolated values of «. This latter kind of discontinuity, again, may occur in Various ways. In the first place, the function may become ‘infinite’ for some particular value (#,) of a The meaning 22 INFINITESIMAL CALCULUS. [cH. I of this is that by taking « 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 1/x, which becomes infinite for 2 —0, and tana, which becomes infinite for a=4n, &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 fora=7. A eraph of this function is appended. Periods ie) _ Fy Skee py ease meme ay Soe eeprom ape pane eer eee 30 Amplitudes Fig. 11. Again, it may happen that as # approaches a, the) function tends to a definite limiting value » (see Art. 26), whilst the value actually assigned to the function for #= a is different from i. | Consider, for example, the function defined as equal to 0 for! 2 — 0 and equal to 1 for all other values of a. | Moreover if 2, lie within the range of the independent variable, it may happen that the function tends to different limiting values as # approaches 2 from the right or left respectively. In this case the value of the function fol 12-13] CONTINUITY. 23 “=x, (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. Velocities Oo Times Fig, 12. Other more general varieties of isolated discontinuity are imaginable, but are not met with in the ordinary applications of the subject. 7 A sufficient example is afforded by the function sin —. a This is undefined for «=0, but whatever value we supply to complete the definition, the point «=0 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 «=0, however short, since the angle 1/# 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. (eH: = 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 « Then d(ut+v)=(ut dut+vutdv)—(u+v) = du + dv. From the definition of continuity it follows that, whatever the value of c, we can find a quantity e such that for | dx|< we shall have | 8u| a>0, and negative outside this interval. From the second form of y it appears that for «=+0 we have y=— 4. We further find, as corresponding values of # and y: oa mre i bye eh, Die 2, 3, +a, eee 01, — 70, — 1, —1°5, Fo, “5, 0, —°25, —-33, —°5. The figure shews the curve, 30 INFINITESIMAL CALCULUS. mae i opt Ex. 2, y Se l+a oes Here y=0 for «=0 and w=1, and y=+o for w=—1. Also y changes sign as « 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 «=+0, and above it for Pf SEA So ‘The figure shews the curve. 4 mee ee rn en ene an we en wn we Fig. 15, eee 1 ee 15] Kx. 3. CONTINUITY. Sl Qa fin BE Here y vanishes for x=0, and for x =+, and becomes infinite for x«=+1. Again, y is positive for 1>a>0 and negative for z>1. Also, y changes sign with 2. Ex. 4. As in the preceding Ex. y vanishes for « =0 and e=+, and changes sign with a. But the denominator does not vanish for any real value of x, so that y is always finite. + xe PAN 1. Draw (1) (2) (3) (4) (5) x i Ere INFINITESIMAL CALCULUS. EXAMPLES. I. graphs of the following functions* : Te ae (e—1)(a-2), a%—w+1. a(l—a«x)?, a (1—«)* 1 1 1 ae 8 ee x a x 1 Je, Ja Si) ee Rese eee mae oe ee (e—1)(e-2) (@-1) (e-3) a : wx —2 ‘- a Fred 1 — 2’ 1 +a?" l-ax+ 2 Ll+aet+a? x (l= 2)"" 3. Prove that the equation Qa? + Da? — 5a —-3 =0 [cH. I has a root between —o and —1, another between —1 and 0, and a third between 1 and 2. 4. Prove has three real roots, and find roughly their situations. 5. Find roughly the situations of the roots of that the equation 20° + Tx’ + 32-5 =0 Qa3 — 3x” — 36x + 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 tableg 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 sinz, cosa, tanw, We, whose definitions and properties are given in books on Trigonometry. The function sin w is continuous for all values of a For 6 (sin #) = sin (w + 6x) — sin & = 2 sin $6x.cos (w + 462). The last factor is always finite, and the product of the remaining factors can be made as small as we please by taking 6x small enough. In the same way we may shew that cos # is continuous. This result is, however, included in the former, since cos w = sin(#+ 47). Again, since tan ¢= : the continuity of sin # and cos a involves (Art. 13) that of tan w, except for those values of # which make cosa =0. These are given by #=(n+4)7, where n is integral. In the same way we might treat the cases of seca, cosec x, cot a. The figures on p. 34 shew the graphs of sin w and tan a. The reader should observe how immediately such relations as sin(—#)=—snwv, sin(w—#)=sing, sin(e+7)=—sina, tan(w+7)=tane can be read off from the symmetries of the curves. 1. 3 [cH. INFINITESIMAL CALCULUS. Fig. 18. | =} Nu Fig. 19. 27. eis 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 a“ coe a a COANE eb ee Eby: 2 Yo see that this series is convergent, and has therefore a definite ‘sum,’ for any given value of a, we notice that the ratio of the (m+1)th term to the mth is a/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 geometrical 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 Ei (a), 1t may be shewn that Lay Rel (y) SL YY decatbsscse (2). The proof involves the rule for multiplication of absolutely convergent series. Let Uy + Uy + Ug +t 10. $Unt... \ and Vy + V, + Uy +--+ UAt... | be two such series, whose sums are U and JV respectively ; and consider the series Wane at, Ar oie Wry Eso ate TE ee} (4), where w,, consists of all the products of ‘weight’ * m which can _ be formed by multiplying a term of one of the given series (3) by a term of the other ; viz. WD, = Uyy Wy = UyVy + UV, oeese Wy, = UUy + UyVy—1 + UqUn_o + ++ + Uy —101 + Ug¥orsrccness (5). | We will first shew that the series (4) will be absolutely con- vergent, and that its sum will be UV, the product of the sums of _ the two former series. Let us write Vin = Vo + Uy + Ug + coe + Ups Oy, = Up + Uy + Ug + v0. $F Un, Wy = Wy t+ Wy + Wet + eof * The ‘ weight’ is the sum of the suffixes, 36 INFINITESIMAL CALCULUS. [cH. I Then it is evident, from the actual multiplication, that the product U,V, includes all the terms of W,, and others; whilst W., includes all the terms of the product U,V,, and others. Hence in the case where the given series (3) consist entirely of positive terms, we shall have W,. U,V iy ieee (7), it is evident that the upper limit in question cannot be less than UV. Hence the series (4) is convergent, and its sum is UV. It also follows that if we write where Ry = (UVp + UgVy—y + 0 HUY) + ee + Un Up, coereeee (2); then F, can be made as small as we please by taking n great enough. We have still to examine the case where the series (3), one or both, contain negative as well as positive terms. With the above definition of #, it appears that if both series be absolutely convergent, the limiting value of &, will be zero when all the letters are made positive. That is, the sum of the absolute values of the several terms of /,, can be made less than any assigned magnitude by taking m great enough. It will be @ fortiori true that the absolute value of the actual quantity #,, which includes both positive and negative terms, can be made as small as we please by taking m great enough. It follows from (8) that the limiting value of W,, will be equal to that of U,V, 2.e. it will be equal to UV. | In the application to the exponential series, we put n n ee Y Uh, See ee eer i es dhe St: with the convention that u,=1, », =1, so that T= h(x), Va Big), | 17] CONTINUITY. 37 Hence w,=1, and (for n> 0) ge” gn-l y gn-2 y Cy yet y" eet it (aly h t@Sal ah hl @ Tt a 1 n n- n(n—1) n-2,2 1 yn "| ao {eete ieee treet aes Utrera Poss _(a+y)" pia? * by the Binomial Theorem. The several terms of the series (4) coincide therefore with those of the series for H(x+y). Hence the result (2). It follows from (2) that E(a)x E(y) x E(2)=E (@ty) x E@=E(e+y+2) Seer (10), and so on for any number of such factors. Also, since E(x) x E(—2)=H(O)=1........00.. (11), we have E(-2)= RiGee een (12) The function F () is continuous for all finite values of z. For, writing h for 6v, we have E(a@+h)— E(«)=E (a). E(h)— E(@) = E(x) {E (h) — 1}. hile Now E(t)-1=h(1+ 5)+5)+--): It is easily seen that the series in brackets is absolutely convergent, and that its sum approaches the limit 1 as h is indefinitely diminished ; whilst the factor h can be made as small as we please. Hence # (x+h)—H (a) can be made as small as we please by taking h small enough. When «z is positive, every term of the series (1) con- tinually increases with x, and becomes infinite for v=+o. The same holds @ fortiort for the sum #'(x). Also, in virtue of (12), it appears that if 2 be positive (—«) is positive and continually diminishes in absolute value as # increases, and vanishes for v=o. Hence as w increases from — 0 to +o, the function # («#) continually increases from 0 to+o, and assumes once, and only once, every intermediate value. 38 INFINITESIMAL CALCULUS. — [cH. I The accompanying figure shews the curve y= Ef (@). A column of numerical values of the function H(#) is given in Table E at the end of the book. Fig. 20. 17-18] CONTINUITY. 39 18. The number e. If we form the product of n factors, each equal to (1), we have, as in Art. 17, (10), {F(1)\"=H(1+1+... to n terms) =H (n) ...(1). It is usual to denote the quantity 4 (1), or Leal l+ltsitsit RC ATORE ine ede parma (2), by the symbol e. Its value to seven places of decimals is é = 27182818. With this notation, we have, if n be a positive integer, Again, if m/n be an arithmetical fraction (in its lowest terms), we have { (=) E (= eee esto terms) = 1 (m) = e”, n n mn eeseeseeetesens seste and therefore iL =) = (4). Hence, if « be any positive rational quantity, integral or fractional, we have It follows, from Art. 17 (12), that = hoes —2# i ( £) = ee Sagas so that the formula (5) holds for all rational values of 2, whether positive or negative*., * The investigations of Arts. 17,18 are due, substantially, to Cauchy, Analyse Algébrique (1821). 4.0 INFINITESIMAL CALCULUS, [cH I The actual calculation of e is very simple. The first 13 terms of the series (2) are as follows: ese 2 ~ .000 198 413 1 1 a: = = 000 024 802 1 1 5) = ‘166 666 667 57 = "000 002 756 1 1 7 = 041 666 667 5] = 000 000 276 1 .008 333 333 1 = --000 000 025 5s Tide 1 1 =, = ‘001 388 889 Fj = 000 000 002 The sum of these numbers is 2°718281830. The error involved in neglecting the remaining terms is An es 131 1a (oe which is less than Ea 4 Le em \, oF am i ( + 73 7 [3a 798 to eee 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 e*, when z 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 # (a). The advantage of this definition is that the notation serves to remind us of the algepraices laws to which the function is subject. Thus we have &.v=H(«#)xH(y)=L(et+y=e", whether a 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: : ‘hd : sinh a=} (e—e*)=a2+ 5455+ Sa Tee She cosh «=4(e* +e) =] a yet ates Eas sinh a 1 EEL cosh a’ Jae De cosh =| 2) tha = Cue Y cosecha= : | ii aie | ees sinh we? ~ sinha We notice that cosha, like cosa, is an ‘even’ function of w; we. it is unaltered by writing — for 2; whilst sinh a, like sin a, is an ‘odd’ function, z.e. the function is unaltered in absolute value but reversed in sign by the same substitu- tion of — x for a. The continuity of cosh x and sinh # follows from that of e* and e~* by a theorem of Art. 13. The figure on the next page shews the curves ae — Oe, together with the curves y=cosha, y=sinha, which are derived from them by taking half the sum, and half the difference, of the ordinates, respectivelyt. * 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. + The series are added according to the rule proved at the end of Art. 6. ~ The curve y=coshz is known in Statics as the ‘catenary,’ from its being the form assumed by a chain of uniform density hanging freely under gravity. 42, INFINITESIMAL CALCULUS. Y Fig. 21, |cH. I 19] CONTINUITY. 43 Since sinha and coshz are continuous, whilst cosh « never vanishes, it follows that tanhz is continuous for all values of z. Fig, 22 shews the curve - y = tanh 2. This has the lines y = + 1 as asymptotes *, N's Y’ Fig. 22, It is evident from the definitions (1) that cosh v+sinha=e*, cosh«—sinha=e~ ... (3), whence, by multiplication, cosh? # — sinh? a=1 ..... peeeer eres (4). From this we derive, dividing by cosh? and _ sinh2az, respectively, sech? = 1 — tanh? a, ( cosech2 2L = coth? 2 v4 it eeeeeeeeeeeeeeseees e The formule (4) and (5) correspond to the trigonometrical formule . RE SUAS SE lead We nr ane (6), sec? x= 1 + tan’ x, ) cosec? «= cot?#+1 f * The numerical values (to three places) of the functions cosh z, sinh z, tanh x, for values of x ranging from 0 to 2°5 at intervals of 0-1, are given in the Appendix, Table E. AA INFINITESIMAL CALCULUS, [cH. I Again, from (1) and (8) we easily find sinh (a + y) =sinh w cosh y + cosh # sinh y, (8), cosh (w + y)=cosh w cosh y + sinh wsinh y, }° * ” whence, as particular cases, sinh 2% = 2 sinh « cosh 2, cosh 2a = cosh? # + sinh? , These are the analogues of the trigonometrical formule sin (7 + y) = sin x cos y + Cos x Sin y, (10) COS (a + y) = cos x cos y + Sin x Sin y, euseieleie ie ‘ and Pee ee 1] COS Qa = cos? Gi sin? An ( )s respectively. 20. Inverse Functions in general. If y be a continuous function of «, then under certain conditions # will be a continuous function of y. This will be the case whenever the range of # admits of being divided into portions (not infinitely small) such that within each the function y steadily increases, or steadily decreases, as # increases. Let us suppose that as x increases from a to b the value of y steadily increases from a to 8. Then corresponding to any given value of y between a and 8 there will be one and only one value of x between a and b. Hence if we restrict y Fig. 23. 19-21] CONTINUITY. 45 ourselves to values of x within this interval, w will be a single-valued function of y. Also, if we give any positive increment ¢ to #, within the above interval, y will have a certain finite increment c, and for all values of dy less than a, we shall have da1, cosh # = log {a +4/(@?— 1) eee eee (6). Hither sign is here admissible; the quantities «+ ,/(a?- 1) are reciprocals, and their logarithms differ simply in sign. It appears on sketching the graph of cosh™’« that for every value of x which is > 1 there are two values of y, equal in magnitude, but opposite in sign. * The mode of calculating this quantity will be indicated in Chapter XIII. 22-23] CONTINUITY. Sunilarly, we should find, for a? <1, l+a tanh™ « = 4 log —— and, for a? >1, cotha=4 3 log = alas EXAMPLES. II. 1. Draw graphs of the following functions; (1) cosecx, cotx, cota+tan 2. (2) cosechz, cotha, coth«—tanh zx. (3) sin?a, tan? a. (4) sinaw+sin 2x, sin #+cos 2a. (9) log sina, log,tanz, from x=0 to x= dr. (6) sina’, sin ,/a, sin. 2. Prove that the equation sin «—a2 cos x=0 has a root between wr and 3a 3. Prove by calculation from the series for & that 1/e=°367879, cosh 1=1-5430806, sinh 1 = 1-1752019. 4. Prove that Je = 1°6487213, 1/,/e = 6065307, cosh $= 1:1276260, sinh 4 = 5210953. o. If |a|<|6| the equation a cosh «+ 6 sinh x=0 _ has one, and only one, real root, 6. Shew that the function tanh 1 2 is discontinuous for x =0. Draw a graph of the function. Vs 4 49 INFINITESIMAL CALCULUS, [CH. I EXAMPLES. III. Prove the following formule: ir 2. 8. 9. then and 10. ll. 12. cosh 2a = 2 cosh? «—1=1+2 sinh? a. sinh 2a = gos teat le tee cosh 2% = ie BU AUR tanh es ~ 1 — tanh? x’ 2 tanh a 92=——__——_—_. ran se 14+ tanh? x cosh? cos? w + sinh? a sin? w= 4 (cosh 2x + cos 2x), cosh? # sin?.z + sinh? a cos’ # = 4 (cosh 2a — cos 2a). cosh? a cos? z — sinh? # sin? a = 4 (1 + cosh 2a cos 2a), cosh? # sin? z — sinh? « cos’ a = 4 (1 — cosh 2 cos 22). cosh? w -+ sinh? v = sinh? w + cosh? v = cosh (wu + v) cosh (w—v , cosh? wu — cosh? v = sinh? w ~ sinh? v = sinh (w + v) sinh (w—7). sinh7! a=cosh~ ,/(a?+1), cosh-t#=sinh ” ,/(#— 1). —o 1 1l+a sech !a= log Le vl a: ) 3 cosech™ 'a=log Ls le ° ay x tanh~' “= =lor a If x == log tan (47+ $y), siahg=tany, cosha=secy, tanha= siny, ; tanh 4=tan $y. si.h w+sinh v=2 sinh }$ (w+) cosh $ (w—), sinh w—sinh v= 2 cosh § (w+) sinh $ (w—»), cosh w + cosh v =2 cosh } (w+) cosh $ (w—), cosh u— cosh v =2 sinh 4 (w+) sinh § (w—%). sinh wu cosh w—1 coshu+1 sinh zw lee ffl mS ee: sinh 3u=4 sinh*w +3 sinh w, cosh 3u=—4 cosh®w— 3 cosh wu. tanh 4 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 f. 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, ze. one not exceeded by any of the rest; but there will in any case be an ‘upper limit’ to the magnitudes of the assemblage, 7.e. there will exist a certain magnitude w such that no magnitude in the assem- blage exceeds yw, whilst one (at least) can be founc exceeding any magnitude whatever which is less than w. And if pw be not itself one of the magnitudes of the assemblage, then an infinite number of these magnitudes can be founc exceeding _ any magnitude which is less than p. 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 will in ary case be a ‘lower limit’ X such that no magnitude in the assemblage falls below 2, whilst one (at least) can be found below any magnitude whatever which is greater than >. And if d be not itself one of the magni- tudes of the assemblage, an infinite number of these magni- -* can be found less than any magnitude which is greater than 2X, 4__2 , 52 INFINITESIMAL CALCULUS. [Ono 1 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 y be a function which is continuous from «=a to «=b, inclusively, and if w be the upper limit of the values which y assumes in this range, there will be some value of w in the range for which y=p. 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. | } { t i The following is an outline of such a demonstration. In the geometrical representation, let OA =a, OB=b. If at A the value of y is not equal to the upper limit. p, it will be less than mw; let us denote it by y%. We can form, | in an infinite number of ways, an ascending sequence of magnitudes . Yor Yrs Yayeee whose upper limit is w. For example, we may take y, equal - to the arithmetic mean of y and p, y, equal to the arith- | 1 | | ] ; : metic mean of y, and yw, and so on. Since, within the range — AB, the value of y varies from y, to any quantity short of 4, | 25] CONTINUITY. 53 there will (Art. 10) be at least one value of x for which y assumes the intermediate value y,. Let «a, denote this value, or (if there be more than one) the least of such M, M,M;M, M Fig. 25. values, of 2 Similarly, let #, be the least value of a for which y=y2, and so on. It is easily seen that the quantities L1, Ha, Vy yoee (which are represented by the points /,, 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 = np, OH = yy, ON, = y;; ON i= Yas.5. 2) a0, 1D tne mode of forming the sequence Yoo Yio Yoo-- which is suggested (as a particular case) in the text, N, bisects HK, N, ' bisects N, K, N; bisects N,K, and so on. : a 54 INFINITESIMAL CALCULUS. (eH y differs from w 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 p. 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 0 let the value. of the function be (sin x)/a, and for «=0 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 |v] small enough; but it never actually attains this limit. As another example consider the function defined as equal to 0 for all rational values of «, and equal to sin za for all other values of «. (We have here an instance of a function which is discontinuous for every value of x.) 26. Limiting 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 2,. Let us suppose that, as # approaches the value 2, y approaches a certain fixed magnitude 2 in such a way that by taking |vw—~2,| sufficiently small we can ensure that for this and for all smaller values of |#—~«,| the value of |y—”| shall be less than o, where o may be any assigned magnitude however small. Under these conditions, X% 1s said to be the ‘limiting value’ of y as # approaches the value a, from the side in question. | The relation is often expressed thus: limger, Y=A, but in strictness the side from which w approaches the value x, should be specified. If we compare with the above the definition of Art. 9 we see that in the case of a continuous function we have lima <>. (0) = Gy wee eee (1), or the ‘limiting value’ of the function coincides with the value of the function itself, and that if x, lie withen the range of the independent variable this holds from whichever side — # approaches a,. If, on the other hand, 2, coincides with 25-27 | . ' CONTINUITY. 55 either terminus of the range, # must be supposed to approach x, 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 unlimited on the side of « positive. If as « is continually increased, y tends to a fixed value ® in such a way that by taking w sufficiently great we can ensure that for this and for all greater values of x we shall have |y— | less than o, where o may be any assigned positive quantity, however small, then » is called the limiting value of y for c= , and we write tia A. There is a similar definition of hi ess Y, when it exists, in the case of an independent variable which is unlimited on the side of # 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 w, v be two functions of x, and let us suppose that as a approaches the value a,, these tend to the limiting values U,, v,, respectively. If, then, we write U=U,ta, v=v,+f8, 56 INFINITESIMAL CALCULUS, [cH. I a and B will be functions of « whose limiting values are zero. Now (w+ v) —(u,+%,)=a+B, UY — UzV, = av, + Bu, + af, U UW avy,—Ppuy, vo % 2%, (%, +B) And, as in Art. 13, it appears that by making 2 sufficiently nearly equal to a, 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 «, of the independent variable a, for which the function exists, is simply the value of the function itself for «=2,. 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 « 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 0 and zr, but it does not exist, in any strict sense, for the extreme values 0 and 7. 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. Ex. 1. Take the function . Ai UE) — x The algebraical operations here prescribed can all be performed for any value of « between +1, except the value 0, which gives to the fraction the form 0/0. Now the definition of a quotient es. } 27-28} CONTINUITY. 57 a/b is that it is a quantity which, multiplied by 6, gives the result a. Since any 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— 2), put the function in the equivalent form Are _ and for all values of « between + 1, other than 0, this is equal to 1 1+ /a- x”) : Since this function is continuous, and exists for =0, its limiting value for «=0 is }. Fx. 2. Consider the function J(1 +a) — fx. As # is continually increased this tends to assume the in- determinate form o—o, But, writing the expression in the equivalent form ] we see that its limiting value for x= is 0. Hz. 3. To find hinige cen) This ass"~es the indeterminate form © x 0. But since 2 ole=1/ (24 1+ sit Rite)s we see that the limiting value for «=o is 0. _ If we write z for e~”, and therefore —log z for x, we infer that : lim,» 2 log z= 0, In the same way we can prove that lim, pec"enf = 0; | Where m is any rational quantity. 58 INFINITESIMAL CALCULUS, 29. Some Special Limiting Values. The following examples are of special importance ir Differential Calculus. | 1°. To prove that gm oo qi” in,.: w=a L2—-a y.(1), for all rational values of m. If m be a positive integer, we am —qm fet lini; a he = Lind 5 (ans +¢ Lae i Fae 10 kines yy since, the number (m) of terms palig finite, the nee 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/q, say, we put a= yf, = be and therefore wm — gm yi — bay? — o—-a. yl—bt yt—ht" This fraction 1s equal to y? — b4 y—b yi — bf y—b denominator. is ght, by the saa required limit is Peg ee a a fh? 1=* ari = mar q q as before. If m be negative, =—n, say, we have Oh ae 1 eat L— a o-a go? eee 29] CONTINUITY. 59 If n be rational, the limiting value of this is roll a m1 — m—1 a na? ,=—na’*, = ma", by the preceding cases. 2°, To prove that limp—» ave i, limg_, bee we 1 Pe ann (2). id 0d 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(7’P + TQ) will be the perimeter of the corresponding circumscribed poly- gon. Now, if G=2P0A='r/n:; we shall have . chord PQ PN osmé jarc PQ “arceLA Pee IP+TQ ‘PT. Ve arc PQ arc PA’ Va Hence the fractions and sin 0 and 122 6d id 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 sub- multiple of 7. But, whatever the value of the angle POQ in the figure, we have chord PQ < arc PQ, and TP + TQ > are PQ; _ te. (sin @)/@ is less than 1, and (tan 6)/@>1. Hence these _ fractions must have respectively an upper and a lower limit; _ and it follows from the preceding that neither of these limits _ ean 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 0 is — continually diminished. n 6/3 (sin @)/6 (tan 0)/0 4 |.) +95 90032 1:27324 Bo 20 93549 1:15633 LOP a2 410 ‘98363 | 1:03425 20 | -05 ‘99589 1-00831 40 025 99897 100206 | co 0 100000 | 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 Kes liga Lee et —] h h Ph We have Wi =145 (1+5+3qt-): 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 (¢” —1)/h and 1 can be made as small as we please. If we put h=log (1 +2), the theorem (3) takes the form Ze linge: _ ee 1, log € =o = ; z or limy,-. 7 log (1 + = oe ) n whence - hin; (1 as =) at Ot Se es (4). 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, VA and A7' are ultimately equal. Tor, by similar triangles, Ole Oe. ON? OF? OP—ON OT-OP and therefore SONS OP oP sh Sa ORL ql ATE OPS and the limiting value of the ratio ON /OP is unity. It is sometimes convenient to distinguish between diffe- rent orders of infinitely small quantities. Thus if w, 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 aswu. 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 w. More particularly, if the limit of v/w™ be finite and not zero, v is said to be an infinitesimal of the mth order, the standard being w. Thus, in the figure referred to, V7 is an infinitesimal of the second order, if the standard be PN. For PN2ON ENT, ie Saree PVs Os OA | 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 _ mito 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. whence in the limit. | | 62 INFINITESIMAL CALCULUS. (6H. 1 A quantity which in any process finally exceeds any assignable magnitude is said to be ‘infinitely great.’ And if one such quantity w be taken as a standard, any other v is said to be infinitely great of the mth order, when the limit of v/u™ is finite and not zero. EXAMPLES. IV. 1. Shew geometrically that the sequence Le Nt | PG oe has the upper limit 1+ ; é 2. Find the upper and lower limits of the magnitudes 2n n+ 1 where n=1, 2,:3,.... 3. If aand 6 be positive, and a>, the function ae” + be-* e+e” has the upper limit a and the lower limit 6. 4. Find the limiting values, for «=0, of sin ax sinh ax and x 5. Trace the curves sin x sinh x cat hope Be Sage : [= 008 2 ye 6. Prove that li oe a eee x 7. Prove that hm 2 (sec #— tan a) = 0. 8. Prove that lim,_, V(l+ x) — J(l— «) =I x 9. Prove that lina 2. {/(a? + +1)—atb=4. Z 7 30] CONTINUITY. 63 10. Prove that im ene =, =00 x and hence shew that lim n=l. 1]. Find the limiting values, for «= 0, of sin” a tan™ x and P x 12. A straight line 4B 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 AS, and Q the point where AB is met by- the bisector of the angle XOY, then AP=QB. 13. Through a point A on a circle a chord AP is drawn, and on the tangent at A a point 7’ is taken such that AZ7’=- AP. If 7’P produced meet the diameter through A in Q, the limiting value of AY 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, OY a triangle AOB of constant area. Prove that the limiting position of the intersection of two consecutive positions of AB is the middle point of AB. 15. A straight line AB of 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 W the foot of the perpendicular from O on AB, then AP= VB. 16. Tangents are drawn to a circular are 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 PCP’ be any fixed diameter of an ellipse, and QV any ordinate to this diameter; and if the tangent at Q meet OP _ produced in 7’, the limiting value of the ratio 7'P : P V, when PV 1s infinitely small, is unity. CHAPTER IL. DERIVED FUNCTIONS. 31. Definition and Notation. Let y be a function which is continuous over a certain range of the independent variable x; let da be any incre- ment of « such that # + 6 lies within the above range, and let dy be the consequent increment of y. Then, # being regarded as fixed, the ratio will be a function of da. Ifas dv (and consequently also dy) 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 w, and is denoted by the symbol 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 dy, dx. The latter ratio is indeterminate, being of the form 0/0. | The symbol dy/dx 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. ae - 31] DERIVED FUNCTIONS. an 35) When we say that the ratio dy/da tends to a unique limiting value, it is implied that (when @ hes within the range of the independent variable) this value is the same whether dz approach the value 0 from the positive or from the negative side. It may happen that there is one limiting value when 6x approaches 0 from the positive, and another when 6x approaches 0 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 dy/dx 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/dz. The derived function is often indicated by attaching an accent to the symbol denoting the original function. Thus if | EAA CAL SS PRON S OR wre rete (3), the derived function may be denoted by 7 or by ¢’ (#). ae ay _ (wt Br) ~$(e) | ye O@ we have, writing h for oz, | $" (a) = lim, ea hae) is ee The operation of finding the ioe coefficient of a siven function is called ‘differentiating’ If « be the inde- dendent variable, we may look upon d/dz as a symbol lenoting this operation. It is often convenient to replace L. 9) 66 INFINITESIMAL CALCULUS. [CH. II this by the single letter D; thus we may write, indifferently, dy d ax ’ da Y; Dy, for the differential coefficient of y with respect to «. 32. Geometrical meaning of the Derived Func- tion. In the annexed figure, let OM=2,.0ON=a2+6e, _PM=y ON Fo), (Sh eae x Fig. 26. and draw PR parallel to OX. Let QP produced cut the axis of g in 8. Then sy QR PM <= pp gy = tat PSK ee, (1). As 6 1s indefinitely diminished, Q approaches P, and if follows that if the derived function exist the line PQ tends to a definite limiting position P7, such that 3 d tan PTX = os I (2). It appears then that the assemblage of points (Art. 11) which represents any differentiable function has at each Ss 5 i s 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 a. 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-line varies continuously as we pass along the assemblage. That is, it is implied that the ordinate y is a differentiable function of the abscissa w, and that the derived function dy/da is itself a continuous function of 2 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 «. Fig. 27. _ If this angle be negative, the gradient is negative. If the angent-line be parallel to the axis of «, 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, different from one another, then on the corre- sponding curve there are two branches making an angle with one another. The value of dy/dx is then discontinuous. The figure illustrates 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 s of the point from some fixed origin in the line of motion will be a function of the time ¢ reckoned from some fixed epoch. The relation between these variables is often exhibited graphically by a ‘curve of positions,’ in which the abscisse are proportional to ¢ and the ordinates to s; see Fig. 7, p. 14. If in the interval é¢ the space és is described, the ratio 6s/d¢ is called the ‘mean velocity’ during the interval dt; ve. a point moving with a constant velocity equal to this would accomplish the same space 6s in the same time o¢. In the limit, when d¢ (and consequently also 5s) 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 # In the notation of the calculus, therefore, this velocity v is given by the formula In the graphical representation aforesaid, v is the gradient of the curve of positions. Again, the velocity v is itself a function of ¢. The curve representing this relation is called the ‘curve of velocities.’ If dv be the increase of velocity in the interval 6, then 6v/dé 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 o¢ is indefinitely diminished is ! called the ‘acceleration at the instant #.’? If this acceleration be denoted by a, we have 32-34] DERIVED FUNCTIONS. 669 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 0 be the angle through which the body has revolved from some standard position, the ‘mean angular velocity’ in any interval 6¢ is denoted by 60/dé, and the ‘angular velocity at the instant ¢,’ by do ee (3). Again, if w denote this angular velocity, the ‘mean angular acceleration in the interval 82’ is denoted by 8w/6¢, and. the ‘angular acceleration at the instant ¢’ by dw Again, the length of a bar of given material is a function of the temperature (0). If « be the length at temperature 0 of a bar whose length at some standard temperature (say 0°) is unity, then 62/60 represents the mean coefficient of (linear) expansion from temperature 9 to temperature 6 +60, and dx/d@ represents the coefficient of expansion at temperature 6. As another example, suppose we have a fluid which is free to ‘assume a series of states such that the pressure (p) is a definite ‘function of the volume (v) of unit mass. If the volume change from v to v+dv, the fraction —dv/v measures the ratio of the ‘diminution of volume to the original volume, and gives therefore the ‘compression.’ The ratio of the increment of pressure dp required to produce this compression, to the compression, is —vdp/sv. The limiting value of this when év is infinitely small, iviz. —vdp/dv, is defined to be the ‘ elasticity of volume’ of the fluid under the given conditions. _ 84. Differentiations ab initio. 1 Before investigating general rules for calculating the Jlerivatives of given analytical functions, we may discuss a few examples independently from first principles. Eu. 1. If y=, we have dy= dx, and therefore Oy dy _ se 1, whence Te ry: 70 INFINITESIMAL CALCULUS. [CH. II Ex, 2. Let Ear (1). eeoeceeeeeeeeeeee eee eee eS Seeoeoe We have, writing / for da, dy (a@+h)— a = a se i x+h Proceeding to the limit, when h=0, we find dy Fics 20 s.o san crete anes eee (2) 1 fx. 3. Let Y =o cee tetenneeeeenenneeeceneneae: (3). 1 1 h We have | Ye Tk oo ee oy 1 and therefore Nap te hca dy é 1 a ST EE a tee ee 4). Hence te lim, 5 a (eth) (4) The negative sign indicates that y diminishes as « increases. Ex. 4. If Y = [Wace cvsesssoaeee nr (5), h we have = /(at+h)— J/e= e+e Sy _ 1 jn (ath) + a" Proceeding to the limit ( = 0), we find dy 1 EXAMPLES. V. 1. Find, from first principles, the derived functions of neo yo ae i Aloo: 2 = xt+a u— a Pres: 3. Also of J (37 + a), A 4. Also of cot x, secx, coseca. J 34-35 | DERIVED FUNCTIONS. 71 5,. Also of ‘sin*2, ‘ cos*#,. sin 2%, . cos 2a. 6. If, in the rectilinear motion of a point, s=ut + hal’, where wu, a are constants, prove that the velocity at time ¢ is u+at, and that the acceleration is constant. 7. If the pressure and the volume of a gas kept at constant temperature be connected by the relation pv = const., the cubical elasticity is equal to p. 8. If the radius of a circle be increasing at the rate of one foot per second, find the rate of increase of the area, In square feet per second, at the instant when the radius is 10 feet. 9. If the area of a circle increase at a uniform rate, the rate of increase of the perimeter varies inversely as the radius. 10. If the volume of a gramme of water varies as (9 — 4)" 144000’ where @ is the temperature centigrade, find the coefficients of eubical expansion for 9=0° and 6= 20". 1 35. Differentiation of Standard Functions. fee if TEAL EE aig ay OORT (LY, dy (e+ dx)™— a™ Sa (a+ 6n)—a@ It has been shewn in Art. 29, 1°, that, for all rational values of m, the limiting value of this fraction when 6a =0 is ma”™—, Hence we have (Ba. If m=2, dy/dx=2x; if m=4, dy/du=}a74. Cf. Art. 384, er Af ASSEN Om tat Ue love ae Sey (3) we have, writing h for dx, oy sin(e@+h)—sina _ _ sin gh ae h Th cos (a + sh), 72 INFINITESIMAL CALCULUS. [cH. Ir If the angles be expressed in ‘circular measure, we have | by Art. 29,2°; and the limiting value of the second factor is cosa Hence dx Be The student should refer to the graph of sinw on p. 34, and notice how the gradient of the curve varies in accordance with this formula. 3°. It Y = COS.@ csomecnaae ae ee Oe we have oy ees (2+h)—cos x Ox ip sin $h Th .sin («@ + 3h); the limiting value of which is, on the same understanding as before, 2 =— SID @ . sce ee (6) Norte 8 8 y= tal ©... ee erat (TY; we have dy tan(#+h)—tanaw_ sin(#+h)cos#—cos(«+h) sing oa h % h cos @ cos (@ +h) _ sink 1 : h cosxcos(x+h)- Hence, in the limit, dy _ ——__— == SCC .W vec epen eee ; aap Sec? @ (8) This shews that the gradient of the curve y= tana, between the points of discontinuity, is always positive; see Fig. 19, p. 34. eae bi Y= Oe Ae (oy jy CaS Gore et — 1 Ox h es) we have 7 35-36] DERIVED FUNCTIONS. 73 It was shewn in Art. 29, 3° that h lim;=9 i : Pale dy, Hence | a (10). More generally, if I ie de OAD BEET BoE CEL: we have dy a See RA a caeet Bel : ekh ripe Aes lim; =o Sparse = lim;z,=0 kh ke wh eee Pave ot Balle aati y meee tres geri (12). In particular, if J COA IT SE EEE Oe (13), we have u seh aa ts Soe ra See dees Shea (14). Again, if a be any positive quantity, we have by the definition of Art, 22 (2), q* = evloga Hence if LS Tage ei ata 3 haar AR Fae dy 1 ee 1 and therefore era F = a (5) The ambiguity of sign in these results is to be accounted for as follows. We have seen that if y=sin-!a, then y is a many- valued function of x; viz. for any assigned value of x (between the limits +1) there is a series of values of y, and for some of these dy/dx is positive, .for others negative. See Fig. 28. Similarly for cos~! a. If, when x is positive, we agree to understand by sin~!z the angle between 0 and $7 whose sine is 2, we must write see en 1 Wes sin” ¢%=-F Jd-#) occ c ee cee ccc ceecee (6). Similarly if, a being positive, cos-!a be restricted to lie between 0 and 37, we have d —. )— 1 — . Fp 08 Misbagioe) Tere ree (7) | per) PCNA es ea. setts ede ken evion oo ha as (8), we have 2 = tan 7, ue = sec? ¥, Y dy g dy 1 1 and therefore fay aie (9). There is here no ambiguity of sign. For each value of « there is an infinite series of values of y, but the value of dy/dax is the shy 88 INFINITESIMAL CALCULUS, [CH. II same for all, the tangent lines at the corresponding points of the curve y=tan~’« being parallel. See Fig. 29. Qn eee ew en ee ee en ce wwew wn Been Ae ee ee ee Sees we ee ee ewww wee ee on own oe Omen wc ee ee ee ewe ee ew ww ee ae ee ee ee ee eeeeees ee te Or ee OP ee ee BEBO OF OWE Ue Y OB Oe ee Re ee ee me owe od — CEES ES SS OS Ow we mn ee ee ee ee es ee ee eeests ue eae — lr Fig. 29. a eal tie. A. .-Let y = sin Jia (10), or y=sin-'u, where w= se J(1 + #) 41-42] DERIVED FUNCTIONS. 89 dy dy du _ 1 du We have da dudx J(1—w) dx’ We easily find 1 du 1 a OF og oe RE Aer ey IER Ee V1 — ai) = Ri lpeeias Vee cee ( Lec a 8 whence 3 = os je 6 blab ofalaiuia'siarsteiave @ nie eleidierelenn (11) It is easily proved (putting x= tan @) that ies | oe Lea es | sin Wire +a) tan~* a, so that the above result is in accordance with (9) above. He. 2, Let y = tan“ a arostteaneeman carne kd (12). 2 If we write ri aml l—-v4+2 dy 1 du we have SP ay ee We find Taf 2 (1 + 3a? + a‘) Dut _ 2(1-2") (l—-xw+2a*)? > dw (1—-w%+27)?’ wane whence EDs AS (13). 42. Differentiation of a Logarithm. cae be UNOS secs ses Daaavareceersseeed: (1), da we have v=e, —-=@=2, dy : dy _1 and therefore eye (2). This diminishes as 2 increases, so that the representative curve becomes less and less inclined to the axis of w% See Fig. 24, p47. olf [DEA Fe hee ae oe (3), we have x= da, 90 INFINITESIMAL CALCULUS. [CH. II i] aS 7; @ 7 ee eceeeeesessesseoseeoe 4 e whence da logea a (4) For instance, if ¥ = lO ip @ =-ncouseapenenaeseee ea (5), dy _p we have | ee (6), where p= ‘43429... as in Art. 22. 3°. If 9) = log Ub. ss'.ics'sos teen een (7), where w is a given function of z, we have, by Art. 39, dy _ dy du_1du 8) We duds ude (8). Pgs AE y = log sina, 2. ge eee ee (9), we have hips Sas . DSi & = Cone ee (10). dx sin Similarly, if y = log sec 4 =— log. cOS es arene eee (It), dy we find Aas AD 0 21... serene (12). ie. 2, Tf y = log-tan $2... <<. ee (13), we have Ls leat Lip aid de” nae Oe SOC SIs 1 ce nas Peer corer see eee roeerersarerseesenseesesses res (14) Similarly, if y = log tan (dir +30) eee eee (15), dy__t we should find dn cose es (16). l+x Ex. 3. Let y =t log Dang VS riitittteeesnes (17), =4log(1+a)—4 log (1-2) dy 1 1 1 a eek ae pS Hence det lant ?loe toe (18). Ex. 4. Let y = log fet [Ge £1) ee (19). 42-43] DERIVED FUNCTIONS. tit d: 1 : We have EERIE EAN Ook oe J (a? + 1)} 1 x “aa seen Ut Jersty 1 * 5 Eran pe ee ER a ee (20). 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 we have logy = log wu, + log u.+ log ws + — log v, — log v, — log v3 — ...(2), and therefore, by Art. 42, 3°, 1ldy Adi, Lduy 1 dus, yda ude wu,de wu; dx late Lede Fld: —-— —-— —-— —-....(3). v1,dce v,dxe v, dx This is a generalization of the results of Arts. 37, 38. on (a +x) (b+ 2) Fe, If Y= sf aan orat we have | log y = 4 log (a + x) + $ log (b + x) —4 log (a— =) — 3 log (b —@). Hence et ast iet astral ie ae b (a + b) (ab — x”) a” Poa (a? — a”) (6? —x")’ dy _ _ (a+b) (@ (ab — a2”) daz (a—a)8 (B— 2) (@ + a)h(B + a)} INFINITESIMAL CALCULUS. [CH. Ir EXAMPLES. IX. Verify the following differentiations : 1) = 108 a, Dy =1 + log x. 2) ny) = doa, Dy =a" (1 + m log a). 3. y=logsing, Dy = cot x. 4. y=log cosa, Dy =—tan x. 5. y=log tan a, Dy = 2 cosec 2a, 6. y=log sinha, Dy = coth x. 7. y=log cosh a, Dy =tanh x. 8. y=log tanha, Dy = 2 cosech 22. ot 1 or Ue lee ae OY cee l—«z 2 tO TEAR age 0d aes ae 1 LL, -y=1og Westy’ Dy =a G41) 1+ /x 1 bs = oO D — SA AY TS ae (2) Je 13. y=log {/(x+1) Dy= 1 + /(#—1)}, 2 /(2—1)" 1 1 Ve eee J(a? + 1)— 2’ Dy aaa 22-1 e+] 15. PMR Degatye dos ass. os tote _ 2(1=—at) 16. Yee | ; 1 ees YE ere 25 = ee 17. y=sin1(1-2), Dy Je =a 18. y= sin Ye. Dy = sin7 x + fesse Jae" Ww 24, 32. DERIVED FUNCTIONS. = =i y=cot * 2x, ae —1 ¥y = Sec" 2, = sal 7—cosec* x, y=sin-! x +sin-! /(1 — 2), y =tan-e+ tan y =sin-1{2e ,/(1—2*)}, 2a #—tan-? fog? y=tan-{ J(a?+1)—2}, Dy=— y = sin~' (cos &), — 9 T4a?? y-[) ie J(1+2*)—2x J(1 + 2) + x’ y = cos~! ——. _ V(l+a)+./(1—-2) Sir ~ J(1+a)—/(1—-2)’ _ (+08) +J(1-a) pe (ira) —/(1— ay © 44. Differentiation of the Inverse Functions. ih we have If 93 1 tare he 1 UA? Sage aaa 1 se as G1) Dy C Dy =0 2 TO =) 2 leer 1 2 (a? +1)" Dy =-—1. 2 ee ere 2x YU= ea iy -2 dm ses eee) EA Sp ous wo FIs x ‘ 2 2 I= la) Hyperbolic Sed Wert: ery ee cacy Src eee tL); 94 INFINITESIMAL CALCULUS. [CH. II | dy age | and therefore Ag = Vd + #) oe : or . se cvceveces (2). There is no ambiguity of sign, for cosh y is essentially positive. Jeo AL y= cosh” Hess eee (3), we have x = cosh y, 7 = sinh y = + /(#’—1), dy _ 1 whence Ap = + Va? — 1) covceceseesccceess (4). For any given value of x, greater than unity, there are two possible values of y, one positive, the other negative, and for these dy/dx has opposite signs. [Cf. Fig. 21, Art. 19, inter- changing «x and y.| 3 hee 64 y = tanh" 2,0. (5), we have e = tanh y, 7 =sech? y =1— 2, dy __ 1 and therefore dn Da gh ee (6). This agrees with Art. 42, Ex. 3. It is to be noticed that y is real only when «<1. See Fig. 22. Similarly, if y = coth a 3.5 (Tea dy 1 we find oh =— WG ek (8), x being necessarily > 1, if y is real. EXAMPLES. X. Verify the following differentiations : tee = sech a, Dy =— Pee ° Ju Y= eosechs am, Dy =— aa : 38. y=sin7'(tanh 2), Dy = sech a. 4, y=tan7! (sinh 2), Dy = sech x. 5. y=tan-!(tanh de), Dy = 4 sech a 6. y=tanh-(tan4x), Dy=4}secu, A 4.445 ] DERIVED FUNCTIONS. 95 EXAMPLES. XI. 1. What is the geometrical meaning of the theorem S45 (Bae) = hp! (ha) 2. If y=tan®, vDu —uDv prove that Dy = iat ee GA 1 — 9"! 3. Assuming that eee L+v+aort+... +2", deduce, by differentiation, the sum of the series 1+ 2e+ 3274+ ...4+ nx, and test the result by putting v= 1. Hence shew that, if |x|<1, 1+ 2u + 327+ 4a? +... to o=(1—2)-%. 4. If, in the rectilinear motion of a point, v? be a linear function of s, the acceleration is constant. 5. If v be a quadratic function of s, the acceleration varies as the distance from a fixed point in the line of motion. 6. If the time be a quadratic function of the space described, the acceleration varies as the cube of the velocity. 7. If | waded, ; the acceleration varies inversely as the square of the distance 'from a fixed point in the line of motion. 8. If s* be a quadratic function of ¢, the acceleration varies as 1/s°. 9. If the pressure and the volume of a gas be connected by the relation pv" =const., the cubical elasticity is yp. _ 45. Functions of two or more independent vari- ables. Partial Derivatives. Although in this treatise we are concerned mainly with functions of a single independent variable, it will occasionally | | ig fe 96 INFINITESIMAL CALCULUS. (CH. 11 be useful to have at our command ideas and notations borrowed from the more general theory. One quantity wu is said to be a function of two or more independent variables x, y,..., when its value is determined by those of the latter, which may be assigned arbitrarily and independently within (in each case) a certain range. Thus if P be any point of a given surface, and a perpendicular PN be drawn to any fixed horizontal plane, the altitude PV is a function of the coordinates (a, y) of the point IV. Fig. 30. So again, in Physics, the pressure of a gas is a function of two independent variables, viz. the volume (per unit mass) and the temperature. The functional relation is expressed by an equation of the form U = (L,Y...) eaees cae (hy In particular, in the aforesaid case of a surface, if we denote the altitude PN by z, we have Z=h(k, Y). Fitnacaeaae ee (2). Let us now suppose that all the independent variables Save one (#) are kept constant. Then the function u may or ee —— j 45-46] DERIVED FUNCTIONS. 97 may not be a differentiable function of #; if it is differenti- able, its derived function with respect to «# is called the ‘partial differential coefficient’ or ‘partial derivative’ of w with respect to #, and is conveniently denoted by du/oz. Thus Ou dh (a + da, y,...) — b (a, Y, «++) ba They Ans = lims,=0 In like manner Ce. h(x, y+ dy,...)-— 6 (a, y,«.-) ANE telah ee eco k. fie res + (4). In the case of the surface (2) it is plain that the partial derivatives iz 0s ov” oy are the gradients of the sections (HK, LM, in the figure) of the surface by planes parallel to the planes ZOX, ZOY respectively. He. 1. If Let 12! Vaca iy dad gt Cogan al SE (5), 02 mM—1 pn Oz m 1 te A ere INE Of cept ae, hae, oer 6). we have a mam ty”, ay nay (6) Ex, 2. Assuming that in a gas the pressure (p), volume (v), and temperature (@) are connected by the relation Ld op hd ap we have Rs Care yA) 46. Implicit Functions. An equation of the type ‘in general determines y as a function of x; for if we assign any arbitrary value to wz, 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 iD: 7 ie 98 ; INFINITESIMAL CALCULUS. [CH 1 real. The term ‘implicit’ is applied to functions determined in this manner, by way of contrast with cases where y is given ‘explicitly’ in the form y =f (8). sie (2). = ch (&, Y) o. aes ae ee (3) as the equation of a surface, then (1) is the equation of the section of this surface by the plane z=0. If the plane xy be regarded as horizontal, the sections z=(, where (’ may have Hiforat constant ones are the ‘ contourlines? If we regard 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 sufficient, for the present, to consider the case where ¢(a#, y) is a rational integral function of # and y, 2.e. it is the sum of a series of terms of the type An »w™”y”, where m, n may have the values 0, 1, 2, 3,..... Since, by hypothesis, ¢ (a, y) is constantly null, its derived function with respect to « will be zero. Now by Arts. 35, 87, 39, we have 2 (aay) ies ini te =. Te unas = f Hence, if 6 (2, y)= Am nt™Y® ....0 ee (4), we have YA» pma™—y™ + 2A nanan 9b == (Gene (5). In the notation of Art. 45, this may be written a aD = 0 Cove cee nee neneseeverccee Bo: It will be shewn in the next Chapter that the results (6) al (7) are not limited to the above special form of ¢ (a, y); b the present case is sufficient for most geometrical applicatio 46] DERIVED FUNCTIONS. EXAMPLES. XII. 1. Sketch the contour-lines of the surface az= 0+ y?, and describe the general form of the surface. 2. Also of the surface az = «xy. 3. If prove that and give the geometrical interpretation of this result. ee Lt prove that 5. If prove that 6. If _ prove that me If prove that eit prove that 2=f(%+y), Oz Oz da ay” r= J/( + y"), az ay 02 ax a x+y? ay wry” e=f(ety, ad al pp dx Oy vs ax? + 2hey + by’? + 2gxu + 2fy+c=0, dy axthytg dx het+by+f 7—2 99 CHAPTER IIL. APPLICATIONS OF THE DERIVED FUNCTION. 47. Inferences from the sign of the Derived Function. as If y=¢(a), and if dx, dy be simultaneous increments of a2 and y, the limiting value of the ratio dy/da when dx is indefinitely diminished is, by definition, ¢’(#). Hence, before the limit, we may write oY _ 4! (x) +e ae Settee neens (1), where o is an ultimately vanishing quantity. A numerical example of the manner in which the ratio dy/dx approximates to its limiting value may be of interest. We take the case of y=log, x, for the neighbourhood of «=1. The limiting value is here @Y _ 43499... dx « The numbers in the second column are taken from the printed tables. ox oy dy /dx "1000 =| -041393 41393 (0500 | -021189 ‘42379 0100 | -0043214 ‘43214 0050 =| 0021661 43321 0010 | -00043408 43408 ‘0005 00021709 43419 ‘0001 000043427 ‘43427 oe a 47] APPLICATIONS OF THE DERIVED FUNCTION. 101 Let us first suppose that dp’ (x) > 0. Since the limiting value of o is zero, we can by taking 6x small enough ensure that fp (4)+a>0. That is, by (1), 6y will have the same sign as 6 for all admissible values of da which are less in absolute value than a certain magnitude e. In the same way, if $ (x) <0, dy will have the opposite sign to 6x for all admissible values of da 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 «= OM, where M is a point within the range considered, we may say that if ¢’(«) be positive there.is a certain interval to the right of MW for every point of which the value of the function ¢(«#) is greater than its value at M, and a certain interval to the left of MW at every point of which the value of the function is less than its value at M. If ¢’(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 a, the intervals referred to lie of course to the right or left of I, respectively. It follows that if ¢’(«) be positive over any finite range, the value of #(#) will steadily increase with # throughout the range; 2.¢. if #, and a, be any two values of « belonging to the range, such that x, >, then (2) > $ (a). For ¢(«), being by hypothesis differentiable, and therefore continuous, must have (Art. 25) a greatest and a least value in the interval from a, to a (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 ¢ (a) must occur at the beginning of the interval. In the same way it appears that if ¢’(x) be negative 102 INFINITESIMAL CALCULUS. + (OH. THE over any finite range, then ¢(«x) will steadily decrease as # increases, throughout this range; 2.¢. if a, and a, be any two values of « belonging to the range, such that a, > a, then p (#2) < p (a). The geometrical meaning of these results is obvious. When the gradient of a curve is positive the ordinates increase with a ; when the gradient is negative the ordinates decrease as a Increases. The graphs of various functions given in Chapter I. will serve as illustrations. The converse statements that if ¢(#) steadily increases with w throughout any range, ¢’(#) cannot be negative for any value of « belonging to this range, and that, if d(#) steadily decreases as x increases, $’(#) cannot be positive, follow immediately from the definition of ¢’ (@). Again, even if $’(x) vanish at a finite number of isolated points, provided it be elsewhere uniformly positive, ¢ (x) will steadily increase. Suppose, for example, that ¢’(#,) =90, and that with this exception ¢’(#) is positive in the interval from x= x, tO © = 4, where 4, >, The least value of ¢(a#) cannot then occur within this interval, or at the upper extremity (c©=a,). It must therefore occur at the lower extremity (w=2,). Hence p (22) > ) (x). The same conclusion is arrived at if $'(#) is positive from “=X, to =a, except for r=«a,, where it vanishes. ay; oul = \ 47-48] APPLICATIONS OF THE DERIVED FUNCTION. 103 In the same way, if ¢’(«) vanish at a finite number of isolated points, but is otherwise negative, ¢ (zx) will steadily decrease. Ex. It y =tan «— a, da we have °F — sec? x —1 = tan? a. dx Hence dy/dx is positive, except for «=0, 7, 27, .... Hence y steadily increases with x throughout any range which does not include one of the points of discontinuity (# = 37, $7, ...). Tt easily follows that the equation tan «—«=0 has no root between 0 and 37; one, and only one, root between 4m and 37; and so on. These results may be verified by a graphical construction. If we draw the lines y=tane, y=2%, their intersections will determine the values of a which make tan c=2. 48. The Derivative vanishes in the interval be- tween two equal values of the Function. If @(#) vanish for s=a and «=b5, and if ¢(z) be finite for all values of « between a and 6, then ¢'(z) will vanish for some value of # between a and 6. For, either ¢ (#) is constantly zero throughout the interval from a to b, or it will have (Art. 25) a greatest or a least value for some value (#,) of « within this interval. In the former case we shall have ¢’(#)=0 throughout the interval; in the latter case $'(a#) cannot be either positive or negative (Art. 47) and must therefore vanish, since it is by hypothesis 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 on p. 34; also Fig. 13, p. 27. 104 INFINITESIMAL CALCULUS. Meeks Th d (x)= (x — a) (x — b), we have d' (x) = 2a —(a + b). Hence ¢'(x) vanishes for «=3(a+6), which lies between a and. 0. | | sin a Pie ae cae d (x) = aa x COS # — Sin x& we have Pe) — aan Here ¢ (x) =0 for «=m and #=27; hence ¢’(«) must vanish for some intermediate value of x This is in agreement with Art. 47, where it was shewn that the equation «= tana has a root between a and 37. It is to be carefully noticed that, in the above demonstra- tion, the conditions that ¢(#) and ¢’(«#) should each have a definite (and therefore finite) value throughout the interval from x=a to «=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. Fig. 32. A slightly more general form of the theorem of this Art. is that if ¢(#) has the same value (8) for =a and #=b, then under the same conditions as to the continuity and. finiteness of ¢(w) and ¢’(«), the derived function ¢’(#) will vanish for some intermediate value of w. This follows by the same argument, applied now to the function ¢ (a) — B. 49. Application to the Theory of Equations. If $(«) be a rational integral function of a, then (a) and its derivative ¢’(x) are both of them continuous (and finite) for all finite values of a Hence at least one real root of the equation 48-49] APPLICATIONS OF THE DERIVED FUNCTION. 105 will lie between any two real roots of PAU Or Ie: pis ecte eke set (2). This result, which is known as ‘Rolle’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). He. 1. Tf (x)= 40% — 212? + 18x + 20, we have d (x) = 12a? — 420 + 18 = 6 (2” - 1) (w— 3). Hence the real roots of ¢ (x) = 0, if any, will lie in the intervals between — « and 3, 3 and 3, 3 and + w, respectively. Now, for By 3, +0, the signs of ¢ (x) are — +, -; +, respectively, so that ¢ (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 ¢ (#). xH=—-D, 4 t ' 4 1 ’ ' ‘ ‘ t ' ' t ' t ! 1 t { t ‘ ' r ‘ { ‘ t 1 t ' ' '‘ ! ‘ If by continuous modification of the form of $(a), for example by the addition or subtraction of a constant, two roots are made to coalesce, the root of ¢'(#)=0 which lies between must coalesce with them. Hence a double root of $(z)=0 is also a root of $’(x)=0. 106 INFINITESIMAL CALCULUS, [CH. TIL More generally, an r-fold root of ¢(x)=0 being regarded as due to the coalescence of 7 distinct roots, the equation d’ (z)=0 will have r—1 intervening roots which coalesce. This suggests a method of ascertaining the multiple roots, ‘if any, of a proposed algebraic equation. If a be an r-fold root of ¢(«x), we have tb (@) = (8 —4)! 1a) oe (3), where x («) is a rational integral function. Hence p (a) =(@ — a)" {ry (@) + (@ — 4) xX (@)f + (A); 1.€. (e—a)'— will be a common factor of $(a) and ¢’ (a). And it is easily seen that (#—a)’ will not be a common factor unless ¢(x) is divisible by (#—a)’. Hence the multiple roots of (a), if any, are to be detected by finding the common factors of ¢(#) and ¢’(«#) by the usual alge- braical process. Ti 2k db (x) = x4 — 9x? + 4a + 12, we have Pp’ (x) = 403 — 18a + 4. The usual method leads to the conclusion that #— 2 is a common factor of (x) and @’ (x); whence we infer that (w— 2)? is a factor of @(x). The remaining factors are then easily ascer- tained; thus we find f (x) = (« — 2)? (@ + 1) (w + 3). Ex. 3. To find the condition that the cubic e+ galt 7 = 07... 1 ae ee (5) should have a double root. 3 The double root, if it exists, must satisfy 3a°+g=0 or 2 =+ /(— sq) (6). Substituting in (5), we find r=+32j/(—4¢), or ¢=—-770 ee (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 @(#) is a maximum for 49-50] APPLICATIONS OF THE DERIVED FUNCTION. 107 “=x,,if two positive quantities, « and e’, can be found such that ¢(#,) is greater than the value which ¢ (x) assumes for any other value of # in the interval from w=a,—€ to e=2,+e. Similarly for a minimum. Since the comparison is made with values of the function in the immediate neighbourhood only of a, a maximum is not necessarily the greatest, nor a minimum the least, of all the values of the function. See Fig. 33, p. 105. We will limit ourselves for the present to the case, which includes all the more important applications, where ¢(#) has a determinate and finite derivative at all points of the range considered. The argument of Art. 47 then shews that if f(x,) be a maximum or minimum, ¢’(#,) cannot differ from zero. For if it be either positive or negative, there will be points in the immediate neighbourhood of #, for which ¢ (x) will be greater, and others for which it will be less, than (ax,). Hence, in the case supposed, a first condition for a maximum or minimum value of ¢(w) is that ¢$’(«@) 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 «, there is a certain interval throughout which ¢’(z) is altogether positive or altogether negative*. Now if ¢’(a) be positive for all values of # between 2,—e and 2,, (x) will (Art. 47) steadily increase throughout the interval thus defined; and if ¢’(x) be negative for all values of w between a, and a,+¢€, $(«) will steadily decrease throughout the corresponding interval. Hence if both these conditions hold, ¢(#,) is a maximum. And it is evident that if the signs be otherwise, ¢(#,) cannot be the greatest value which the function assumes within the interval extending from x,—e€ to 4+. We may express this shortly by saying that the necessary and sufficient condition in order that (a) may be a maximum value of ¢(#) is that $ (#) should change sign from + to — as & increases through the value a. * That is, we exclude cases where ¢’ (x) changes sign an infinite number of times within any interval including x,, however short. The point «=0 in the function 2? sin 1/x is an instance. 108 INFINITESIMAL CALCULUS. [cH. 111 In the same way we find that the necessary and sufficient condition in order that $(a,) may be a minimum value of o(«) 1s that ¢'(#) should change sign from — to + as & increases through the value ay. 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. Ex. 1. The distance (s), from an arbitrary origin, of a point moving in a straight line is a maximum when the velocity (ds/dt) 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 s=ut—tgt? dete u—gt. a ols AP g Hence ds/dt changes from positive to negative as ¢ increases through the value u/g. ‘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 2a, the lengths of two adjacent sides may be taken to be « and a — is hence we > have to find the » maximum value of the function C(G = X) cies. voeean eee ee (ly); The derivative of this is a — 2x, which changes sign from + to — as x increases through the value 4a. 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) = £e8 — 21a? + 180420 (2). We have p(x) = 12 (2-4) (8 - 3)" (3). This can only change sign when « passes through the values 4 and 3. Now when «@ is a little less than 4, the signs of the second and third factors are —, —; whilst when & is a little greater than } they are +, —. Hence as x increases through the value 4, ¢’ (x) changes sign from + to —. In a similar manner we find that as « increases through the value 3, ¢’ («) changes he 50] © APPLICATIONS OF THE DERIVED FUNCTION. 109 sign from — to +. Hence ¢ («) is a maximum when # = $, and a minimum when «=3. If we substitute in (2) we find that the maximum value is 244, and the minimum value —7. See Fig. 33, p. 105. Ex. 4, If | p(x) = Lag ccc (4), 2(1-2 we find d' (#) = ay Sea eee (5). This can only change sign for «=+1. As & increases (alge- braically) through the value — 1, 1 — a? changes sign from — to +. As x increases through +1, 1 —-«* changes sign from + to -. Hence for x =—1 we have a minimum value —1 of ¢(«), and for «=1amaximum value 1. See Fig. 17, p. 31. 2a ex. 5. If dp (a) = ‘ioe Alsace im ole dietete oltie ele nlcllatsrenaiss (6), i) 2 we have d' (x) = aa Bt Mites. +: Veet (7). Here q’ (x) is always positive, and the function ¢ (a) has no finite maxima or minima, See Fig. 16, p. 31. zx. 6. To find the right circular cylinder of least surface for a given volume. If x denote the radius and y the altitude, the surface is Qa? + Inay, and. if the given volume be 27a, we have ey Hl, Hence, eliminating y, the expression to be made a minimum is a? > 5 Oe | the derived function of which is aire 2(w- “): This changes sign as « increases through the value a, and the change is from —to +. Hence «=a makes the surface a mini- mum; and since y then = 2a, the height of the cylinder is equal to its diameter. _ The reader may verify that with these proportions the surface is 1-:1447...0f that of a sphere of equal volume. 110 INFINITESIMAL CALCULUS. [CH. III Whenever the derived function ¢’(#) vanishes, the rate of increase (Art. 33) of the original function ¢$(z) is momentarily zero, and the value of («) is said to be ‘stationary.’ As already stated, a stationary value is not necessarily a maximum or minimum, for cases may of course occur in which ¢’ (x) vanishes without changing sign. Ex, 7. The simplest instance of this is furnished by the function ch (00) 0. eter ee (8). This makes ¢’ (x) = 3x*, which vanish s, but does not change sign, as « increases through the value 0. Hence ¢(«), though ‘stationary,’ is not a maximum or minimum for x=0. Fig. 34 shews the graph of the function «*. ny. In most cases of interest, the derived function ¢’(@) is ea 50-52] APPLICATIONS OF THE DERIVED FUNCTION. 111 continuous as well as determinate (and finite). 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 q¢’ (#), though generally continuous, becomes discontinuous for some isolated value of «; and if the discontinuity be accompanied by a change of sign as 2 increases through the value in question, we shall have a maximum or minimum, by the same argument as in Art. 50. Ex. If (eas OU te OE Nes forsee al (1), we have i a 5(*)' obs eee eee (2). As « increases through the value 0, this changes from —@ to +0. Hence ¢ (x) isa minimum for#=0. See Fig. 35. Y x’ fo) x Fig. 35. Again, in Fig. 32 there occurs a point where ¢q’ (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. III Ex. 1. Thus, in the problem of Ex. 2, Art. 50, we have x (a — x) = ta? — (x — ha)”. | Since the last term cannot fall below zero, this expression has its greatest value (4a?) when x = $a. fix. 2. The expression 2a? — 3x + 2, may be put in the form 2 (a? — $a + 1) =2 (ew —#)7 4-2. Hence the expression has the minimum value 4, corresponding to ge 2. Again, the solution of many important problems comes at once from identities such as ay=t {(etyl—(a—y/y} ....seceee. (1), (2 + y= (@— YP HABY 00sec cee enceceeeee (2), e+y=t {(x +yP+(a—y)} ...... ees (3). Thus: The product (xy) of two positive magnitudes, whose sum (a+) 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 2a, 2y be the sides, we have to make xy a maximum subject to the condition that «? + y?= a’, where a is the radius of the circle. Now Jay = 0 + y? — (# —y)? =a®—(x—y)?............ (4), which is obviously greatest when «=y. Hence the greatest inscribed rectangle is a square. Ex. 4. To find the minimum value of a'cot:0 +} tan 0 ....c.ce (5), for values of 6 between 0 and $7. The product of a cot 6 and 6 tan @ is constant, hence their sum is least when they are equal, 7.e. when tan = (a/b)... 20. pitee ee ee (6). The minimum value of the sum is therefore 2a#6}. 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 be generated by the revolution of the curve about the axis of x, then if 2 be the length of the axis, and « the abscissa of the end of the cylinder nearest the origin, the volume of the cylinder is Ea ven ae o(,_¥" 8 my? (h—-x)=7y? (h fa) ce (8). ams Now the sum of the quantities y? and 4ah — y? is constant ; their _ product is therefore greatest when they are equal, 7.e. when 9 Soe SOF) Qt Ab acadaepuns ve eves e's (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) $ (#) =(#—a)” (w — 6)”, 24 ab (2) $(2)=log oy» 6) 4@-2 9-9), 2. Prove that the curves y = x4 — 6x? + 9a’ + 4a - 12, | and y =e — x — 32° + 5x —- 2, touch the axis of x, and find where they cut it. Trace the curves. 3. Prove that when « increases through a root of ¢ (x) =0, | $(a) and ¢'(x) will have opposite signs just before, and the | same sign just after the passage. Does this hold in the case _ of a double root? 2 gett, tor a> 27> 0, $ (x) =——a, L. 8 114 INFINITESIMAL CALCULUS. [CH. Ill a? and, for «>a, p(x) =a—-—a, whilst for «=a, (x) = 0, prove that ¢ (a) and ¢’(«#) are continuous from «=0 tow=a. Trace the curve y = ¢ (2). 5. Prove that the expression (x—1)e*+1, is positive for all positive values of a. 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 $=acos ne. 7. Find the maxima or minima of the function at — 892 + 2997 — 245 8. Prove that the function 2a? — Ba? — 362-410 is a maximum when «=—2, and a minimum when «=43. 9. The function 4a°— 1822+ 27”%-—7 has no maxima or minima. 10. Find the stationary points of the function ae — Sat + 527 +1, and examine for which of them the function is a maximum or minimum. 11. Prove that the function 102% — 120° + 15a* — 202? + 20 has a minimum value when x=1, and no other maxima or minima. 12. Prove that the function a (aa at) is €a Maximum when x= 3a. (a +x) (b+2) is a maximum when # = ,/(ab), and a minimum when «=— ,/(ab). 13. The function 52] APPLICATIONS OF THE DERIVED FUNCTION. 14. Prove that the function (a — 1)? (x +1)? has a maximum value ,4,, and a minimum value 0. 15. Prove that the expression lixvta l-xv+2 has a maximum value 3, and a minimum value }. ene incion 7 es at — a? +1 _ has a maximum value 2, and a minimum value — 2, 2] 17. The function ge (isah) xt — a + 1 has two maxima, each = 4, and two minima, each =— 4. 18. Prove that cos 0 + sin 6 is @ maximum when 6 = dr. 19. Prove that sin (@—a) cos (6 — f) is @ Maximum or a minimum when 6=4(a+f)+4r+hnz, according as n is even or odd. 20. Find the maximum ordinate of the curve y =xe~*, Trace the curve. 21. The curve Ga vor we has a minimum ordinate — -3678.... Trace the curve. 115 22. Prove that the ratio of the logarithm of a number (x) to she number itself is greatest when «=e, 23.. Prove that the expression a cos 6+ bsin has the maximum and minimum values + ,/(a? + 6”). af 3 116 INFINITESIMAL CALCULUS. [cH. 11 24. Prove that if a> ob the expression acosh « +b sinh x has the minimum value ,/(a?— 6”), but that if a ey mh eo 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 2, y, z.... We have ua Bx + 5 by + 52 82 + se riadtg (13), 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 ¢, due to errors 6a, 66, dC in the observed values of the two sides and the included angle, is to be found from 3 (0) = 1 ba SEI 5 EO (1), which gives cdc = (a —b cos C) da + (b — a cos C) 86 + ab sin C8C, or dc = cos Bda + cos Ad) +a sin BOC .......scceeceneees .(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 C, we have BSC) SILC Beate ce Saxe bho oe (3), whence log A =log $ + loga+ logb + logsin@ ......... (4). Hence, differentiating, dA da db py tN Mies IEA eo COU COU. natures ee (5). This gives the ‘ proportional error,’ ¢.e. the ratio of the error (dA) 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 dw in terms of da, dy, 62, .... 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. Lee ens. if Ranta PGT Rate he gm ape age where a, y are given functions of a variable ¢, we have, ultimately, du op da dd by Bt. du St Dy dt SP ts ee (2), du db dx , ab dy or di du di’ dy dt mie ietelelcfeta el ate’s! scm siete (3). This may be applied to reproduce various results obtained in _ Chap. 1. To conform to previous notation we may write Y= (u, v), 140 INFINITESIMAL CALCULUS. [CH. Il where w, v are given functions of x; the formula (3) then takes the shape dy dpdu | 0 dw Ts = Bat i IP ag ae slong at stare eerste veneane (4). Thus, if oh (U, 0) SUN... ceca eee (5), we have dp/du=v, Ap/dv=u, : d(uv) du dv and therefore da dat dye (6), in agreement with Art. 37. Again, if fb (u,v) =" oe te (7), we have dp/du= vu, dp/dv =u". log u, by Art. 35. Hence 7 ae a ey BES 2 oe dv a 6 Je OUT earaas log i (8). 2°. Again, if y be an implicit function of w#, defined by the equation b (&, y) = 0.22. ..3. ee (9), then differentiating this equation with respect to #, we have Op Ga Oe Cee oc da Oy dae. oe CP OY _) ae (10). 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 4 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 6a in the observed elevation is dh =a sec? ada. If a=100 feet, a=30°, and the error in the angle be 1’, prove that 64=°47 inch. 62] APPLICATIONS OF THE DERIVED FUNCTION. 141 3. Ina 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 (x, a’) of a point on the axis of a lens, and of its image, from the lens, are connected by the relation u a 1 eel Be a ea f! prove that the longitudinal magnification of a small object is (a /x)?. 5. Verify the theorem of Art. 56 in the case of o («) =a8, 6. Prove that if ¢@(«#) be continuous and differentiable, except for «=2,, when it becomes infinite, then ¢’(a,) 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, 6 is given by OS da h 5b Pte Metres ie 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 sind 0a 6b dc ~ sin B sin C aioe SaGecs 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 pNee om CuOd) eer 0 kOl A a@ asinB asin@’ Also, verify this result geometrically. 10. Ifa triangle ABC be slightly varied, but so as to remain inscribed in the same circle, prove that fa Shin, OGe ie cos A cosB cosC 142 INFINITESIMAL 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 6W, dW’ in these weighings is : bs VOW éW’ a. WWW Wee 12. A crank OP revolves about O with angular velocity o, 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 w.OR, where # 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 » 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 like manner from the results of Examples 14 and 16, above? 22. The polygon of m 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 Baar can be sent is a cylinder 2 2 feet long and 4 feet in girth, and that its volume is 2°546 cubic feet. — CHAPTER IV. DERIVATIVES OF HIGHER ORDERS. 63. Definition, and Notations. If y be a function of w, the derived function dy/dax will in general be itself a differentiable function of 2 The result of differentiating dy/dx 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 d/dx as a symbol of operation, the first, second, third, ... nth derivatives may be denoted by FB OOL: Ge -Y; ae © Yy wee (7 -Y; respectively. The more usual forms are TUE eg) ha eh d”y which may be regarded as contractions of the preceding, although (historically) they arose in a different manner. Again, writing D for d/dx, as in Art. 31, we have the forms Dy, Dy, Dye If y = $ (2), the successive derivatives are also denoted by $2), $'(@), 6°)... 6 (@). Occasionally it is convenient to adopt the briefer notation / y'; y", y a oe y™, r 63] DERIVATIVES OF HIGHER ORDERS. 145 There are a few cases in which simple expressions for the mth derivative of a function can be found. The more im- portant of these are given in the following examples. He 1. If a Ee ME lt SO I oe Os GL); we have Dy = A, +2A.~+ 3A,27+...4mA,0" 4, yy = 2.14,+3.24A,+... +m (m—1) Aya”, Peta, Pee eee eee sees eee ese Fees e ese FFF eee eese eee seer eeeeeseneses D™y =m (m—1)(m—2)...2.14,, and therefore Whe SPEAR IO A6 ('6- GC lence | Pret PEARS (3). Hence the mth derivative of a rational integral function of the mth degree is a constant, and all the higher derivatives vanish. aoen, ~1f poo OER BER CEE BEE (4), we have Diype keer yee. 2. ; and, generally, EASE SI ond Sar AAR Sa rd ey (5). Hence, putting 4=log a, we have DFAG = LO ONS Cran Evan neces one enews elds (6). ix. 3. If Di Ty otc Sryct AR On tiene (i); wehave Dy= BeosBx, D? mea D*y = — B cos Ba, Liye 5) Beste Gay and so on. Otherwise, we have Dy = B sin (Bx + 47), and therefore D*y = 3? sin (Ba + dx + $7), and, generally, De Tfies: Fe PALL AGG ita TUT hy hess kee dele #0 8s (9). Hx. 4. If TEESE SE obey Sater CROCE OT PETER OF (10), we have Dy=—fsin Bx, Dy =— B’ cos Ba, (11) D'y= fsinBa, Dty= B*cosBu,\ : and so on, Or, Dy = B cos (Bx +47), whence D*y = B? cos (Ba+ 42+ 47), and, generally, = Dy = B™ cos (Ba + $277) voeceeessseeseeeens (12). L, 10 25 146 INFINITESIMAL CALCULUS. [CH. IV Heo Ds eT oe 00S BU 1. cee epee 625.8" we find Dy =e” (a cos Bu — Bsin Bx), > Bia (14). D*y = e* {(o? — B?) cos Bu — 2a8 sin Ba} Similarly, if Y =O SiN BI..ceccreesers EMeeg ne (15), we have Dy = e* (a sin Bx + B cos Bax) oe (16) D*y = e* {(o2 — 8?) sin Bx + 208 cos Ba} General formule may be obtained, in these cases, by putting a =r cos 6, B=rsiiG eee (17% This makes D.e* cos Bu =e (acos Bx —B sin Ba) = re” cos (Ba + 6), and by repeated application of this result we find D" e™ cos Bx =1"e™ COs (S710 eee (18). Similarly, DD". ¢” sin Ba = r"e™ sin (Ge 4 m0) ee (19). Hix, 6.5 TF Y = OG Wivacasc saa ane een (20), we have | Dy ea, Dy =— EF BPy=— l= feeds and, generally, 64. Successive Derivatives of a Product. Leib- nitz’ Theorem. If u, v be functions of x, we have by Art. 88 (20), D (uo) = Du 0+ De (J). If we differentiate this again, we have D? (wv) = D (Du.v) + D(u. Do). Now, by the rule referred to, we have D (Du.v)= Deu.v+ Du. Dy, D(w. Dv) = Du. Du+u. D0, whence D? (wv) = Du.v+ 2Du, Dv Ds... (2). 63-64] DERIVATIVES OF HIGHER ORDERS. 147 The general formula for the nth derivative of a product 1s n(n — ae +nDu. aie SE iD Venton (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 wv. Using the accent notation, we have D* (uv) = D®u.v + nD""u. Dv + — 2D) pen, Dut. TEAS) (Sg TONER wie Sry eee RE Cr Er (4). Differentiating this again, D? (uv) = uv + wv’ +we + uv" SiO DUO UD a5 00b conten oes (5). The next differentiation gives D*(uv) = w'"'v + 2u''v' + u'v" + wy’ + Qu'v" + wo" ....044..(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 ~~ Duv)sw'v + 3u’v' + 3u'v" + Ww” oc ce eee (7) It appears that the numerical coefficient of the rth term in (7) is the sum of the coefficients of the rth and (r— 1)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+0; and since the coefficients of D(uwv) are the same as those of the first power of a+, it follows _ that the coefficients in the expanded form of D"(uv) will be the same as those of (a + 6)”. el, If gsc al npee neh ee othe Ain pee iar (8), we have Dry =x2D"u + nDx. Du BSG Bla) gOS 8 sca Tee a ae (9), since D’x=0. re 148 INFINITESIMAL CALCULUS. [cH. Iv Thus if y= 0 SID! Bott ae ee (10), we have Dy =xD* sin Bx + 2D sin Ba =— S'asin Bz + 2B cos ae (11). Again, if Y = LOS .escereeceveseereesenes (12), we have Dy = xD" logx+nD"" log « a oa b =(-)"- eae)! Dun +(e oe - 1 by Art. 63 (20). Hence n—2)! D"y =(-)" ( ae ite ee. (13) Re 22 kos y= 60h, «isc oe (14), we have | DX —. 402 dD D ax D1 n(n — 1) D?. av D"-2 y=". D'w +n. Der. 6 5 eae. uU+ =e™(D"u + naD"™-*u + hee oF DP eee ee (15). Thus, if ye sin Bx... ee (16), we have D*y = & (D* sin Ba + 2aD sin Ba + a? sin Bx) = e*{(a” — 6’) sin Ba + 2a8 cos Ba} ......... (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 formule ds dv v= Wi ) a= dt veo eerceresee ere veeees ( 1) Hence, in the present notation, we have 4 (it). 2 sen dt\dt/ dt? t.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 dw dO dt = dé @eeoeee eee eee eeesreeseeseeooe ene (3). Ex. 1. If s be a quadratic function of ¢, say ES Wy 2b Talon Oa ener rrr een. (4), ds we have ae 2At +B, d’s wat DA Cee aaPe es case weer tiee cavers (D), i.e. the acceleration is constant. Hx. 2. In ‘simple-harmonic’ motion we have Birch COGN SLG a ©) ever Olea Wee to tre okt (6), ds : whence a ne sin (nt +), 2 & =— N74, COS (nt + €) =— NS... ...eseeee (7), i.e. the acceleration is directed always towards a fixed point (the origin of s) and varies as the distance from that point. ero. If a= A cosh nt +B sinh Nbc .0is0% 0.08.08 (8), we have a = nA sinh nt + nB cosh nt, ihe Pea aes > qua cosh nt + n7B sinh né = ns......... (9); a.e. the acceleration is from a fixed point, and varies as the distance, EXAMPLES. XVIII. Verify the following differentiations : y= (1-2), Dy = 2 — 120 + 1227. y = tx? (J - 42), D’y = p (l- x). Y = gypx(l-ax)(P+la—2*), D°y =hpux (x — I). Y = gape? (3P — Ala + 20°), Dy =4py(x— 4). ca gl ed es 20. INFINITESIMAL CALCULUS. (cH, IV -1 Jute 1 1 Y= sy Dy =3 |e Be) 1 1, 24 (1 — 10a? + 5x4) Ese) OY ee 2a > 40 (a + 3) YS ae Dy = (i ae y =(1—2)-", Dy =m(m+1)...(m+n—1) (1—a)-™™. l+a . 2.7! es ee YY aay Fos we, Dy = 2 cos 2a: 9) = COS" GO, D"y = 2"-| cos (2a + 4nz). a == RCC a, Dy = 2 sec? x — sec x. Y= 0 SI, D°y = 4x cos # — (a? — 2) sin x, y = sin? x cos a, D*y = 6 — 60 sin? « + 64 sin‘ a. y= sin «sinh x, D?°y = 2 cos x cosh a. y = Cos « cosh a, Dy =—2sin «sinh «, y = sin « cosh a, D*y = 2 cos x sinh a. y = cos x sinh a, D*y = — 2 sin x eosh x, —amn-l oe a : y sn es Dd ae The first five derivatives of tan x are 140, 21+), 2(14+3°)(14+#), 8¢(24+30)1+?), 8 (2 + 15+ 15/4) (1+#), where ¢= tan x. 21. 22. 23. 24. x — 3 o y= log—, Y= 10s Hz, ate a Y= Xe"; aT pe. ee ge? Dy ae (-)""? £ Dry = (x +n) &. Dy = \x? + Inx +n (n — 1) e*. 66] DERIVATIVES OF HIGHER ORDERS. 151 25. By applying Leibnit~ Theorem to the differentiation of the identity gin 2 ge = git - prove that , r(r—1 r(r—1)(r—2 My + 1 Mp_y NM + — ‘ 2 ) Myp—gNq + ( ee Gs ), Mp3 is + 00 +N, = (M+ 1)p, where m,=m(m—1)(m—2)...(m—r+]). 26. The equation oo hE + nts =0 is satisfied by s = Ae~* cos (ot + €), for all values of A and e, provided ee 1 ne d’s ds Ta BS ge 27. The equation pt tats 0 is satisfied by s=(A + Bt) en”. « 28. If n=6(), y=x(0, dx dy dy@« Cy dtd? dt dt dai ee (ai 66*. Geometrical Interpretations of the Second Derivative. prove that In Art. 56 an important property of the derived function was obtained by a process which consisted virtually in a comparison of the curve Pee eee eta bees (1) with a straight line Ba ELE Waene weet arth es tcaed +s (2), the constants A, B being determined so as to make (1) and (2) intersect for two given values of a. We proceed, in a somewhat similar manner, to compare the curve (1) with a parabola Ti eats RR or eR 04 ctaak ese ri a re (3), * Arts. 66, 67 can be postponed. 152 INFINITESIMAL CALCULUS. [cH. IV where the constants A, B, C are determined so as to make (1) and (3) intersect for three given values of a. 1°. We will first suppose these values of # to be equi- distant; let them be a—h, a, a+h. The equations to determine the constants are then A+B(a—h)+C(a-—hy=¢(a—h), A+ Ba + Ca? = (0), eeee eee ae (4). A+B(ath)+C(at+thP=¢(ath) Let us now write F(@®)=d{(a)—-(A+ Bay?) (5), i.e. F(a) denotes the difference of the ordinates of the curves (1) and (3). By hypothesis, £’(«) vanishes for «=a —h, and for =a; hence, by Art. 48, the derived function F’’ (2) will vanish for some intermediate value of a, that is EF (a—0,))=07 ee ene where 1>0,>0. e@Again, since f(x) vanishes for «=a, and for c=a+h, we shall have FE’ (a+ 0,)h)=0..5 8 iene a neta (7), where 1 > @,>0. By a further application of the same argument, since the function F’ (x) vanishes for s=a—6,h and for c=a-+ 0,h, its derived function F(x) will vanish for some intermediate value of a; we have therefore F" (a+ th) =0 (ee (8), where @ is some quantity lying between —@, and @,, and ad fortiort between +1. Since, by (5), EY (a) = b (0) —20 Bee (2 it follows that, for some value of 8 between + 1, hb’ (a+ Ch) = 20 i ee (10). Now from (4) we find b(a+h)—2¢(a)+¢(a—h) =2CP ...... (11), and therefore p(ath)— t@) +f (42) ge oh ae 66] DERIVATIVES OF HIGHER ORDERS. 153 Hence ee a) =” (a)...(18). In the,same way we could prove that (a+ 2h) 26 (ath) +$(a)_,, eee a ean Cb yest ee (14). lim; —» lim;_, If the difference $ (a +h)— 9 (a) be denoted by dy, the expression {p (a + 2h)— $ (a+ h)} —{6 (@ +h) — 4 (a)} may be denoted by 6 (dy) or oy. Hence the formula (14) is equivalent to To interpret the theorem (13) geometrically, let, in Fig. 42, : OA=a, OH=a-h, OH’=a+th, and let AQ, HP, H’P’ be the corresponding ordinates of the curve (1). Join PP’, and let AQ meet PP’ in V. Then VA=1(PH+PH) =}${d(at+h)+¢(a—h)}, 154 INFINITESIMAL CALCULUS. and therefore VQ=VA-QA =2{p(ath)—2¢ (a) +6 (a—h)} Hence the eoea (13) asserts that VQ=ths’. OG) ae (16y ultimately. | It appears that the chord is above or below the are according as $”(a) 1s positive or negative. 2°. We will next suppose that two of the three points at which the curves (1) and (8) intersect are coincident. More precisely, we suppose that for «=a the curves not only intersect but touch, and that they intersect again for c=a+h. The conditions that, for c=a, y and dy/da ~ should have the same values in the two curves, are A+ Ba+ Ca? = ¢ (a), B+ 2Ca= ¢’ (a) while the third condition gives A+B(ath)+C(at+thyP=¢(ath)...... (18). With the same definition of F’(#) as before, we have F(a) =0, To . cpa ease cence (19), and therefore F' (a+ 0,1) =0 oe (20), where 1 >6,>0. Again, since £” es vanishes for «=a, and for e=a+ 6, h, we have EY (a + Oh) = 0. sateen (21), where 0,>0>0. Now from (17) and (18) we find p(at+h)—(a)—hd' (a)=Ch? .........(22). Hence, by (9) and (21), | d(ath)=¢(a)+hd' (a) + th? $” (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. 7 Or 66] DERIVATIVES OF HIGHER ORDERS. 15 From (23) we deduce a+h)—¢(a)—hd¢’ (a i _ GDI a AOMTED Cerny hm,=9 ~-——-—_—,, —_—_—— = 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=d(ath) VH=¢(a)+ hd’ (a). Fig. 43, Hence (24) asserts that Ue aL AD) cloths ec see sess (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 $” (a) +0, QV does not change sign with h, and the curve in the immediate neighbourhood of P lies altogether above, or altogether below, the tangent line, according as gp’ (a) is positive or negative. The formule (16) and (25) have an interesting appli- cation in the theory of Curvature. See Chap. x. 156 INFINITESIMAL CALCULUS, [CH. IV 67. Theory of Proportional Parts. Let us make the curves and y= A+ Be +0. coe (2), intersect for «=a, w=a+zh, c=at+h, where 1 >z>0. We find (1—z)d$(a)+2¢6(at+h)—d(at+zh)=20 —2)VC...38); and consequently, by the method -of the preceding Art., (1-2) $ (a) + 2b (a +h)— $ (a+ zh) =$2(1— 2) hg” (a + Oh) 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 (a) is a function which has been tabulated for a series of values of w 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>z>Q. In the method of ‘proportional parts, the interpolation is made as if the function increased uniformly from «=a to c=at+h, ve. we assume (at zh)—p(a)_2 (5) G4) 6) 0) or d(a+z2h)=(1—2) $ (a) +26 (at+h)...... (6), The formula (4) gives the error involved in this process, which is equivalent to assuming that the are of the curve (1) between =a and «=a+h may be replaced without sensible error by its chord. The maximum value of z(1— 2) is 4, by Art. 50, Ex. 2. Hence if A denote the greatest’ value which ¢” (#) assumes in the interval from «=a to «=a +h, the formula (4) shews that the error 67-68] DERIVATIVES OF HIGHER ORDERS. 157 Ex. 1. Ina seven-figure logarithmic table, the logarithms of all numbers from 10000 to 100000 are given at intervals of unity. Now if GENE OS Gi COA aon. Rane mae mPa e (8), we have p” (x) =— . nos aa prayer (9). Hence, putting h=1, in (7), we find that in the interpolation between log, and log, (z7+1) the error involved in the method of proportional parts is not greater than 05429+n? ..... Reena neice (10). Thus for 7=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 ¢$” (x) is large. The differences are then said to be ‘irregular.’ 1 ee AOA GIT oo cs vs,nesrad gidto dente (11), we have re AGl feanes PACOSOC CLIO 5 sVeh vs eaices oa oe (12). Hence, putting ce raaD ~ -000291, we find th>p” (x) =— 00000000460 cosec? @ ......... (13). Since cosec? 18° = 10°47, 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. Concavity and Convexity. Points of In- flexion. Just as $’(#) measures (Art. 33) the rate of increase of d(x), so $” («) measures the rate of increase of ¢’ («). Hence if $” (x) be positive the gradient of the curve Tee Tg CS aos Pear SOP Ee EON (1) increases with #; whilst if 6” (@) be negative the gradient decreases aS # increases. If $’(«)=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,’ we. a point at 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 ¢’(#) is greater than at P. Hence, by Art. 47, the value of $”(x) 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 #”(a#) at P cannot be positive. It follows that the curve is concave upwards when ¢” (2) is positive, and convex upwards when ¢”(a) is negative. This result may be inferred also from Art. 66 (25), which shews that QV has the same sign as 6” (a). It appears, moreover, that at a point of inflexion, where the curve crosses its tangent, $” (~) cannot be either positive or negative, and therefore (since it is assumed to be finite) must vanish. This condition, though essential, is not sufti- cient. It is further necessary that ¢” (~) should change sign 68] DERIVATIVES OF HIGHER ORDERS. 159 as « 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 eurve both to the right and to the left, in the immediate neighbourhood of P, at which the gradient is greater than at P, ze. 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 ig a maximum at P, and ¢”(#) changes from positive to negative. ee 1. If [Tee NO a Rare Re enn Bega (2), we have ae One ‘This changes from — to + as increases through 0. Hence we have a point of inflexion ; see Fig. 34, p. 110. Ex, 2. ES oe ote CR LS Pert eee (3). » _ 4a: (a? — 8) ~ (L+aiy * This makes 160 INFINITESIMAL CALCULUS. [CH. IV Hence there are three points of inflexion, viz. when «=0 and when == ,/3. See Hig. 17, ‘p. 31. Ex, 3. In the curve of sines ie we have y =—— sin -=—-. 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 Yy = 08s ic eee race (5), we have yf” = 12a, This vanishes, but does not change sign, when «=0. Hence we have a stationary tangent, but not a point of inflexion in the strict sense. It is in fact obvious, since a* 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 ¢(#) can also be expressed in general in terms of the second derivative $” («). Since $” (a) is the derivative of ¢’ (@), it appears that if, as x increases through a root of ¢' (z)=0, $" (a) 1s posite, ¢’ () must be increasing, and therefore changing sign from —to+. Hence ¢(2) is a minvmum. Similarly, if 6” («) is negative when ¢’ (x) =0, ¢’ (#) must be decreasing, and therefore changing sign from + to — Hence ¢ (a) is a maximum. The connection of these results with the criterion of concavity and convexity (Art. 68) is obvious. He. 1. In rectilinear motion, the distance (s) from the origin, is a@ maximum or minimum when the velocity (ds/dt) vanishes, according as the acceleration (d’s/d¢t?) is then negative or positive. Hx. 2. Let d (x) ——. 68-70] DERIVATIVES OF HIGHER ORDERS. 161 We have seen, Art. 50, Ex. 4, that ¢$’(#) vanishes for «=1 and «=—1. Also from the value of $’(«) given in Art. 68, Ex. 2, it appears that ¢'(1)=-1, ¢’(-1)=1. Hence the former value of x gives a maximum, and the latter a minimum, value of d(x). See Fig. 17, p. 31. It may happen, however, that a value of # which makes ¢' («)=0 also makes $” («)=0. It 1s easily shewn that in this case @(#) is in general neither a maximum nor a minimum (cf. Fig. 34, p. 110), but it 1s hardly worth while to continue the discussion here. The complete rule will be given later (Chap. XIv.) as a deduction from Taylor’s Theorem. 70. Successive Derivatives in the Theory of Equations. ; The successive derived functions play a great part in the Theory of Equations. We have seen (Art. 49) that, if ¢ (x) be a rational integral function, at least one root of ¢’ (#) = 0 will occur between any two roots of ¢(x)=0. Similarly, at least one root of g(x) =0 will occur between any two roots of ¢’ (x) =0, and SO on. Moreover, since an 7-fold root of ¢(#)=0 is an (r—1)- fold root of ¢ (x)=0, it will be an (r—2)-fold root of _ (x)=0,..., and finally a simple root of 6° (x) =0. Hence _ the necessary and sufficient conditions for an 7-fold root of @(a)=0 are that the functions eee) DA) jn veep DNL) ost ner ses (1) should simultaneously vanish. Ex. It (x) = 20° + Sat + 4a3 + 2a? + 204 1, | we have ' (x) = 10a: + 2023 + 120? + 4a + 2, | $” (x) = 4 (1008 + 152% + 6x + 1). These all vanish for 2=—1, which is therefore a triple root of $(«)=0. We find, in fact, that f (x) = (a + 1)? (2a?-a + 1). > 162 INFINITESIMAL CALCULUS. [cH. IV EXAMPLES, XIX. Prove that, in a table of natural sines at intervals of 1’, the error of proportional parts never exceeds 0000000106. 2. Shew that in a table of natural tangents the method of proportional parts fails for angles near 90°. Also prove that the limit of error for angles near 45°, when the tangents are given at intervals of 1’, is 0000000423. 3. Shew that in a table of logtangents the method of proportional parts fails both for angles near 0° and for angles near 90°, Shew also that the maximum error involved in the method is least for angles near 45°, 4. Prove that the curve y=logx is everywhere convex upwards. 5. Prove that the curve Y= 10l es is everywhere concave upwards. Trace the curve. 6. Find the maximum ordinate, and the point of inflexion, of the curve of =a. Trace the curve. [The maximum ordinate corresponds to w«=1; the inflexion to #=2.] 7, Shew that the curve y= e-® ; : 1 has inflexions at the points for which w= + ap ; and trace it. 8. Find the maximum and minimum ordinates, and the inflexions, of the curve y= ae, Trace the curve. [The maximum and minimum ordinates are given by w=+ ,/$; the inflexions by «=0, + ,/3.] DERIVATIVES OF HIGHER ORDERS. 163 9. A certain function ¢() is constant and ke LEP 7 el = 3 (0° —a’) : a for 0b. Prove that ¢(«) and ¢’(«) are continuous, but that ¢’ (x) is discontinuous. Trace the curve y= ¢ (a). 10. Shew that y =x? (3—2) has an inflexion at the point (1, 2). Trace the curve. 11. Shew that y = a7 (1 — 2:7) has inflexions at the points (35 5a) . ‘Trace the curve. 12. Find the points of inflexion of the curve be 2 EC [e==a/,/3.] an 13. Shew that Yy = (x—a)? has a point of inflexion at (— 2a, —2a). Trace the curve. 14. Find the points of inflexion of the curve : a | IO Pa? , and trace the curve. fe=0, + a,/3.] 15. Shew that the curve l-« vege? has three points of inflexion, and that they lie in a straight line. Trace the curve. 16. Prove that the equation a — 10a? + 15a—6 =0 has a triple root. 1l—2 164 INFINITESIMAL CALCULUS. [CH. Iya 17. Prove that the equation x® — 5a? + Sat + 9a? — 14a? - 4 + 8=0 has a triple root ; and find all the roots. 18. If PN, P’N’ be two neighbouring ordinates of a curve y=(x), and if MH, any intermediate ordinate, meet the chord PF’ in J, prove that QV=1NH. HN’. 4" (0), ultimately, where c is the abscissa of some point between WV and 19. Shew that in the formula b (ath)=¢ (a) +h¢' (a + Oh) of Art. 56, the limiting value of 6, when / is infinitely small, is in general 4. 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 difference of the differences of successive entries). CHAPTER V. INTEGRATION. 71. WNature of the Problem. In the preceding chapters we have been occupied with the rate of variation of functions given @ priort. 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 where ¢(#) is a given function of x, subject to the condition that for some specified value (a, say) of w, y shall have a given value (6). For example, the law of velocity of a moving point being given, and the position of the point at the time ¢é, it is required to find its position at any other time ¢ This is equivalent to solving the equation ds eens Shee, Ga Oa een eer eee 2 a =$(i) (2), where ¢ (¢) is a given function of ¢, subject to the condition that $= 8, (say) for ¢ = ty. If we can discover a continuous function y(a#) such that (a) = 6 (#), 166 _INFINITESIMAL CALCULUS. [CH y the equation (1) becomes Hence if, as is the case in most practical applications of the subject, y be restricted to be continuous, we have, by Art. 56, where C' is a constant. The precise value of C is indeter- minate, so far as the equation (1) is concerned; Cis therefore called an ‘arbitrary constant.’ Its use is that it enables us to satisfy the remaining condition of the problem as above stated. Thus if y=) for «=a, we must have b=W(a)+C, whence y—b =p (@)— a @).. eee (5). kx. Given that the velocity of a moving point is w+ gi, we have S =uU+ gt= - (ut + £90?) vi... een os. (6), whence g=ut + $90? + Co imeee (i. Determining C’ so that s=s, for t=¢,, we have S$ —s =U (t-t)) + hg (€7— ty?) ...ceebee eee (8). If, as in Art. 31, we use the symbol D for the operator — d/dx, the equation (1) may be written and its solution may, consistently with the principles of algebraic notation, be written y= D6 (#2) (10), the definition of the ‘inverse’ operator D™ being that D{D“${2#)| =¢ (@) ee ee (11). The function D6 @).... (12), when it exists, is called the ‘indefinite integral’ of ¢ (#) with respect to a. It is more usually denoted by 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 C. 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 D~ (a) or fd (a) dx of any given continuous function ¢(zx). 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 (in a finite form) in terms of the functions, whether algebraic or trans- cendental, which are ordinarily employed in mathematics. The following are instances : ae sin x dz [e da, [= dx, Ce =) ; and the list might easily be extended indefinitely. | INFINITESIMAL CALCULUS. [cH. Vv ‘ 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, differing by a constant. The student should make himself thoroughly familiar with the following results, which are fundamental : @ ah ano, [erde= same, (AY [except for n=—1], | d 1 dx 7p OS B= Ee, [= log «, (B) a . Ek& = Ipeke | eh dan : el (C) dx 7 kas sin # = cos &, | cosada=sin#, - (D) © cosa =—sin L, [sin ada = — cos a, (L’) Se . tan & = sec? 2, i sec? ada = tan 2, (F ) . cot # = — cosec? 2, i cosec? da = — cot «, (@) d ie 1 da ey ie aa sin ee Gea | cece = sine (11) d 1 vee da. «2 Aan dg ¥80 ae Ora’ laverzere (1) = sinh a = cosh 2, | cosh dt Se Gye ee cosh z = sinh 2, i sinh «dz = cosh a, (K) * Ags to the question of sign, see Art. 41. ° 72-73] INTEGRATION. 169 4 _tanh # =sech? a, | sech? «dx = tanh a, (L) S .coth #« = — cosech? 2, | cosech? dx =—cothaz, (MM) if gee 1 afin Ne oes 2s Fp’ sinh AaACE (+ a)’ | iD) Beas — = Sfp Aa (N) a ad cosh ar [ager pron da’ a (#—a*)’ J V(a—a?*) : = log? t=)» (0) a d St ae da u Poe dx as a @—a’ a? — a? ae oo a _1),,0+2 ge) Toy G—2’ (P) d x a da 1 x a pote 2’ | ae oth™ : mia | L— a [a? > a?] Por! ae (Q) 73. Simple Extensions. To extend the above results, we first notice that the addition of a constant to « makes no essential difference in the form of the result (cf. Art. 89, 1°). Thus, obviously, on em [@+a) Cad cea Ayres Jaza7 heer Mt Aon Sere ee (2), mene 0 fon =| Ta eray sin ....(8), and soon. Some further illustrations occur in Arts. 74, 75. * As to the sign, see Art. 44. 170 INFINITESIMAL CALCULUS. [cH. Again, if # 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 | sin kada=— : C08 he so ck See | (4), da 1 3 apa) 7g OB e+) vette GB), and so on. Again, we have the theorems f Cuda = O fade. ...cc.cccceces.1.. (6), f(utot+twt...) de=fude« + fode+fwda+...... (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 Aa + Ayo +... bt Apa (8), 1 at A amt + = Ayo +... + 4Am 127+ Am®......(9). Again, suppose we have a rational fraction of the form ME) By division this can be reduced to the sum of a rational integral function and a fraction The former part can be integrated as above, and the integral of (11) is A log (6+ 0) i. (12). Ex.1. f(a—1)8de=, : 5 1)i+1 = 2 (a1), dx Pas. ar 4 log (2x —1). Ex. 3. fin? a dx =4f(1 — cos 2a”) dx = 4a—4sin 2a. Ex. 4. ftan? x dx = [(sec’ «— 1) dx = tan a — a, 73-74] INTEGRATION. 171 eo 1 ee = filg?+t cial EA 2 SRY kx, 5. lea fee +4044 See dx EXAMPLES. XX. i Lyd 1,2 1 = Gu + 3 + Gx + zy log (2% 1). Find the indefinite integrals of the following expressions * : 1 i peat (Sn): 1 aaa \s 3. («—1)%, Gants F 1 1 J(2+2)? f(38 — 22x)’ 1 —2x 24+a te Ste Coa , ite 1-3 l-aw’ l+e2 besarte ay Lf 13. (cos x#— sin x)”. 15. tanh?a, coth? x. 2. 4, 1 1 esl Qe-1y 1 l+z2 NE AE l+a l+a Yd a? (+5) (e+5) e+—), L+— x a in a" Ute Lae cos? 2, cot? x. cosh? aw, sinh? a. fay am om+1 l—-« l+2 - _— l+2a l-«z 74. Rational Fractions with a Quadratic Denomi- nator. We next shew how to integrate any expression of the form a + pa +q eeeneoe where F(z) is rational and integral. ee el) If necessary, we first divide the numerator by the denominator until the remainder is of the form az+b. We thus get the function (1) expressed as the sum of a rational integral function and a fraction ody he ane ia ean (2). * 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 The form of the result will depend on whether p? = = 4q. If p? < 4q, we have a+ pu+g=("+op)+(q— ap) =(@—ay + B, where a, 8 are real. Now ik eae 7G or Ss se eae by an obvious extension of Art. 72 (1). If p?= 4g, we have w+ pe +q=(e@+ 3p), dx alt and (@+ip) eth If p?>4q, we have a choice of methods. In the first place, writing e+ pet+q=(a+ spy — Gp —-g)=(@-ap— where a, 8 are real, we have by Art. 72 (Q) 1 ee ao ae ae i] Le af = 98 log 2 aE ale Savala eral etetetakenene (7).* 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 o + pe +q=(e—a)(x—B), where a=a+B, pf’ =a— 8. With a proper choice of the constants A, B we may then put 1 he! B (8) (c—a)(a@—P’) “w—a@ 8 9 * It is assumed that t>a+ 8. The modifications necessary in other cases may be easily supplied. 74] INTEGRATION. 173 viz. this will be an identity, provided RA Yeas 8 He (ie 0) od eccke oe (9), i.e. provided Ab), Ap + Ba =—1)...::....... (10), 1 1 or aera B” ae, Eyres Uhh): Hence uF da zoe {lee - lsh (a—a’)(w7—f') a—P Va-a jes 1 : : rors Law loe a—a’ =— Bi 8 oR chetereelesis\sjaie vicieiaisis'- (12), 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’, and afterwards put =a’. This gives the value of A. Again, multiplying both sides by #— §’, and afterwards putting « = 6’, we find B*. du da 2 =F Ex. 1. et eee es 173 i y} eee — da dx jh ta ae 2 eee (Qaeslyaueo Og 1: da da a+i Kx. 3. Se ceactoe Pe TY hee ee eee -1 2 hPa. aaa ere 2 tanh Otherwise, assuming 1 A se (I—a)(@+2) Ie" a+2? * Hence the simple rule: To find A omit the corresponding factor in the denominator of the expression which is to be resolved into partial fractions, and substitute a’ for x in the expression as thus modified. Similarly for B. 174. INFINITESIMAL CALCULUS. [CH. Vv we find, by the method just indicated, A=}, B=. dx 1 ? Hence } os oes log (1 — x) + 3 log (a + 2) e+ 2 =} log >—. Proceeding to the more general case (2), we observe that, by a proper choice of the constants A, w, we can make ac +b=n (20 +p) + ws -tcree ae (bop viz. we must have | A=4a, p= b—40d-. eee (14). Hence [—2. de=r ey do + w [= (18) v+ px+q a? + pu + q a+ pu + q Of the two integrals on the right hand, the former is obviously equal to log (a? + pe +) ..s1..ceen eee (16), and the latter has been dealt with above. Les a 4—4(2x-1) Ka. 4. fF aieee e ree. dx ° se dx 4 2a—1 ae 44 =e ye tan™ te 1 ee "coe ) gee When the denominator can be resolved into real factors the integral on the left-hand side of (15) can be treated more simply by the method of ‘ partial fractions.’ Thus, we have ax+b A iB @=a)\(a-8) enc oe eoceccces (17), provided ak+b=A(«a—-')+B(a-a’, ue, provided A+B=a, AP’ + Ba'=—-b........... (18), ‘+b ap’ +b | A Set B= 7 7 19 or Aa: Be (19) iv | A i= Ver 74] INTEGRATION. 175 It is unnecessary, however, to go through this work in every case, as the values of A, B can be found more simply by the artifice explained on p. 178. The integration of (17) then gives Goo dx = A log (a—a')+B log (x—f’)...(20). Ex. 5. To integrate eee x. ov, Ol 2 (a — 2) (aw +1) . : ‘ A B Assuming that this = aD Puranas we find A=t, B=. The required integral is therefore 3 log (« — 2) + $ log (a + 1). EXAMPLES. XXI. 1, ie dx = tan « + log /(1 +2). 2 e. dx=42?—-log /1+2 ' ae = 7v — og ,/( + x"), a0 2 2 3, iF ~~, da =— $22 — log J(1 - 2°, da - sip 4. a = tan (2x Ly 5 da loo 22+! aes oe tel ne 3a +7 eee ae She i St Fe 8 ep eee: Pe Ds c+! loo 97's + 2% +3) qoern 2 et eae 2 anal 8. Joan 228+ bos seal) are tan vom 9 x+l1 2 c—l [Gp deals @—)- 176 INFINITESIMAL CALCULUS. [CH. Vv. | ade... (w+4)? 0. fspeerB 8 ee ine [ecy@ry @ 8-3 le (x — 2) + 8 log (a — 3). 12, = dee = @ + (8,/5 + 1) log (22/5 -1) -- (2,/5 — 1) log (2% + ,/5 — 1). 13 eG ee ee pace: : pe v= 46) 5 iene ne gm -1 1— (~)" gem . 14. Ss dx =" — 407 + $05 -—... + (m4 ae ae qo ax +b Saint oe JAa® + Br+O. A somewhat similar treatment can be applied to functions of the above type. 1°. -If A be positive, the form is equivalent to axe +b Tet peg ce (1). Consider, in the first place, the form 1 JG pebg nee (2). By completing the square, the expression under the root- sign may be put in one or other of the shapes (a—a) + Now, by Art. 72, (1), (0), dx 1 Oa eae = sinh B Corre ecccnes (3), d . EE. and | HWenaic ai = cosh} = B et (a These functions have the alternative forms, loo VT AEM Ie 4 ERY : B d ro] INTEGRATION. 177 a—-at+ V(e?+ p2t+q) or log ars eR ges cee (5)5 ef. Art. 23. _In the more general case (1), we assume AG + D=N(HHED) TM rccreccecreceeer (6), which is satisfied by 7 1 ed hee 10 RPP PPP er (7). Hence ax +b x+hp dx —_—.—_—— dx = | ———*4_—, dx+ | fe Ve+patgy)— IN@+petqg) | "SV@+ pe+g) eit de (8). The former of these two integrals is obviously equal to (2 + pa + Q)s 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 SNR V(q+ pr— 2’) Unless the quadratic expression be essentially negative, in which case the function would be imaginary for all real values of #, it can be put in the shape 6? — (a —- ay. da 5. op — a N lan — —1 Oe, Oe 10 > ce N= @=a 7B aK In the more general case of the function ax +b er LE); V(q + px — x’) oy) | we assume At+D=NA(AP—L)+M...cecccceeeeee C12); or ASH — A, PHD. ARNG... cceeeee, (13). 178 INFINITESIMAL CALCULUS. = ~~ [ Hence eet by 3 | ee Es Kae pane) q+ pea) tH ee The former of these two integrals is equal to V(q + pu — 2), and the latter has been treated above. Ex, 1; | l+2 (e+4)+4 J(1l-—#- J -#- oe latths a +8 TER Heo =— (1 -a— at) +} sin! S48 = : 22 +1 ae Loy ga Le J(1 -#—-«#?) +4 sin RE Oh ay rr | (w+ )t+h_ ‘ererar oe J@+e+Io" +4 dai. 3 au+} [aang J(e +041) +041) J{(a+ 3) + F b = /(2?+ 041) +} sinh*——2 at 2 22+1 se. es 2 1 a es | Fan e Ja? +0e+4+1)+4 sinh Je Bes / (zo) a= [as -Saa=a-S= =sin- a +,/(1— «’). Te, en Ng | Oe eee Re . $ Hid ae Ap Sale Re Sia SARA 75-76] INTEGRATION, 179 EXAMPLES. XXII da hee a V ie sea (/32). dx LO ei WY ares a: | Jagan = 50 (,/2m). du . lee Tey we sin-1 (2a — 1). da S Seam Agel 5. [yee {3% +1 4 Jarseera a miss = oe _,3x-1 | Fire > 8 aL dx an x 8. | Jeazan= 0" (1-2). 2—*) , 24-4, 9. [JC ) die = Jf (a=2)} + Jasin™ no 2 10. [JE ~) d= sin” a — ,/(1 — x’) 11. WES (5) dix = ,/(a® —1) + cosh™ # 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 TiO MC Ah Ra renee 2 here ..(1) from # to t, where x is a given function of t, we have, by Art. 39, du __dud« da Fe pe ge PUB) ap eeveeeeeeee sees (2), 180 INFINITESIMAL CALCULUS. [CH. Vv and therefore, by the definition of the inverse symbol f, u=[6@ Gat Hence | [6 (@) do = [$6 (@) Godt cerrseceeene (3)* - Conversely, whenever a proposed integral is recognized to be of the form | d (u) AL. 0s eee eee (4), we may replace it by Sh (u) du ...c cee (5), which is often easier to find. The following are important cases : 1°. So(e@+a)da=fd(u)du............ (6), where u=@+4. 20, fb (har) der => fg (ti) Utbevevvesseee (7), where u= ka. These results have already heen employed in Art. 73. 3°. Sb (@) ada = 4) (u) du ..........20%... (8), where u = 2”. The following are examples of (8). de xx du 1 1 |e SS eee aa ae tau) * Hence the rule: After the sign | replace da by s dt. La 76] INTEGRATION, Bx. 2. [a> =t/s"7 bit) 181 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 Ga Lit, dx/dt = —1/t.* Thus | da: -loees dt ate 1 ‘i vf (a + a2) ti(a+t) aslv(?@+ a) =— u sinh at a Be ue te a x 1 i x a Pate +a) Similarly, i aos =— : cosh — x gk es a? aia 2?) * The substitution is equivalent to writing dt dx — — for —. t 2 dt mat)) 182 INFINITESIMAL CALCUL ie. [On ¥ da ee ie and lege = Vas ° So dachiniray (11). More generally, the integral eee da ————_—, = re 12 @t a) Nd + Ba 0) | a is reduced by the substitution z+a=I1/t to one or other of the forms discussed in Art. 75. Again, the substitution «= 1/t gives latan~—lae re -—laeery (a? + a) Jae +t>jF J (1 +a) ees @(1 + ae?) 1 x Bese 4 Cees tere er oeeeveerveees eee (18). ae u% Sunilarly i a -, alae) (14), du 1 x é and Neto =— a? (a — a?) sv ale ieleleyags)ateis (1 By) da fe a EEE OTE? aie ee The form (A+ Bato) (16) can, by ‘completing the square, be brought under one or other of the preceding cases. 77. Integration of Trigonometrical Functions. £2) fran ada = ee ” da COs & _ [d(cos x) _ =— log COS & COS @ = 10G SCC & iss snes gue eens eee (1). Similarly {cotada =log sin 7... (2). ¢6-77] INTEGRATION, 183 Again, by the same artifice, | sint , __ (d(cos 2) costa! jaca a ie oA (3) In a similar manner | — AGA = COSCO fics sstece eee ees (4). er. Art. 38, 2°. 90 [s da: f sina | 2sin 4a cos 42 oy + se? tadx [d(tan 4a) tan 4a tan 4a SOLO Ea eas ss ac eca screebcr yes tens (5). From this we deduce a -\aaeTD = log tan (f{7 + $2)......(6). The formule (1) to (6) rank almost as standard results, and should be remembered. 30 | dx: # dx: " Jatbcosa J(a+6)cos?4a%+(a—b)sin? da > sec? 4ada (7) EE DSe esp tant ta ee If we put tan 4x = u, this takes the shape { du 9 I EEG (8), and so comes under one or other of the standard forms (JZ), (P), (Q) of Art. 72. Similarly, with the same substitution, dx : du Jextame=? lap merae nO 184 INFINITESIMAL CALCULUS. dx setade is 10) Se eee a 4°. iF ens =|; = ...++- (10), 2 cos? w + b? sin? & + b? tan? If we put tan 7 = u, du d (bu) 1 _, bu we get fase 3 b Ja? +(buy =e j, *8 an a, ea SYD pee: tan an) «cate ee eee (11); The analogous results involving hyperbolic functions may he noted. We easily find ftanhadz=logcosha, f coth x dx =log sinha... (12), (eae dx = — sech x, [ Saeed =— cosech @...(13), cosh? sinh? x dix | nh p= 108 tam BO ceeveeerereese (14), - da e" dx es [aren ? [aw earl ays st co (15). Similarly the forms dx dx ey larvae can be integrated by the substitution tanh 4a = wu. 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 /(a? — x”) suggests the substitu- tion v=asin@, or «=atanhu; that of /(a — a*) suggests xz =asec 0, or e=acoshu; that of /(a + a?) suggests a=atan @, or «=asinhu. 77-78] INTEGRATION. 185 Ex. 1. To find ie) (Geert) 0G) eee sis a (i). Putting x=asin@, dx=acos 6dé, we find SJ(@ — 2) da = a*{cos? 6 dé = ha’J(1 + cos 26) dé = 4a? (6+ 4 sin 20) = 4a? sin™ = + 4x,/(a? — x) ...... (2). 2 2 Fx. 2. To find | eis Sard oe A At Ge (3). Putting x=asinhu, dx=acoshu du, we obtain the form fcoth’ u du, which =(1 + cosech? w) du =u — coth u “Ajit SPSL eRe (4) - ttttetteseese teres dx Ex. 3. To find | or . (L—2) /(1 — 2) If we put x=cos@, dx=—sin 6dé, the integral becomes dé - | “7 —2 | smzqo 008 39 o 1 + *) az G ery EXAMPLES. XXIII. 1 fs erie 1 1—2 1-2 | os [EE =p tana a 3. [7 * de = 4 (log x). 4, fsinxcosadx= sin? x 5. Na Cb =e (SiN t)%. eo J (k— #*) i: / 10. al. 22. Ce INFINITESIMAL CALCULUS. 6. O° | reins ards dx = log (1 + sin #), + sin x sin x 1 fe de =— Flog (a+b cos 2). fsin x cos*x dx =—4 cos* a. sin 2 COs & iE : B ccotoa Fae die = =.—— log (a cos? x + b sin? 2). a cos’ « + b sin? x 2 (b—a) ftan® «dx = 4 tan? x + log cos a. sin 7 : fax dx = + sect x 11. fsect «dx =tan «+ $ tan’ x. {(sec «+ tan x) dx = log € came? J (sec « — tan x) dx = log (1+sin 2). xv dix = == COSCO 0 CUb ars __—_—— = — cot%— cosec x. 1+cos x 1—cosz dx 0 ——_—.——. = tan @ — sec 2, =, = tan & + sec @. 1+sin « 1—sina dx RT MEAS Ren tan x — cot a. sin? x cos? x dx . | see ee los tan eae sin x cos? x 2 dix ————.,- =} sec’ x + log tan a. sin # cos? x | ade + Flos (cos + sin 2). pe ae : tan7 (= tan ne l+cos?a ,/2 J/2 fac /(u? + 22°) doe =} (a? + a)8, lawman Le ne ota + 1) oan 1 Evaluate f{,/(a?+a?)dx and f,/(a®—a*) dx by hyperbolic substitutions. JL +2" 23, ae 79] INTEGRATION. 187 edie Tae a” ,/ (a? — 2") ax x dec 25. Ja +2) = 4 (30 — 2a?) Ne tr x). 79. Integration by Parts. The second method referred to in Art. 76, viz. that of ‘integration by parts, consists in an inversion of the formula d dv du FE ee ae aie &\ sietslwie i's e\s.aielee/e-0 CL); given in Art. 37. Integrating both sides, we find dv du w= [uF d+ | oF de, whence [u dx = w — [e = OS elle ares te (2). This gives the following rule: If the expression to be integrated consists of two factors, one of which (dv/dz) is by itself immediately integrable, we may integrate as if the remaining factor (w) were constant, provided we subtract the integral of the product of the inte- grated factor (v) into the derivative (du/dx) of the other factor. A very useful particular case is obtained by putting @—2 in (2). Thus a Cee eia\. Jude SW ea | The following are important applications of the method. Ne flog ede = slog a — fo. = deo AHS OYE2 DS TAOS anne Ne (4). 2°, To find f/(a? — x?) da. * If we write v for dv/dx, and therefore D~'v for v, this takes the form D™ (uv) =uD~v - D~“! (Du. Dv), 188 INFINITESIMAL CALCULUS. [CH. V Putting w= /(a? — x*) in (3), we have ada | Viet —a8) da = 0 Maat) +] 7m (5). But [ V(a? — a”) da = hea dx: wda (Ge eas a?) /(a? ss V(a? — 2) 2d atk sin — aa rere Adding to the former result, and dividing by 2, we find Jv — x) d*«=ta/(0—a2)+4e sin .» ise cf, Art. 78, Ex. 1. In exactly the same way we should find i /(a2 + @) de = ha/(a+2)4407emh RIS s18 [vce — a?) dx = $4 /(a — a’) —4a? cosh -—.... (9). 3°. To find the integrals P= fe cos Badz, Q=fe*sin Badz..... (10). Putting u=cos Bx, V= fe in (2), we find Pao cos Be [er .(—Bsin Be) de 1 OX B = —¢ cos Ba t+ PO (11), Similarly Qa ee sin Be —|~ e* Bcos Bede 1 ak ot B | ES? =e sin Bx me ir tieescessees (12), 79-80] INTEGRATION. 189 Hence aP — BQ =e cos Ba, ‘is BP + al) = et sin Ba a6 Oh o/ae aw v win ove ; and therefore le cos Bada = P = 8 sin Ba + a cos Bx ast ee as ROS: le sin Bade = Q = asin Re Boos be ‘ 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 CD SIE ela iil ee cee PE CE), We have Un, = : et yr — | ; et nada id arerr — = DR ee ee ee Ae eA (2). If n be a positive integer, we can by successive applications of this formula obtain wu, in terms of Uy, = [ede, ae Dee hae ey. (3). a Ex. 1. Thus, if TEN IHS 1 bone er teers ie erere (4), we have Fete had maakt SY 7) | aa ane ence, SaaS (5). For example, u,=— ae" + Su, =— ae 7 + 3 (—- ae” + Qu) =— xe-* — 3a7?e-* + 6 (— xe + UH), or fate" das = — (2° + 3a + 6a + 6) e~*. 2°. Let Un = fa” cos Badz, (6) Pei abde | So We find Un = 2 sin Ba. a2” — | a sin Bx. nada B B Lins. n SSI O Are i Use sehen cents es Che) OU INFINITESIMAL CALCULUS. [CH. Vv and Un, = — F cos Ba. a” — | (- 308s Be) na” dar = = 5,008 B.A HF tg eens ieee (8). If nis a positive integer, these formule enable us to express Un and v, in terms of either wu, or v, which are known. Ex. 2. Thus, if B=1, we have Un =" SINL—NVy 1) Vn = — HL" COSU+ NUy_ 3 «ss (9): For example, u, = sinw—3v, = x sin «— 3 (— 2’ cos x + 2m) = «sin x + 3x’ cos a — 6 (wsin x —%,), or fa? cos x dx = (a? — 6x) sin x + (3a? — 6) cos a. Sy eee be u, = [tan 0d0 422. (10) = { tan” @ (sec? 6 — 1) dé = ftan” 6 d (tan 0)— f tan" 6 dé, i i tan” 0 =U,» 6 (11). we have Me n Hence if » be a positive integer, u, can be made to depend either on uw, =Jftan@d?,=log sec? or on Uj, = fae, = 2.0... (13), according as n is odd or even. : Similarly, if Un = [ cot” 0d’... eae .... (14), 1 =— r1G— olka einiolataloiea aitee . we find Vn See cot”! 8 — Un_» (15) 81. Reduction Formule, continued. 1°. Let Un = [ cos" 000.2 ae (1). We have u,=fcos" 6d (sin @). =sin 0cos”—6—/fsin 0.(n—1)cos"-?6.(—sin 0) d0 = sin 8 cos”—! 6 + (n—1) fC —cos? 0) cos”? 6dé 4 = sin 6 cos” 6 + (n— 1) (tn_2 — Un). 80-81] INTEGRATION. 191 Al t- posing, and dividing by n, we find n—1 iy Un—2 ( ) y successive applications of this formula we reduce the by 2 at each step; until finally, if n be a positive er, the integral wu, is made to depend upon either thy of COD COO se SID O05 cae ves'e cae (3), TH et Be To re ena CDR ding as n is odd or even. Eke Un =~ sin 6 cos" 6 + °. By a similar process, if eee GEL Dare ee. std eo) ste (er), a eee ee (6). n this way v,, when 7 is a positive integer, is made to nd either on n—1 nd m= — - cos 6 sin”! 6 + °. ‘The same method can be applied to the more general than =) BIL tO COST OULD 3 cscs esind onthe (9). e have =fsin™ @ cos” 6 d (sin @) = aise sin™+1 9 cos”—! 8 m+1 1 | = in™+ ee n—2 te 7/32 1@.(n—1)cos"? @.(— sin 0) dé = é sin™t @ cos” @ m+1l 2 m+1 sin” 0 cos"—? 6 (1 — cos? 0) dé Mee n—1 =a sin™+ eos”! 9 + weed (tm, n—2 — Um,n)- 192 INFINITESIMAL CALCULUS. - (Cray: Clearing of fractions, transposing, and dividing by m +n, we obtain —l m+n tie os Ug ; sin™+! @ cos" 6 +— ee : ., In a similar manner we should find : nu—l = ; Joe Soa marie sin™— @ cos”! 6 + mn Uitte AT). By successive applications of (10) and (11) we can reduce either index by 2 at each step, so that finally, if m, n ar positive integers, the integral wm,» 1s made to depend on on or other of the following forms: U,1, =f sin 6 cos 0dé, = $sin? 8 .. selene CO thy, 09 = | CO, = 0 so ncnn (13), ti, = | COS O00; = sn Ove ene Ree u, 9, = {sin 0d0, =— cos 0. (15). The investigations of this section are important as leading to some simple and practically very useful results in definite integrals. See Art. 95. ; EXAMPLES. XXIV. face* dsc = a (a2 — a) e*!", fa log « dx = 42? (log «— 4). a 1 m+1 (log 2 — 5) : foc sin « de =— a cos a + sin @. fa™ log «dx = fa cos x dx =x sin & + cos.a, Jasin « cos «dx =— «cos 2x ++ sin 2a, ese eT hee OO Mm SiN Mx COS NX —N COS Ma sin nx fcos mx cos nadx = a OO m —n : A nm sin mx cos nx — m cos mx sin nx 8. fsin me sin nada = rr EC EE m—n ‘ ee COS Mx COS NX + N SIN Mx sin aN 9. fsin ma cos na da = ot ep ae m — nr 10. fsin-'«dx=a sina + ,/(1—2’). * . 81-82] INTEGRATION. 193 11. ftantaedx=e tan x-— log /(1+ x’). weia, [sec™* x dx =2 sec"! a— cosh a. 13. fatan-? dx = 4 (1 +27) tan“ x— $x. pe 14. fausec? «da =x tan x + log cos a. m5. iE Samad dx = «tan 400, 1+cos2z ! asin-!2 At de 16. eye NF) sin le+a. 17. fcosh «cos w dx = } (sinh x cos # + cosh & sin 2). 18. fsinh x sin x dx = 4 (cosh x sin « — sinh & cos 2). 19. fcosh x sin w dz= 4 (sinh w sin « — cosh x cos 2). P20. fsinh x cos x daz = 4 (cosh x cos « + sinh x sin 2). 21. fe® sin x cos w da = 4, (sin 2a— 2 cos 2a) &”. 22. fare-*da = — (x + Bat + 20x? + 60x? + 120% + 120) e. 23. fat sin w da = — (a*— 12a? + 24) cos x + (423 — 242) sin a. 24, If U,= fa”. coshadx, v,= fa™sinh «dx, rove that u,=2"sinha—nr,_}, ,=2" cosh x—nuy_}. Deduce the values of wu, and 2. [w,= (at + 12a? + 24) sinh w— (42° + 242) cosh a, U4 = (a* + 122° + 24) cosh w — (42° + 24a) sinh a. | 25. If wu be a rational integral function of a, prove that vhere D = d/dz. 82. Integration of Rational Fractions. We return to the integration of algebraic functions. here are certain classes of such functions which can be reated by general methods. We begin with the case of rational functions. By (2 > dx Ex. i he To find loge <0 6 6's 6,0 oieieleteneieslorertaerreenerere (3). ] AB C We assume 2 (1 — a) = Fs aE a oi T=. stelelo:s sli e/cte alsis areTaae (4). If we multiply both sides by 1-2, and then put x=1, we get C=1. Again, multiplying by 2’, and then putting «=0, we find B=1. The constant A remains to be found in some other way. If we multiply both sides of (4) by x, and then put r=0, we find 4—~C=0, whence A4=1. An equivalent method is to clear of fractions and equate the coefficients of 2. Again, we might assign some other special value to «; for example, putting x=—1, we find —-A+B+}C=}], which, combined with the previous results, gives A =1. dx IB | 1 Hence 2 (1—2) (=+a+p=)# ‘ 1 = log x —~ — log (La) ieee (5). 22+ 1 Ex. 2. To find (e+ 2) (@—3) OY vsvauesetee seer (6). Assume eee A) ee “Ce (7+2)(#-3)? w2+2 «2-3 («#-3) The short method of determining coefficients gives —4+1 3 6+1 7 (2-37 705° = oa Also, multiplying by x, and then putting «=o, we find A+tb=0)" Of ope = 25° A= 83-84] INTEGRATION, 197 The integral is therefore — 3°; log (w+ 2) + 3 log (a — 3) — MCLE ee (8). Ex, 3. To find Ga Ty Nee ee (9) We recall Art. 76, 3°. Regarding «?/(x?+1)? as a function of 2’, we find (by inspection) oe (#+1)-1 1 1 re eel Aye oP G41)" Le x da a dat ee laect~ trey ] Kea es Hence = 4 log (a? +1) + 84. Case of Quadratic Factors. The preceding methods are always applicable, but if some of the roots of f(#)=0 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 into partial fractions, we have (a) for each simple factor #—a which does not recur, a fraction of the form (b) for a simple factor 2-8 which occurs r times, a series of r fractions, of the form B, B, B,. bee ered ee a el oh ae ash 3); eae G@=ay *@—ay 198 INFINITESIMAL CALCULUS. [cH. V (c) for each quadratic factor a+ pa +q which does not recur, a fraction of the form Ca + D KL" + Du + (d) for a quadratic factor #?+px+q which occurs r times, a series of partial fractions, of the form C,a + D, Ca + D, Ca + D, Che ee es oR ae a ee e+ pe+g (a+ pet q) (a? + px + q) 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 C,0+D, (x* + px +q)s can be found. The case s=1 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, p so that Ost De _, 2e+p : ie (7) (a +pet+q) (a +putq) (a +put+qy ae viz. we have A\=10,, p= D,-WO (8). The integral of the first term on the right-hand of (7) is r 1 sl @apeeg (9), and it remains only to find ea ele dt lage or aay ace'e wiare,Sinerelea alee (10), where t=¢+hp, ¢=¢—- Looe (11). * The investigation which follows is given for the sake of completeness, but it is seldom required in practice. The student will lose little by pose ing it. Another method of integrating expressions of the type (10) 3 indicated in Ex. 2, below. 84] INTEGRATION. 199 Now, by differentiation, we find d t 1 i? it (Pro (Prep 8) yep 2 1 (f+c)—c¢ Bega >) eee =~ (28-8) ey : Apel Sas ax mo (12). _ Hence, integrating, t dt (f+) sa Ps lear c)*- Sy ar easy 2s — Sf 1 het a cy 257 2)e (P+ yet 95 — 53 rar +0) ae or Returning to our previous notation, we have = ] c+ 4p | eeporay TE= DG) Ore + ae 2s—3 “2-1 @-) exge eres Cs which is the formula of reduction required. By successive applications of this result, the integral (10) is made to depend ultimately on which is a known form (Art. 74). Ew, 1. To find Raa t+? + 1 The denominator has here two quadratic factors, 2?+a“+1 and x’?—x2+1, which are not further resolvable. We _ therefore assume, in conformity with the above rule, 1 Av+B Ca + D (17) o+ot+l a+etl at—et 1 A or 1 = (Aw + B) (a? —a@ +1) + (Ca + D) (a? + + 1). Equating coefficients of the several powers of x, we have A+C=0, -4+0+6+D=0, A+C—B+D=0, BaD = 1; 200 INFINITESIMAL CALCULUS. [CH. V Hence A=—C=}, B=Detpi | (18). The integration can now be effected ae the method of Art. 74. We have x+1 laSn 24+] =i fae -3 [=e 2e+1)+1 (2%¢—1)-1 aha See! =] e+aet+1 fede a? —x2+1 ee =} log (2? +«+1)- eee me +1 “fey tfaxy ye: ran ye +a+1 773 _,2¢4+1 —) = #8 eel | 23" @+et+1 1 / 3a as SF —_——_————_ SN pe —1 =a ee. ehele 6p .@ 78/166 1016) et 6-8, te mar wR gp low (19) dx Ex. oh To find (142)? 0-0-0 eisle ee os el elaverelpiateretecete terete (20) This comes under (14), but may be treated more simply as follows. If we put ee tan, we get dx +a = Joos 6d0 = 4/(1 + cos 26) dé = 40+4sin 20 =t¢tanta7+4 i PR See (21). EXAMPLES. XXV. J da x 1 alo 7 98 (1-2) © f(1 a)" | 22+3 lon (x — 1)3 a (a — 1) CEE es ” a (a + 2)8 a dae SGN E= es = 4 log (w—1)— 4 log (wa —2) + 2 log (a —3). 84] INTEGRATION. 201 2x—3 = Ne (20 +3) = $ log (x + 1) — 7p log (w— 1) — 32 log (2a + 3). ah r= | da a 1 oe a— I] a4Saet—4° 3 (a+2) ® P+ 2° i i i a pee log oa) I e-NE=- J” BEI +O Gra Be ) + ayer es e-9) 8. eat) — eee leraere- r= Sarat 10. eee, : leeer a8 orl eel Oe dx e-l 32 i Vee loge oo on al | @ nip eas 81 Mi BV Eselle Sart PG Pag ees a 22. ree MS aca L 1-2 jee ee 25. | a - = Flog — oat ae 7g ton ag 26. BOY =} log ae a ‘ art <3 tan} =F 27. ile aa ca = 2 log (1 +2) — Flog (I +8) +} tana, 28. a eee log (1+ a) +4 log (1 + #) —4 tana, (l+a)(l+a*) ? ; - j 29. [arg = ts? | Pee eet 2 SP eel 28 foe 3 30. [pmo (+2) — flog (2 +1). ac” alae 1 o 1 o (a2 pe gaa 31. oe Pate) = 28 @-1)- blog @+1)— gE: a | e—xt+1 ie lap Btls ee ee 33. [Pap = torte 2 ae a da 1 + 22? at | eee ee 85] INTEGRATION. 203 VG od Save ae Jeter Qx(l+a) 2 ™ jp 1 a Mint 2 i fee at a SBOE G GINSES VaR RED Win i Ba ada 1—w,/2 +2 1 de. : firearzp 2 State a 273" Tw 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 TALES ag) bid keh Soma [ck Lae ee a ae (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 2. 2°, Any rational function of « and X, where A RI Ue) vet AC ude ences <0 (2), can be integrated by the substitution Ca Diba Oy OI t — 20D iors will tend, as & is diminished, to some definite limiting value S, in the sense that by taking & small enough we can ensure that & shall differ from S by less than any assigned magnitude, however small. If the function ¢ (x) admit of graphical representation, & will be represented by the sum of a series of rectangles, whose bases 86] . DEFINITE INTEGRALS. 209 ee IR) ade h, make up the range }—a, and whose altitudes are ordinates of the curve at arbitrarily chosen points in these bases. And the limiting value S, to which § tends as the breadths of the rectangles are indefinitely diminished, is known as the ‘area’ included between the curve, the axis of x, and the extreme or- dinates x=a, x=b. See Fig. 46. Y Fig. 46. The sum which we have denoted by © is more fully expressed by DU OREOL ED UR) O Bln vo01ea, and x a es Maxwell, Theory of Heat, c. v.; Rankine, The Steam-Engine, rt. 43, , ay is not implied that a mathematical formula for the integral can be ound. 14—2 212 INFINITESIMAL CALCULUS. [CH. VI that (x) steadily increases as x increases from a to Db. Consider any particular mode of subdivision jis laces ha eae (1), of the range b—”, we shall have Fig. 47. 87] DEFINITE INTEGRALS. 213 In Fig. 47 the quantity >’ is represented by the sum of a series of rectangles such as PN, and >” by the sum of a series of rectangles such as SN. Hence the difference >” —%’ 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 $(b)—¢ (a), and if k be the greatest of the bases, ze. the greatest of the intervals (1), we shall have SS IU aN Cd) ean ee (5). Now, considering all possible modes of subdivision of the range b—a, the sums %’, being always less than B ( —a), will have an upper limit, which we will denote by S’, and the sums }”, being always greater than A (b—a), will have a lower litnit; which we will denote by 8”, and it 1s further evident that S”« S’. It follows, from (5), that the difference S” — S’ must lie between 0 and & {f(b)— $(a)}; and since, in this statement, & may be as small as we please, it appears that S’ and S” cannot but be equal. We will denote their common value by S. Finally, it is evident that |\2—S|< 2”- >’ a. If ba. We have b b b | ude — | yde =| (eee In virtue of (4), every term of the sum, of which the latter integral is the limit, will be positive. Hence b b | yda < i UNL va cinca pte emer wee (7). 89-90] DEFINITE INTEGRALS. 219 b b Brrailerly | yda > i eer (8). If b f oh (2) dn 4, By dividing the range 5 —a into » intervals such that the abscissee of the points of division are in geometric progression, and finally making n infinite, prove that b 1 m = M+1__ qmti allis’ R i ae Oda ar (b a™*1), — (Wallis’ method.) 90. Differentiation of a Definite Integral with respect to either Limit. b Let r=[ VORP ee (1). Evidently, J 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 oe limit 6. We have [+6l= ae d (x) dx =| s()de+ | Ws (2), by Art 89, 2°. Hence ; b+8b sr= | 6 (x) de = 8b. Oe (8), by Art. 89, 3°. This shews that 6/7 vanishes with 60, so that J is a continuous function of b. Also, since él sp = 0 (6 + 8 0b) epee eeneewenae (4), we have, on proceeding to the limit (6b = 0), Fig. 53. In the figure, 0A =a, OB=6, BB’ = 8), and SI is represented by the rectangle having BJS’ as its base. 90-92] DEFINITE INTEGRALS. 221 In the same way, if we regard the upper limit 0 as fixed, and the lower limit a as variable, we find that J is a continuous function of a, and that dl Th =— d (a) Wieietevedsleleisieiets ececeieisveretars (6). 91. Existence of an Indefinite Integral. We can now shew that any function ¢(«), having the character postulated in Art. 87, has an indefinite integral, i.e. there exists a definable (but not necessarily calculable) function y(«) such that Alea (Gs Ve=iCoi i ure cscs ss .. cer are (1), or MCh Sd BS 02) a an ey (2). For if-we write whe | OWE pest (3), 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 Ae = Ge lagntiialtys ste res ceeen ike (4). The lower limit of integration in (3) is, from the present point of view, arbitrary, and the function yw (é) 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 [/$(@) de to the right-hand side. Cf. Art. 71. 92. Rule for calculating a Definite Integral. Whenever the analytical form of a function (x), which has a given function (a) as its derivative, has been dis- covered, the value of the definite integral 279 INFINITESIMAL CALCULUS. [CH. VI can be written down at once. For, if we regard a as fixed, we have, by Art. 90, | =r (0), ... cee reese enn ae (2), by hypothesis. It follows by Art. 56 that J and yf (0) can only differ by a ‘constant,’ ae. a quantity independent of 6; thus [¢ («@)de=1 Ye (3). To find the value of O we may, since it does not vary with — b, put b=a, whence ¥(a)+O=]|"$(@)de=0 ee (4). Hence OC =— (a), and 6 [-$@) @e=4O-FO vores) 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 D1™¢(a). The reason why this inverse function is usually denoted by CO Camere oe (6) is now apparent. ‘The form (6) is simply an abbreviation for ip (a) dei... (7), where a is arbitrary. We have seen that a change in a is equivalent to the addition of a constant. b The notation | 6) | voeesnss tear eet eeenenied (8) is often used as an abbreviation for ap (b) — (a). Ex. 1. To find | = DG iiss iss Aaa tea ee (9), Here $(0)=0, (2) =4e whence I & dx = f(b) — (a) =4 (8? —a) ............ (10). Sn te 92-93 | DEFINITE INTEGRALS. 223 Ee, 2. To find [eax ibe eee (11). Here (x) ae Wr (a) = 27, whence He Pe (00) ccc ces¥ i ont vans « (1 2). Fx. 3. To find I YE Be Geen Seren (13). Here b(2)=2", Y(a)=re, whencé i "ge Ges : Cee ter shee sais (14). The above results agree with those obtained, by much greater labour, in Art. 88. 93. Cases where the function ¢(«), or the limits of integration, become infinite. Before proceeding to further examples, it will be con- venient to extend somewhat the definition of an integral given in Art. 86. It was there assumed that the limits of integration a, b were finite, and also that the function ¢(«) was finite throughout the range b—a. We proceed to explain how, under certain conditions, these conditions may be relaxed. 1°. Suppose ¢ () to be finite and continuous for values of # ranging from a to «, and consider the integral | Tap Ea Veena ara (1), where w>a. If, as w is increased indefinitely, the integral tends to a definite limiting value, this value is denoted by | ; Ae SVE cc ose eee (2), The integral (1) is then said to be ‘convergent’ for w=o. As might be anticipated from the theory of infinite series (Art. 6) it is not a sufficient condition for convergence that [AVA coe cer ied REET Ui Pe Saar (3) ; this condition is moreover not essential, for there may even be convergence when ¢ (x) has no definite limiting value for =o, 224 INFINITESIMAL CALCULUS. — [cH. VI A similar definition of b [e 6 (a) de ry can obviously be framed. Fix. 1. [e-sde= [-ce-=] Mean (5). JO a 0 a As w increases this tends to the limit 1/a. Hence we say that | e-tde = — (6) 0 o da w Ex. 2. eis | log | = logs gc eee (7). This increases without limit with w. Hence there is no limiting value for w=, although lintge. : = Q). 0. tution ete epee (8). 2°. Let d(x) become infinite at or between the limits of integration. It will be sufficient to consider the case where there is only one value of # for which ¢(«)=o0. The general case can be reduced to this by breaking up the range b—a into smaller intervals *. If d(x) become infinite at the upper limit (only), we consider in the first place the integral b-e where e¢ is positive. If, as ¢ is diminished indefinitely, this integral tends to a definite limiting value, this value is adopted as the definition of i : OV A similar definition applies to the case where ¢(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. 220 Ex. 3. The function 1/,/(1 — x) becomes infinite for «=1, but [a5- -2 (1-2) |“ = 2-2 Je ae GLL); : and as e is indefinitely diminished this tends to the limit 2. Hence If ¢(x) becomes infinite between the limits a, b, say for #% =c, we consider the sum aro [_¢@ da rare la) If, with diminishing e¢ (and e’) each of these integrals tends to a finite limiting value, the sum of these values is adopted as the definition of iF h(a) doo. The cases where ¢ (a) becomes infinite, or is discontinuous, at a finite number of isolated points, are dealt with by dividing the range into shorter intervals bounded by the points of discontinuity. 94. Applications of the Rule of Art. 92. We give a few more typical examples of the evaluation of definite integrals. * Cases may arise in which each of the integrals [oo eae and fi gp (a) de a c+e is ultimately infinite, whilst, if some special relation be assumed between the ultimately vanishing quantities ¢, ¢’, 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. Ee 15 226 INFINITESIMAL CALCULUS. [cH. VI 37 dr Bok. ; sin % da = - cos ai = anon eee (i); O 4 1 ho ieee : | cos « de = | sin | eR yes ee (2) ; 0 0 | mit be fee i sin « cos x dx = E sin? x | Be EER Eres (3). 0 , 0 Ex. 2. By Art. 79, we have @ asin Bx+BcosBu _ | i e-eesin Bx de =—| aa e at aod ene asin Bw + B cos Bw a? + fe a? + B? : If a be positive the last term tends, as w is increased indefinitely, to the limiting value 0. Hence \ oo zc $ e B i} ee gin Bos dit = eon nnenee =e (A), Sein ik: | e-20 00s Birdie =" je (5). 0 Ex. 3. We have o d ray i nee = | tan | = tan w — tan7! 0. 0 lia 0 ‘ The function tan™+x is many-valued (Art. 21), but it is im- material which value we take, provided we suppose it to change continuously as « varies through the range of integration. Hence if we take tan-!0=0, we must understand by tan-?w that value which increases continuously from 0 with wo. As w increases indefinitely, this value tends to the limit 47, so that Ex. 4. By Art. 77, 4° we have DT carepcare Lam (Ja) | Now, as 6 increases from 0 to $7, ,/(a/8). tan 6 increases from 0 94] DEFINITE INTEGRALS. 207 to oo, and we may therefore suppose that tan! {,/(a/f). tan 6} increases from 0 to $7. Hence | 7 dé 7 if aim 0 1 Boot O= Tap) (7). 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. Ew, 5. ‘To find | Kas oa) daeTaicodetiyiel ts (8). 0 ve We found (Art. 78), putting «=asin 6, that J J(@ — 2?) dx = a? {cos’ 6 dé = 4a? (6 + 4 sin 26). Now, if @ increase from 0 to $2, x will increase from 0 to a. Hence fe J (a — x) dx = 1a? [oxy sin 26 |" ae yt eae (9). EXAMPLES. XXIX. 1 dz 1 da GE 1, the first part vanishes, since snQ=0, cos47=0. dr =a $0 Hence I cos” 0d0 = ~ : i COS aU OU tics. o.. (3). 0 nm JO Similarly, from Art. 81 (6), 40 | pra dud 0 230 INFINITESIMAL CALCULUS. [CH. VI If n be a positive integer, we can, by successive applica- tions of (3), express bar | cos” 6d0 0 in terms of either i avee eo) ee i di (5), according as n is odd or even. In the same way i a sin” 0d0 can be made to depend either on ip sin 0d = 1 or on ie Pe ea dG 0 0 Ex. 1. ie cos’ d6 = £ a cos® 6d6 0 0 4 2 an 8 After working 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 Air | cos 0d) = 5.3.4. daa aha, 0 The general values of the preceding integrals can be written down without difficulty. Thus, ifm be odd, we have - ee eae _(n-—1)(@=8)...2 . ik cos ado= | sin (00 = vats whilst, if m be even, ae Pat os gen ‘ _(n-1)(m-3)...1 7 ii s odo =| sink 000 — Integrals of this type are of frequent occurrence in the physical applications of the Calculus. 95] DEFINITE INTEGRALS. 231 ir 2S Ae epee rea i gin 8 008" Od ......++.-+.-»-(9), 0 we have, by Art. 81 (10), = 1 1) m4+1 n—1 He nn Festus 8 cos” @ : + n—Il eer Um, n—2 ++ (10). If n> 1, the expression in [-] vanishes at both limits, and we have 47 | sin” @ cos” dd = 0 n—Il m+n ir | sin” @ cos" @d@...(11). 0 In the same way from Art. 81 (11) we obtain, if m >1, Mm ae Lee hie es i sin” @ cos” 0d@ = | sin” @ cos"@d@ ...(12). 0 +nJo By means of these formule, 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: An 4 | Pa feos ddd: | jh ney 0 0 ; (18). | ” sin 7dé,=1; i cos 6d0,=1 0 0 Ex. 2. We have by (12). Again, by (11), [sin 6 cos’ 6d0 = 2 [sin 6 cos 0d6 = 2.4. ly 2 no 3 et 4 2 2 i a oe Hence i sin’ @cos*0d0 =4.%.2.2= 5h. After a little practice, the result can be written down immedi- ately. Thus lr 2 . « i sin’ 0 cos? @d0=% 3.4.23 ly. 0 . 232 INFINITESIMAL CALCULUS. [cH. VI- | The formule (11) and (12), as well as (3) and (4), are often required in practice, and should be remembered. ~ Again, the algebraic integral is reduced by the substitution # = sin? @ to the form hr 9 | sin?™41 9 C08" Od cecccceseee. (15), 0 and can therefore be evaluated by means of the formulz given above, whenever 2m+1 and 2n+1 are positive integers or null. Similarly, if we put « =sin @, the integral ip 2 (1— ade (16) takes the form : i : sin” @ cos™™**: 00.0 7 seems a) Ex, 3. i Be eWeey ries I ™ 1, oe dx n i f+ Ja +1)}" n?—1° [Put #=sinh wu. ] 15. Prove that : ly i (1 +2)" (1 — 2) "doe = gm+nes i ” sin®™+19 cos?) 9 db. 0 16. Prove that Tb a — 2s (a ae, i cosh®?u n—1 Jo cosh”-?u° [Put cosh w = sec @. | * 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 dr T i sin®*1 9d6 < i Pnin® 0d0. 0 0 Hence show that 47 lies between 222,.4.4.6.6...9%. 2n 1.3.3.5.5.7...(2n—1) (2n+1) and the fraction obtained by omitting the last factors in the numerator and denominator. ( Wallis.) 18. Prove from first principles that dir 7 i tan™*+} ed8< | tan” 0d0. 0 0 Hence, using the result of Ex. 5, shew that f * an" 60 0 lies between cine eh pele Cy ce CEN: 19. Shew that I mar Gipband i EL 0 0 are indeterminate. 20. Shew from graphical considerations that ” sin 6 i 7-4 is finite and determinate. 21. Prove that if ¢ (x) be finite and continuous for values of x ranging from 0 to a, except for 2=0, when it becomes infinite, the integral [ $e) ae will be finite, provided a positive quantity m can be found, less than unity, and such that lim,—9 x” > (x) is finite. [Put «=¢".] 238 INFINITESIMAL CALCULUS. [CH. VI 22. If $(x) be finite and continuous for all values of x, the integral [ ¢@ae will be finite, provided a quantity m can be found, greater than unity, and such that lim, 20%” p (x) is finite. [Put «=7-".] 23. Prove that I cos 22da and I sin @? dx 0 0 are finite and determinate. 24. Prove that ° oe) co i we dees | ve “dx =0. 0 —o 25. Prove that the integral co —72 ‘ ane da 0 (where > — 1) is finite and determinate. 26. Prove that, if n> 1, i are dx = £(n — 1) i a era 0 0 Hence shew that, if m be a positive integer, oO i ontle-P dar — 1 mt, G 97. If ia i are de, 0 prove that C= thn-1s n being positive. Hence shew that, if » be integral, n! atti 7 co i ae das = 0 DEFINITE INTEGRALS. 239 28. If f(x) and ¢ (a) be finite and continuous, and if ¢ (x) retain the same sign, throughout the interval from x=a to x=), then [ s@ (a) dx = f[a+ 0 (b—a)] [ ¢@ae where 1>6>0. © 29. Shew how it follows from the equality { nee = log x uae that the sum of 7 terms of the harmonic series L+4+4+... lies between log (n + 1) and 1+ log x. 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 f(x) steadily diminishes, as « increases from 0 to o, the series f (1) + f(2) + F(3) +. is convergent, and that its sum lies between J and /+/(1), provided the integral Vic I f (x) dx, 1 be finite. Apply this to the series iS eeehanere ties eo, (n +1)? (n+2)? (n4+3) ! oa EXAMPLES. XXXI. 1. Prove that if the pressure (y) and volume (v) of a gas be connected by the relation pv =const., the work done in expanding from volume »% to volume 2, is VY Por log —. Vp 240 _ INFINITESIMAL CALCULUS. [CH. VI 2. Prove also that if the relation be pv =const., the work done is = (25% — 1%): 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 » foot-lbs., where ~ 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 VIL. GEOMETRICAL 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 acase. ‘l'o 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 43 (ON.PQ). Now ON is less than the radius, and } (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 4a x 2ca, or 7a?, where a is the radius. Similarly we may shew that the lower limit to the area of a circumscribed polygon cannot be less than zra*, Moreover, the difference between the area of an inscribed polygon, and that of the corresponding circumscribed polygon, is represented by > (PN.NT), and is therefore less than > (PIV). ¢, where ¢ is the greatest value of V7. 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 7a’. In the same way we may prove that the area of any sector of a circle of radius a is $a°@, 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= (“) weld ca cee ee ere (1), the area included between the curve, the axis of «, and the ordinates =a, x=), 1s | : (a) da or | ; ye. 2), it being assumed that ¢ (#) is a function of the type contem- plated in Art. 87. This follows at once from the definition of the preceding Art. and the investigations of Art. 87. lix. 1. To find the area included between the curve the axis of #, and the ordinate =a; we have a M+1 7a Aa™t!} da= A] = ae [vy Ee 0 mri i where B is the length of the extreme ordinate. The area in 97-98] GEOMETRICAL APPLICATIONS. 243 question is therefore equal to the fraction 1/(m +1) of the area of the rectangle contained by the extreme ordinate and_ its abscissa. In the case of the parabola UO UE: ae Re eee eres a (5), we have m= 4, and the ratio is $. Ex. 2. The area of a quadrant of the ellipse ey a res Ue ia Mie ere ee (6) ¢ b f° is given by i y dx = “i a/ (a? — a2) dae. 0 0 The value of the definite integral was found in Art. 94 to be dra*. Hence the whole area of the ellipse is wad. Kx. 3. In the rectangular hyperbola eet Lemire Wats Src dews ot (7), we may put, for the positive branch, Me COSU tes eet) = SINN Uhre so. cs ssa ceee (8), since these satisfy (7), and give the required range of values of x and y. The area included between the curve, the axis of #, and the ordinate defined by the variable wu, is xv U U i y dz = | sinh? w du = a) (cosh 2u — 1) du 1 0 0 =4dsinh 2u—4u .......060.. (9). 244, INFINITESIMAL CALCULUS. [CH. VII This gives the area PAW in the left-hand figure. Hence the area of the hyperbolic sector AOP is 4PN.ON — area PAN = }hu............... (10). We have here an analogy between the ‘amplitude’ (w) of the hyperbolic functions cosh u, sinh uw, &c., and the amplitude (@) of the circular functions cos 6, sin 0, &c. ; viz. the independent variable in each case represents twice the sectorial area AOP corresponding to the point P whose coordinates are (coshu, sinh wu), or (cos 6, sin 6), respectively. In the case of the general hyperbola the coordinates of any point on the positive branch may be represented by a=acoshw, y= sinh@n ee (12), and the sectorial area is dab U. If the axes of coordinates be oblique, making (say) an angle w with one another, the elementary rectangles ydx which occur in the sum, of which the area is the limit, are replaced by elementary parallelograms yéxsinw; the area included between the curve, the axis of #, and two bounding ordinates is therefore given by taken between the proper limits. Fig. 57. 98-99] GEOMETRICAL APPLICATIONS. 245 Ex. 4. The equation of a parabola, referred to any diameter and the tangent at its extremity, is at Re iil | ahr alr eg (14). The area of the segment cut off by the chord x = a is therefore 2 sino [ yde = Aa! isin o | at de = $a'ta* sin w 0 0 PGS GIG) es Meee, oe ct eer lt te (15). Hence the area of any segment of a parabola is two-thirds the rectangle contained by the intercept (a) of the chord on its dia- meter and the projection (28 sin w) of the chord on the directrix. 99. On the Sign to be attributed to an Area. It was tacitly implied in Art. 98 that b >a, and that the ordinate ¢(#) is positive throughout the range of integration. If we drop these restrictions, it is easily seen that the integral a is equal to + S, where S is the area included between the curve, the axis of 2, and the extreme ordinates; the sign being + or — according as the area in question lies to the right or left of the curve, supposed described in the direction Fig. 58. from P to Q, where PA, QB are the ordinates corresponding _ to w=a, x=5, respectively*. If the curve cuts the axis of « between A and B, the integral gives the excess (positive or * Tt 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, (cH. VII negative) of the area which lies to the right over that which lies to the left. [ven with these generalizations, the Perr [o@de=48 Pe io (2) still applies in strictness only when there is a unique value of y, or ¢ (a), for each value of # within the range b—a. If however we replace #, as independent variable, by a quantity t such that, as ¢ increases, the corresponding point P moves in a continuous manner along the curve*, the formula 4 dx ip ya, ut ete é 4's. ,SReeeuarene erabepetemneanenener es (3) will give in a generalized sense the area included between Fig, 59. the curve, the axis of x, and the ordinates of the points P,, P, for which t=t,, t=%, 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 1t 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 taken between proper limits of ¢, 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 are s of the curve, measured from some fixed point on it. 99-100] GEOMETRICAL APPLICATIONS, 247 accordance with the variation of ¢*. 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 =A, is evidently given, with the same kind of qualification as before, by A i RE oS AN ROP ae (5). h The more general formula, analogous to (3), is n dy I, a dt Aber site 1 Sy oy Feeds 3h Bon en (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 aod te} One Bane ere Tete. (1), the area included between the curve and any two radii vectores 0 =a, 0=f is given by the formula B f02d0 or $f" {$6 (Pde 2) For we can construct, in the manner indicated by the figure, an including area S, and an included area S’, each built up of sectors of circles. The area of any one of these sectors is equal to 47°50, where r is its radius, and 60 its angle, and the sum of either series of sectors is therefore given by a series of the type 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 @>a and that each radius vector through the origin intersects the are considered in one point only. If however we introduce a new independent variable ¢, such that, as ¢ increases, the corresponding point P moves in a continuous manner along the curve, the ex- pression will give the net area swept over by the radius vector as ¢ varies from ¢, to t,, ue. the (positive or negative) excess of those parts which are swept over in the direction of @ in- creasing over those swept over in the contrary direction. Moreover if, as ¢ increases, P at length returns to its original © position, having described a closed curve, the expression taken between suitable limits of ¢, gives in a generalized sense the area enclosed by the curve; viz., 1t 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 y= 20 Sin 0 ..5 ese (6) (see Fig. 39, p. 126) is given by } i "2d0 = 2a? | "sin? 0d) =a or (7). 0 0 Ex. 2. The equation of an ee in polar coordinates, the — centre being pole, is 1” cos? @ 2. sin @ : oe = eres + he o's 8 ck acalepeee aeaeeraret eee (8). Tlence the area is dé Qa 4dr was 2#f2 : af a wanes Beara" d I, a” sin® 6 + b? cos? 6 100-101] GEOMETRICAL APPLICATIONS. 249 The value of the latter integral has been found in Art. 94 to be 47/(ab). Hence the required area is zab. 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 C. Let R, R’ be the middle points of PQ, P’Q’, and let RS be an arc of a circle with centre C. Then if the angle PCP’ be denoted by 60, we have, ultimately, area PQQ’P’ = AQCQ’- APCP = 10@.80 — 4CP*. 86 = PQ.4(CP + CQ) 60 = PQ.OR.86 = PQ. BS. Q’ Fig. 61. Hence, if we denote the length PQ by wu, and the elementary displacement of ft, estimated in the direction perpendicular to the moving line, by dc, the area swept over may be represented by The student will notice that the formule of Arts. 98, 100 are particular cases of this result. Thus in the case of Art. 98 (13) we have w= y, do = dx. sin o. It is tacitly assumed, in the foregoing proof, that the areas are swept over always in the same direction. It is easy to ee, however, that the formula (1) will apply without any uch restriction, provided areas be reckoned positive or nega- ive according as they are swept over towards the side of the ine PQ on which éo is reckoned positive, or the reverse. 250 INFINITESIMAL CALCULUS. (CH. VIL For example, the area swept over by a straight line whose middle point is fixed is on this reckoning zero. ; We will suppose, for definiteness, that dc is positive when the motion of £# is to the left of PQ as regards a spectator looking along the straight line in the direction from P to Q. If PQ return finally to its original position, its extremities P, Q having described closed curves, the integral (1) will, on the above convention, represent the excess of the area enclosed by the path of Q over that enclosed by the path of P, provided the signs attributed to these areas be in accord- ance with the rule of Art. 100. i Fig. 62. 102. Theory of Amsler’s Planimeter. A ‘planimeter’ is an instrument by which the area of any figure drawn on paper is measured mechanically. : Many such instruments have been devised*, but the simplest and most popular is the one invented by Amsler, of Schaffhausen, in 1854, This consists of two bars OP, PQ, freely jointed at P, the former of which can rotate about a fixed point at 0. If a tracing point attached to the bar PQ at Q be carried round any closed curve, P will oscillate to and fro along an are of a circle, describing (as it were) a contour of zero area. Hence by the * See Henrici, ‘Report on Planimeters,’ Brit. Ass. Rep., 1894, p. 496. 101-102] GEOMETRICAL APPLICATIONS. 251 Fig. 63. theorem stated at the end of Art. 101, the area of the curve described by Q will be equal to POLO Rents dns eine th coy sical, + ak (1), where / is the length PQ, and fdo represents the inteyral motion of #, the middle point of PQ, estimated always in the direction perpendicular to PQ*. Now if, as is generally the case in the actual use of the instrument, PQ return to its original position without making a complete revolution, the integral motion of & at right angles to PQ is the same as that of any other point /#’ in the line PQ. For if dc, do’ be corresponding elements of the paths of A, L’, estimated as aforesaid, we have do — bo’ = KR. 80, where 66 is the angle between the consecutive positions of R’R. Hence fda = fdo' + R'R {do Vi Gt eres Et rene ee (2), since, under the circumstances supposed, we have fd = 0. Fig. 64. * This is of course not in general the same thing as the length of the path of R. bi | re | ex 252 INFINITESIMAL CALCULUS. [CH. VII In the instrument, as actually constructed, the integral — motion normal to the bar of @ point #’ in QP produced a backwards, is recorded by means of a small wheel having — a its axis in the direction PQ. As Q describes any curve, the wheel partly rolls and partly slides over the plane of the paper — on which the curve is drawn, and the rotation of the wheel isin _ exact proportion to the displacement of #’ perpendicular to its axis. The wheel is graduated, and has a fixed index for the | record of partial revolutions ; the whole revolutions are recorded by a dial and counter. There is also an arrangement for varying the length PQ; this merely alters the scale of the record. EXAMPLES*. XXXII, 1. Ifa curve be such that y™ a”, the rectangle enclosed between the coordinate axes and lines drawn parallel to them through any point on the curve is divided by the curve into two portions whose areas are as m : 7. 2. The area included between the hyperbola ey =e, the axis of x, and the lines w=a, x=), is k? log b/a. 8. The area included between the axis of « and one semi- undulation of the curve y = bsin «/a is 2ab. 4. The area included between the catenary y =c cosh a/c, the axis of x, and the lines x= 0, x= a, is ; ce’ sinh a,/c. 5. The curve wy = x" (“+ a) | includes, with the axis of x, an area ja’. * Some further Examples for practice will be found in the Chapter (rx) on ‘‘ Special Curves,” Rt sate: 102] GEOMETRICAL APPLICATIONS. 253 6. The areas included between the axis of « and successive semi-undulations of the curve y=—e-* sin Ba form a descending geometric series, the common ratio being e~7a/p, 7. Thearea included between the positive branch of the curve y =b tanh 2/a, its asymptote, and the axis of y, is ablog2. (See Fig. 22, p. 43.) 8. The area included between the coordinate axes and the parabola x\t /y\t_ a) aes is {ab sin w, where w is the inclination of the axes. [Put e=asin'#, y=bcos'6.| 9. The area swept over by the radius vector of the parabola 2a "T+ cos 0 is equal to the difference between the initial and final values of a? (tan 40 + 4 tan’ $6). 10. A curve ABS is traced on a lamina which turns in its own plane about a fixed point O through an angle 6. Prove that the area swept over by the curve is 1 (OA? ~ OB?) 6. 11. Prove that, with a proper convention as to sign, the area of a closed curve is given by dy ~sdx Sl E tek ay een 2 [ (« de 4 a a provided the total variation of ¢ corresponds to a complete circuit of the curve. 12. The formula (Art. 79 (2)) for integration by parts may be written fudv = uv — fvdu ; interpret this geometrically in terms of areas. 254 INFINITESIMAL CALCULUS. [CH. VII 13. The area common to the two ellipses 2 SME REE Gb ue is 4ab tan’ b/a. 14. The area common to the two parabolas 2—4ax, x= 4ay sq L672 18 “3 GW. 15. Prove by integration that the area of an ellipse is Tas Sin w, where a, B are the lengths of any pair of conjugate semi-diameters, and w is the angle between these. 16. Prove by transformation to polar coordinates that the area of the ellipse Ax’ + 2Hay + By’ =1 is 7/,/(AB — #1”). 17. A weightless string of length J, attached to a fixed point O, passes through a small ring which can slide along a horizontal rod AB in the same vertical plane with O, and the lower portion hangs vertically, carrying a small weight P. Find the locus of P, and | prove that the area between this locus and AB is L /(? —h?) —h? cosh d/h, where h is the depth of AB below O. 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 PQ (Fig. 63) makes a com- plete revolution whilst the point Q traces out the closed curve? 20. If S,, S;, S, 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 BOS 404A 84 Aa 103] GEOMETRICAL APPLICATIONS. 255 where the lines BC, CA, AL have signs attributed to them according to their directions, and the signs of S,, S,, S, are determined by the rule of Art. 99, 21. If P bea point on a bar AB which moves in one plane, and returns to its original position after accomplishing one revo- lution, prove that aes as, “ts bS, — rab f a+b where a= AP, b= PB, and the meanings of S,, S,, S, are as in the preceding question. Hence shew that if the extremities A, B of the bar move on a closed oval curve S,— Sp=7ab. (Holditch.) 22. If astraight line 44 of constant length move with its extremities on two fixed intersecting straight lines, any point P on it describes an ellipse of area 7. AP. PB. 103. Volumes of Solids. It is impossible to give a general definition of the ‘volume, even of a solid bounded wholly by plane faces, without introducing, in one form or another, the notion of a ‘limiting value.’ It may, indeed, be proved by Euclidean methods that two rectangular parallelepipeds are to one another in the ratio compounded of the ratios, each to each, of three concurrent edges of the one to three concurrent edges of the other; and, more generally, that two prisms are to one another in the ratio compounded of the ratio of their altitudes and the ratio of their bases. In this way we may define the ratio of any prism to that of the unit cube. But it is not in general possible to dissect a given poly- hedron into a finite number of prisms. The simplest general mode of dissection 1s into pyramids having a common vertex at some internal point O, and the faces of the polyhedron as their bases. And the volume of a pyramid cannot be com- pared with that of a prism without having recourse to the notion of a limiting value. b]. See Fig. 70, p. 272. For each value of x( 06); prove that the volume of that portion of the sphere which is external to the cylinder is 50 (a? — b?)8, 264: INFINITESIMAL CALCULUS. (CH. VII x 12. The volume enclosed by two right circular cylinders ~ | of equal radius a, whose axes intersect at right angles, is 46a’. If the axes intersect at an angle a, the volume is 4° a’ cosec a. 13. If the hyperbola ciel 1 6 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 A apart, is equal to that of a circular cylinder of height /# and radius 6. 14. (27y.PQ). Ultimately, PQ is in a ratio of equality to 6s, and y may be taken to be the ordinate of 272 INFINITESIMAL CALCULUS. _ (cH. vir the curve; the surface is then equal to the limiting value of 27> (y. 8s), that is, to Qarfy do ..cssscee ie ees (2), taken over the proper range of s. x. 1. In the case of the sphere the coordinates of any point of the generating curve may be written e=ac0s 0, y= asin 6 eee (3), whence ds/d@=a...... oa cd goannas -+--(4). Hence the surface of a zone bounded by planes perpendicular to x 1S 65 ara" i sin 6d = 27a? (cos 0, — cos 6.) PT == Dire (0, — 24) ee (5). where the suffixes refer to the bounding circles. Hence the zone is equal in area to the corresponding zone of a circumscribing cylinder having its axis perpendicular to the planes of the bounding circles. In particular, the whole surface of the sphere is 27a. 2a, or 47a? Hx, 2. To find the surface of the ring generated by the revolution of a circle of radius 6 about a line in its plane at a distance a from its centre, we may put a=bsind, y=a—bcosd, ds/d0=d......... (6), 111] GEOMETRICAL APPLICATIONS. 273 and obtain Qar i yds = 2b | (a—bc08 8) dO..........000: WA) The limits of 6 being 0 and 27, the result is 2b x 27a, which is equal to the curved surface of a right cylinder of radius 6 and length (27a) equal to the circumference described by the centre of the generating circle. Ex. 3. To find the surface generated by the revolution of the ellipse ; Cee SUT WURTI/ == O COSC sucinate + ifesab es nas ee (8) about the major axis, we have 2m [yds=2n [y de = Qrab | J(l—etsin® $) d(sin $) vee (9), Peeart, 109,5 If we put ¢sin. d= sin 0......06.0. ces sesecets (10), this meee! cos? 6d@ = Be [0 +.sin 6 cos 6] ........4. (11). To find the whole surface we take this between the limits p= Fin, or O=Fsin-'e. The result is 27ab {sin-e + e,/(1—*)}, emg | or Per PAS wtih Ra Tle so (12). é By a similar process we find, for the surface generated by the revolution of the ellipse about its minor axis, the value ee {sinh-!e’ +e',/(1+e”)}, where e’ = ,/(a? — b)/b, or aN are Ina? + Qrab aie BAe tee as eure. (13). This may also be put into the form 1, Ite 2a? + 7b? . a log [oe es (14). 274 INFINITESIMAL CALCULUS. [cH. VII EXAMPLES*, XXXIV. 1. The length of a complete undulation of the curve of sines y =bsin x/a is equal to the perimeter of an ellipse whose semi-axes are J/ (a +0?) and a. 2. Prove the following formula for the length of the perpendicular (p) from the origin on any tangent to a curve. Also prove that the orthogonal projection of the radius vector on the tangent is 8. The surface generated by the revolution about the direc- trix of an arc of the catenary y=c cosh a/e, commencing at the vertex, is 1 (cx + Ys), where x, y, s refer to the extremity of the are. 4, The curved surface cut off from a paraboloid of revolution by a plane perpendicular to the axis is 4 (Ad + 098 — Bi, where A is the length of the axis, and 6 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 «= 3a is 5°7a?. 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 1:219 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 47a? (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 a (a — 2) a’. 9. A variable sphere of radius 7 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 7 = $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 the 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 y, Y2,---, Yn the lengths of the ordinates, the sum of the trapeziums is E(YrtYo) AAS (Yot Ys) b+... + ¥(Yn2t Yn) hh + $(Ynart Yn) h = (S41 + Yot Yost vee + Yn—at Ynat Fyn) h...C1); that is, we add to the arithmetic mean of the first and last ordinates the sum of the intervening ordinates, and multiply the result by the common interval h. 18—2 276 INFINITESIMAL CALCULUS. [CH. VII The value thus obtained will obviously be in excess if the curve is convex to the axis of a, and in defect in the opposite case. Fig. 71. Another method, originally given by Newton and Cotes*, is to assume for y a rational integral expression of degree n — 1, thus y=A,+ A,0+ A, +... + An ee (2), and to determine the coefficients A,, A,,... A»_, so that, for the m equidistant values of x, y shall have the prescribed values %, Ye, ++») Yn» The area is then given by ly dx, = Ax +$A,a°+1A,a?+... +E Agi), taken between proper limits of a. Thus, in the case of three equidistant ordinates, taking the origin at the foot of the middle ordinate, we assume x 2 : y=Ao+ AF +A, (7) os,e 6610 a! plete (4), * See the latter’s tract De Methodo Differentiali, printed as a supplement to the Harmonia Mensurarum, Cambridge, 1722. 112] GEOMETRICAL APPLICATIONS. 277 with the conditions that vi N15 Yo; Ys; ee 6 respectively, These give A,-—A,+A,= Nn; A, = 42, A,+A,+ A= 4;...(6), so that A,=%, A,=4(ys—y:), As=F(Yit+Ys— 2y2)--.(7). h Hence i ydx=2(A,+44,)h ay =L (Yr t4Yat Ys) .ceccsceeeee (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 3 (H+ Ys). 2h and the parabolic segment 2.{4 (y+ Ys) — yo} . 2h; see Art. 98, Ex. 4. In the case of four equidistant ordinates a similar process leads to the formula 3 (4, + BY. + BYg + Ys) ccrcccceceeees (9), whilst for five ordinates we get Ps (Ty, + 82y, +124, + B2y4 + TYs) hb ..00 (10). With an increasing number of ordinates the coefficients in this method become more and more unwieldy*. | gtdt=i9gt, ; t Jy t Jo 1 1 280 INFINITESIMAL CALCULUS. [CH. VII 4.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= 29s, 8, 2 3 8) t i vds = (29)" I stds = 2 (2ys,)} ; §; J0 S75490 zi 2.e. 1b 1s two-thirds the final velocity. Eu. 1. The mean value of sin 6 for equidistant intervals of 0 ranging from 0 to z is : i ” in 0d6 =~ = 6366, 0 Tv Tv Hence .he mean value of the ordinates of a semicircle of radius a, drawn through equidistant points of the are, is -6366a. Tf the ordinates had been drawn through equidistant points on the diameter, the mean value would have been 1 4 hr alee J (a? — x?) dx = ha [cos 6d =47a, or ‘7854a, It is easily seen @ priort why this latter mean should be the greater. Ex. 2. To find the mean latitude of all places north of the equator. The surface of the hemisphere is here supposed divided into infinitely narrow zones of equal area. If z denote distance from the plane of the equator, the area of the zone included between the planes z and z+ 6z is proportional to dz. Hence, if a be the radius, the mean latitude is a dr “| sin™= de= [ 6 cos 6d6 a Jo a 0 liar = E sin 6 + cos a} 0 =47-1, or, in degrees, 90° — 57:296°, = 32°704". The various formule of Art. 112 may be interpreted as giving approximate expressions for the mean value of a function, over a given range, in terms of a series of values of the function taken at equidistant intervals covering the 113] GEOMETRICAL APPLICATIONS. 281] range. For example, in terms of three and of four such values, the mean values, as given by Cotes’ method, are (Yt 4ya+ys) and $(yi+3y2+ 3ys+ ys), respectively. EXAMPLES. XXX’V. 1. Apply Simpson’s rule to calculate log, 2 from the formula idx log, 2 = i) rare [The correct value is log, 2= 693147...] 2. Calculate the value of z from the formula t= | soca: 38. The mean of the squares on the diameters of an ellipse, drawn at equal angular intervals, is equal to the rectangle contained by the major and minor axes. © 4. A point is taken at random on a straight line of length a; prove that the mean area of the rectangle contained by the two segments is 4a’, and that the mean value of the sum of the squares on the two segments is 3a”. 5. If a point move with constant acceleration, the mean square of the velocities at equal infinitely small intervals of time is equal to (Up + %% + %’); where vw, Vv, are the initial and final velocities. 6. Prove that in simple-harmonic motion the mean kinetic energy is one-half the maximum kinetic energy. 7. The mean horizontal range of a particle projected with given velocity, but arbitrary elevation, is ‘6366 of the maximum range. 8. A particle describes an ellipse with a velocity varying as the conjugate diameter. Prove that the mean kinetic energy is equal to the kinetic energy of the particle when at an extremity of an equiconjugate diameter. 282 INFINITESIMAL CALCULUS. [CH. VII 9. (pdV’), and the mean density is given by S(p8V)_lpdedyde gy > (SV) 7 [[[dady dz Ba oe 3). The above definitions relate to the case of ‘volume- density. If a finite mass be supposed concentrated in a surface, or in a line, we are led to the conceptions of ‘ surface- density’ and ‘line density,’ which may be defined in a similar manuer. p = lim Thus if o be the surface-density of a plane film of matter, and 6A an element of area, the mean surfacé-density is ear X(c5A) __ ffodady o = ms) = I[dady Sefceieicis sinters sen (4). And if w be the line-density of a linear distribution of mass, and 6s an element of the line, the mean line-density is _- 4. &(p6s) “fuds : p=lm Se Eads cae. (5). Ex. 1. To find the mean density of a semicircular lamina, whose density varies as the distance from the bounding diameter. Taking the centre as origin, and the medial line as axis of 2, we have eeereereerevrereseaeer rere 0 where @ is the radius, and y= ,/(a?—«’). Putting o=ka, the numerator becomes a dr k J x, /(a? — x") dex = ka® i sin 6 cos? 6d6 = tka’, 0 | 0 L. 19 290 INFINITESIMAL CALCULUS. [CH. VI — and the denominator is of course = dia’, Hence c= = pha ADA lettin density ......... (7). or Ex. 2. If the density of a sphere be a function of the distance 7 from the centre, taking as the element of volume SV =8 (4a) = 4a 8r, we have, if a be the external radius, a a 4a I pridr ~ 3 I pr’ dr 0 eee t: sma Teak ieee Thus, if p« +”, we find that the mean density is 3/(m +3) of the density at the surface. p= Again, assuming that in the Earth 7 on (1-%5) | (9), we find 5 = p,(1— 3h) = (29, +39): attDy, where p, is the density at the surface (r=a). If the above law of density be really applicable (as it appears to be, approximately, ) to the case of the Earth, then since p = 2p,, roughly, we infer that P>=~p1, or the density at the centre is 34 times the density at the surface. EXAMPLES. XXXVI. 1. If the line-density of a rod vary as the square of the distance from one end, the mean density is equal to the actual density at a distance of 1/,/3 of the length from that end. 2. The mean thickness of a circular plate, whose thickness varies as the nth power of the distance from the centre, is equal to 2/(n + 2) of the thickness at the edge. Soc) itoin' a spherical mass whose density p is a function of the distance (r) from the centre, D denote the mean density of the matter included within a concentric sphere of radius 7, then p=D+yre. ae 115-116] PHYSICAL APPLICATIONS. 291 4. If the density of a solid hemisphere of radius @ vary as the distance from the base, the mean density is equal to the density at a distance 2a from the base. 50. A disk has the form of a very flat oblate ellipsoid of revolution, prove that its mean thickness is two-thirds of the thickness at the centre. : 6. (mx) a X(my) _ (mz) eS aA (my ’ ee aye = (im) Pid ws iether the axes be rectangular or oblique. The summations > are supposed to include every particle of the system; for example } (mz) stands for ma + mo, +... + MnLp. * The simplest and best being the vector definition Peas by Grass- mann, Ausdehnungslehre (1844). f 9-2 292 INFINITESIMAL CALCULUS. [CH. VIII If the origin be at the centre of mass we have > (mx) =0, & (my) = 0, S(mz)=0. Castes (2). If the masses ™m, m2, ... 7%, be all equal, the formule (1) reduce to cha galt nee) tyeel Z=—Z (a), Yardy), 2= | & (2) seats (3). If the masses 7, M2, -.. Mp, 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 formule (3) will then still apply. And since incommensurable magnitudes may be regarded as the limits of commensurable magnitudes, the formule (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 6M an element of mass situate about the point (a, y, 2), the formule become 1. »& (0M) .. 4. 2 GoM) 2 ee ee aoe N Vey y =lim 5 (SM) ’ z= lim = (SM) sa (4). or is [Jap da dy dz Fy EE : _ Sfepdadydz "= "Tipdedyde’ 9 [fjpdudyde’ *~ [ffjpdedyde ee: (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 w be the line-density, and 6s an element of the arc, we have fepds __ fypds haga? y= AEE eee (1). We consider only plane curves, and take the axes of a, y always in the plane, so that Z= 0. C= Hx. 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 ¥=0, by symmetry. Also, putting »=1, as we may do, without loss of generality, we have, writing «=acos 6, 8s=a80, sin a Ip acos 6. adé [cos 646 Ce ee eee Sa 0 a= —.a@... .(2), ip adé - * if 2a denote the angle which the whole arc subtends at the centre. As a increases from an infinitely small value to 7, % decreases from a to 0. For the semicircle, we have a= 47, and x pst 637a. T 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 #, we have, if y be the ordinate of the bounding line, lad OST Esme nae hex Cadet Ti MVE e raeanbe (1), the integrals being taken between the proper limits. Ex. 1. For an isosceles triangle, taking the origin at the vertex, we may write y = mz, and therefore h being the altitude. 4 294 INFINITESIMAL CALCULUS. [cH. VUI Ex. 2. For a semicircular area of radius a, we find a i x(a? — x") de 0 a I a/ (a@ — 22") dae 0 The integrations are exactly the same as in Art. 115, Ex. 1; and the result is ee | %= 3s "ADAG Vee costes oten semper (4). Ex. 3. For a segment of the parabola The formulz (1) will apply also to the case of oblique axes, provided the axis of # bisect all chords drawn within the area parallel to the axis of y. For instead of an ele- mentary rectangle yd we have now an elementary parallel- ogram ydx sin w, where m is the inclination of the axes. The constant factor sin w, occurring both in the numerator and in the denominator of the expression for %, 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 2 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 4/37 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= (a), the axis of x, and two bounding ordinates, the 118] PHYSICAL APPLICATIONS. 295 mass-centre of an elementary parallelogram yéz sin » will be at the point (#, dy). Hence _feyde __ ftiyda ae gilad y= TEE ae ES REC (7), the factor sin w cancelling as before. SI Ex, 4. The mass-centre of the area included between the parabola if & be the ordinate corresponding to «=h. In polar coordinates, we may resolve any sectorial area into elementary triangles 47760. The mass-centre of any one of these is ultimately at the point (27, 0). Hence referring to rectangular axes, of which the axis of x coincides with the origin of @, we have [2rcos@.47°d™ i fr°cos 640 = Syd yfr'dd” (10) ie Zr sin@.$°dd 4 fr*sin 6dée Cnt ick ; Sarda fread the integrals being taken between proper limits of 0. Hx. 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 “a 4a? cos 6d6 ; - , sina = 1 Sie reper (11). oe : Oa 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 (o =const.). 296 INFINITESIMAL CALCULUS. (CH. VIII Kx. 6. For example, take the case of a semicircular lamina whose density varies as the distance from the bounding diameter (or, say, an infinitely thin wedge cut from a uniform solid sphere by two planes meeting in a diameter). With the same axes and the same notation as in Art. 115, Ex. 1, and putting o=ka, as before, we have y=0, and [e. ke. 2ydax [, alla —2) da 0 0 Gig ee eee [fe 2c [, aN (@ =a) de 0 0 dr a | sin? §cos?6 dé 3 0 T = = 16 a= 589Ge ee (12). | x sin 6 cos? 6d@ 0 119. Mean Pressure. Centre of Pressure. A system of parallel forces distributed continuously over a plane area will in general have a single resultant equal to their sum. The quotient of this resultant by the area is called the ‘mean intensity ’ of the force over the area. The ‘intensity at a point’ is defined as the mean intensity over an infinitely small element of the area, including the point in question. Thus, in Hydrostatics, we speak of the ‘mean pressure- intensity ’ over an area, and of the ‘pressure-intensity at a point’ of the area. If the pressure-intensity (p) at every point of an area be given, the total pressure is the limiting value of the sum 39:34). (1), where 6A is an element of the area. The mean pressure- intensity (p, say) is then given by 2(p.4) (2). = (6A) In the limit the summations are to be replaced by integrals, thus p= im _ fipdady ~Tdedy es (3). 118-119] PHYSICAL APPLICATIONS. 297 In a homogeneous liquid under gravity, the pressure- intensity varies as the depth below the free surface. It is therefore uniform over any horizontal area. In the case of an inclined area we may say that the pressure-intensity varies as the distance (a) from the straight line in which the plane of the area meets the plane of the free surface ; or Hence, in this case, Tim 2 ee 8A) _ inutnr PS COA) where % refers to the mass-centre of the area. That is, the mean intensity is equal to the intensity at the mass-centre of the area. i tt Seed (5), The ‘centre of pressure’ of a system of parallel forces distributed over a plane area is the point in which the line of action of the resultant pressure on the area meets the plane. Its position may be found, on statical principles, by taking moments about axes #, y in the plane of the area. Thus, denoting the coordinates of the centre of pressure by (&, 7), we have : : == 1} (w.poA) _). (a.pdA) ffapdady Bes SA) = him = $ (5A) = Bldedy a = pn a =(y.pSA) _ flypdady E(p.b4) ~ " p.2 6A) © plidedy The axes here may be rectangular or oblique. In the particular case of a liquid under gravity, if the axis of y be the intersection of the plane of the area with the free surface, we put p=/w, and therefore > (a. dA) —_— lim nN NS Ss Cn). CSAs deco BS (SA) oe or, if we write ydx sin for 64, where o is the angle between the axes of a, y, ea _ fa?yda - fayrda E == aly da > y= Bly da FGI Te OSI a (8). Hx. 1. To find the centre of pressure of a rectangle, immersed in a liquid, with one pair of sides horizontal. 298 INFINITESIMAL CALCULUS. (oH. vit Taking the axis of x along the line which bisects this pair of sides, we have obviously »=0; and if we denote the distances (measured in the plane of the rectangle) of these sides from the free surface by h+a, we have rh+a | avd : fa he (9). Hence the centre of pressure is beneath the centre of figure, at the distance 4a7/h. As h increases, this interval diminishes, as we should expect, since the pressure-intensity over the area becomes more and more nearly uniform. Ex. 2. In the case of a triangle having a vertex in the free surface, and the opposite side horizontal, the origin is conveniently taken at this vertex, and the axis of w along the medial line. We have, then, 7 = 0, and, since the breadth of the triangle at any point varies as 2, where / is the length of the medial line. Ex. 3. In the case of a triangle with one side in the surface, taking the origin at the middle point of this side, and the axis of a along the medial] line, the breadth of the triangle will vary as h—2x; so that Hex, 4. To find the centre of pressure of a semicircular area having its base in the free surface. , This is analytically the same problem as Art. 118, Ex. 6, and the result is é= 3,00 = "50890. 2a eee (12). It may happen that the sum of a system of parallel forces distributed over an area is zero. In this case there is no single resultant, but (unless the forces balance) a ‘couple.’ 119] PHYSICAL APPLICATIONS. 299 An example of this occurs in the theory of the flexure of beams. Ina pure flexure the algebraic sum of the tensions at the various parts of a cross-section is zero, and the action across the section reduces to a couple. See Art. 130. EXAMPLES. XXXVII. 1. Prove by integration that the mass-centre of a trapezium divides the line joining the middle points of the parallel sides in the ratio 2a+6:a+ 26, where a, 6 are the lengths of the parallel sides. 2. The mass-centre of the area included between one semi- undulation of the curve y=bsin x/a and the axis of x is at a distance 47b from this axis. 3. The mass-centre of the area included between the curve ae Tee? and the axis of x is at the point (0, 4a). 4. The coordinates of the mass-centre of the ‘parabolic spandril’ bounded by the curve y*? = Lax, the tangent at the vertex, and the line y=4, are Tels 2 7: a tk, where / is the abscissa of the ordinate &. 5. Prove that the mass-centre of the area of the circular spandril formed by a quadrant of a circle and the tangents at its extremities is at a distance -2234a from either tangent, a being the radius. 6. The mass-centre of a segment of a circle of radius a is at a distance oh sin’ a = —————_——__ & a—sinacosa from the centre of the circle, 2a being the angular measure of the are. 300 INFINITESIMAL CALCULUS. [CH. VIII — 7. The mass-centre of the area included between the co- ordinate axes and the parabola is at the point (4a, 4). [Put «=asin‘ 0, y=b cos‘ 6.] | 8. If the density (c) of a semi-circular lamina be a function of the distance (r) from the centre, the distance of the centre of mass from the base is given by Aad a a== | ord = | ordr, 0 0 where a is the radius. Hence shew that if car, =:4775a, and that if oa 7, x= '5093a. 9. If the density of a parabolic arc whose axis is vertical . vary as the cosine of the inclination to the horizon, the co- ordinates of the mass-centre, referred to horizontal and vertical axes will be B= 3(% +H), Y=E(MWit 4y' +), where (2, ¥;), (#2) Y2) are the coordinates of the extremities of the arc, and 7’ is the ordinate half-way between y, and y,. 10. A uniform rod of length 7 is bent into the form of a circular arc of radius (2) large compared with 7. Prove that the displacement of the mass-centre is ,1,/?/R, approximately. ll. A triangular area is immersed in liquid, with one side in the free surface. Prove that if it be divided into two portions by a horizontal line through the centre of pressure, the resultant pressures on these two portions are equal. 12. The centre of pressure of a trapezium having one of the two parallel sides (a, b) in the surface divides the medial line in the ratio a+ 3b:a+6, where a is the side which is in the surface. 13. The centre of pressure of a rhombus immersed in a liquid with a diagonal vertical and one angular point in the surface is at a depth equal to ;4 of the diagonal. 120] PHYSICAL APPLICATIONS. 301 120. Mass-Centre of a Surface of Revolution. If the axis of x coincide with the axis of symmetry, and if ds denote an element of arc of the generating curve, and y its ordinate, the annular element of surface will be repre- sented as in Art. 111 by 2ary os. Hence if the surface-density be uniform, the position of the mass-centre of a zone of the surface included between planes perpendicular to a is given by Je.2ryds _fayds (1) Pedy Sqpchene ke If the surface-density (c) be a function of x, the formula must be replaced by i fa.o.2myds _foxyds (2) her dusmas (adds see Fx. 1. For a zone of a spherical surface, putting = Beem COSY.) Y= SiN G,4 (OR = GOO. ss pea ess se: (3), B [eos sin 040 ; h tc _ ,, cos? a — cos? B ea fe ~ 27 Gos a — cos B [, sin ao el COS COS (2) ay (05+ a) ss si 2ny ass tete (4), if a, B be the limits of 6, and x,, x, the abscisse of the bounding circles. Hence the mass-centre of the zone is on the axis, half-way between the planes of the bounding circles. For example, the mass-centre of a uniform hemispherical surface bisects the axial radius. These results might also have been inferred immediately from the equality of area of corresponding zones on the sphere and on an enveloping cylinder (Art. 111, Ex. 1). Ex. 2. To find the mass-centre of a spherical ‘lune,’ 7.e. of the area on the surface of a sphere included between two meridians. The ‘angle of the lune’ is the angular distance between the two bounding meridians. We shall denote it by 2a, and the radius of the sphere by a. 302 INFINITESIMAL CALCULUS. [CH. VIII Divide the lune into elementary lunes of equal infinitely small angle. The mass-centre of any one of these will be at a certain distance x from the centre of the sphere. Supposing the mass of each elementary lune to be transferred to its centre of mass, we obtain a uniform circular arc of radius x, and angle 2a. The mass-centre of this is in the bisecting radius, ata distance sina x Qa from the centre, by Art. 117. Now in the case of the hemisphere (a = 47) this distance must be equal to $a, by Ex. 1, above. Hence x= 4a; and the centre of mass of the given lune is at a distance sin a t from the centre of the sphere. Ex. 3. In the case of a thin hemispherical shell, whose thickness varies as the distance («) from the plane of the rim, we have a Aga 2 ie | cos? 6 sin 6d6 x= = Oe Jayde ie cos 6 sin 6d6 where a is the radius. Ex. 4. A simple rule for finding the mass-centre of any portion of a uniform spherical surface may be obtained as follows. Let SS denote any element of the superficial area, « its distance from a fixed plane through the centre. The distance of the centre of mass from this plane will be Now if @ be the angle which the normal to 5S makes with the normal to the plane of reference, we have «=acos0, where a is: the radius of the sphere, and therefore LoS = a COs 65S — aod eoeee @eceeee eleva ials (8), where 83 is the orthogonal projection of the area 59 on the plane: of reference. Hence 120-121] PHYSICAL APPLICATIONS. 303 if S be the total area of the portion of the spherical surface con- sidered, and & the area of its orthogonal projection on the plane of reference. Thus, for a hemispherical surface, projecting on the plane of the bounding circle, we have S = 27a’, 3} = za’, and therefore 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 asina. Hence S=4aa?, } =7za’sina, and therefore as above. 121. Mass-Centre of a Solid. In the case of a homogeneous solid, if the area of a section by a plane perpendicular to # 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 af (a) d ___ jaf (x) dex C= pian he ee SCL), taken between the proper limits of a. 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 w, we have y= 0, and z=0. In the case of a solid of revolution, taking the axis of « coincident with the axis of symmetry, we have J (a) =7y’, if y be the ordinate of the generating curve. Hence 304 INFINITESIMAL CALCULUS. [CH. VIII Ex. 1. Tn the case of a right circular cone, the origin being at the vertex, f(x) « x, so that h I ada . = $h I a? da 0 rm if h be the altitude. Ex. 2. For the segment of an elliptic paraboloid cut off by a plane «=A, since f(x) « «x, as in Art. 106, Ex. 1, we have & = = hy oie (5). [, eae 0 Ex. 3. For a hemisphere of radius a, putting y?=a?—2’*, we have The same formula gives the position of the mass-centre of the half of the ellipsoid which lies on the positive side of the plane yz, since f (x) in this case also varies as a?—a. See Art. 106, Ex. 2. Ex. 4. In the case of the more general formula S(e)=4+ Be the x-coordinate of the mass-centre of the volume included between the planes x= 0 and «=h, is, by (1), JA+4B41C 2A’ + A” a a: Mae Say AR ey A+4B+40C Ae a (9); 22 + Oe tre (8), 121] PHYSICAL APPLICATIONS. 305 where, as in Art. 107, A, A’, A” denote the areas of the sections x=0, x=th, w=h, respectively. The distance of the centre of mass from the middle section is therefore A” —A = 2(4 + 4d’ + A”) h @erelee eisis s scale six seis (10). 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 by f(x) the area of a section parallel to the plane yz, the appropriate element of volume is J (a) 6x sin 2X, where ) is the inclination of the axis of x to the plane yz. The constant factor sin A, occurring both in the numerator and in the denominator of the expression for %, will cancel, and we are left with the same form as before. z— 1h Ha. 5. In the case of a cone, or a pyramid, on a plane base, taking the origin O at the vertex, and the axis of « along the line joining O to the mass-centre G of the area of the base, the area of any section parallel to the base will vary as the square of its intercept x on OG. We are thus led as before to the result Z = 3h, where h now =OG. Hence the mass-centre of the pyramid, or cone, is at a point H in OG, such that OH = 3064. 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. fx. 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 7 is proportional to r?$r. Also the distance of the mass-centre of this stratum from the vertex is, by Art. 120, Ex, 1, + (7 +7 cos a), =r cos? da, 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 a I dr 0 x= Pane dr 0 cos? 4a = fa cos? da............ (11), where a is the external radius. For the hemisphere we have a=%47, and %=%2a, 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 (asina)/a. Since, for a=47, this must equal 3a, we infer that oT == 16 Q, 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. ux _ om sina v6. Se a 0) wer ovens’ d ecnvetesatetansistenetels (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 a, we have, in place of Art. 121, (1), For example, in the case of a solid of revolution, in which the surfaces of equal density are planes perpendicular to the 171-123] PHYSICAL APPLICATIONS. 307 axis (#), we have f(x) = ry’, where y refers to the generating curve, and therefore eoeoeveeeoseeseeeeeeeseee 808 Ex. 1, Thus in the case of a hemisphere whose density p varies as the distance («) from the bounding plane, writing y*? =a? —x*, we have a i a (a? — x?) dx pene eee ae st RL svat pee tents os (3). a i a (a? — x”) da 0 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 swrface 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. 111, equal to 2rlyds, the integration extending over the arc. But if ¥ refer to the mass-centre of the arc, we have 7 alas a Jds* by Art.117. Hence re Nie BSB TT he 0 A ea a (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. VII area multiplied by the length of the path of its centre of mass. If 5A be an element of the area, the volume generated in a complete revolution is lim > (27ry. 8A). But if ¥ refer to the centre of mass of the area, we have Sia OAD eee by Art. 116. Hence lim & (2ary.6A) = 2ary x lim & (6A)......... (2), which is the theorem®*. The revolutions have been taken to be complete, but the restriction 1s obviously unessential. Ex. 1. The ring generated by the revolution of a circle of radius 6 about a line in its own plane at a distance a from its centre. The surface is 2rb x 2ma, = 4rab ; and the volume is mb? x Ira, = 2x? ab’, Of. Art. 105, Ex. 3, and Art. 111, Ex. 2. Ex. 2. A segment of the parabola y*= 4aa, bounded by the double ordinate «=h, revolves about this ordinate. If 2k be the length of the double ordinate, the area of the segment is 3hk, by Art. 98; and the distance of the centre of gravity from the ordinate is 2h, by Art. 118. Hence the volume generated is shk x éarh = temh?k. The theorems may be used, conversely, to find the mass- centre of a plane arc, or of a plane area, when the suzface, 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.p. 300. They were given as new by Guldinus, de centro gravitatis (1635—1642). (Ball, History of Mathematics.) 123-124] PHYSICAL APPLICATIONS. 309 Ex. 3. Thus, for a semicircular are revolving about the diameter joining its extremities, we have ra x Iny = 4rra?, whence y= | Again, for a semicircular area revolving about its bounding diameter, dra? x Iry = $ra', cn whence y=>5-a ar 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 6A, 6A’ be corresponding elements of area at P and P’, we have Mea Zr OA ye 02, (2 OA) Beas SAT = Hess (SH): , since 6A, being the orthogonal projection of 8A’, is in a _ constant ratio to it. Hence the volume of the solid =P eROA. ) Sie Xp LOA): ievies scitecs (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. - [CH. 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. The volume of the wedge-shaped solid cut off from a right circular cylinder by a plane through the centre of the base, making an angle a with the plane of the base, is 4 dra? x ga tan a =2a* tana ; T ef: Art: 114. -Eix,-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 where S is the area, and do 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 0 the angle between ds and the normal to the plane, the formula may also be written [S cos. 0ds ii... ee (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 corollary 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 *876a. 2. The mass-centre of either half of the surface of an anchor-ring cut off by the equatorial plane is at a distance 26/7 from this plane, where 0 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 28: Prove that the distance of the mass-centre of the remainder from the centre of the sphere is 4a sin’ a sec a cos f, 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 Lot 6 6th” where / 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 c from the centre of figure are 3 (atc)? 4 2a+c° 6. By dividing a tetrahedron into plane lamin 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 ‘869qa. 8. 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 0, is. cut round a cylinder of radius a; prove that the volume removed is mab? — Srb?. Also that the surface of the groove is 27° ab — 47}. 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 2rabR + rab?, where a, 6 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 4b/3m from the plane, where 0 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 multiplied 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 7m, m2, ms,... be the masses of the several particles, »,, p2, ps, -.. their distances from the line, and if J denote the moment of inertia, we have T=mp2t+mepe+ mpet...= > (mp)... (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 J/ be the mass of a body moving in a straight line with velocity u, its momentum is Mu; andif / be the extraneous force, we have In like manner, if 7 be the moment of inertia of a body rotating about a fixed axis with angular velocity w, its angular momentum is Jw; and if G be the extraneous couple, we have The moment of inertia of a body about a given line is often most conveniently specified by means of a linear magnitude called the ‘radius of gyration.’ This is a quantity k such that where M is the total mass of the system. 314 INFINITESIMAL CALCULUS. [CH. VIII Tt is evident from the dynamical principles above referred to that, as regards rotation about the fixed line, the body will behave exactly as if its mass were concentrated into a ring of radius & having its axis coincident with that line*, 5] The formula (4) is equivalent to oo bt pl t en M+ Mz + M3+... > (mm) | Hence k? may be regarded as the mean square of the distances of all the particles of the system from the given line. It is easily seen that in calculating k® we may replace any group of the particles by a single mass equal to their sum and situate at a distance from the line equal to the radius of gyration of the group about that line. 126. Two-Dimensional Examples. In the case of bodies whose mass is distributed over lines, surfaces, or volumes, and not condensed into isolated points, the summations in the formule (1) and (5) of Art. 125 must of course be replaced by integrations, We begin with a few simple examples in two dimensions. Hx. 1. To find the radius of gyration of a uniform straight bar about a line through its centre perpendicular to its length. If 2a be the length, we havet -—a The same result evidently holds for the radius of gyration of a rectangle about a line of symmetry, if 2a be the length perpendicular to this line. * Hence the name ‘swing-radius’ was proposed by Clifford, as the equi- valent of ‘radius of gyration.’ t The line-density, when constant, may be put equal to unity since, whatever its value, it appears in both numerator and denominator of the expression for k*, and so cancels. A similar remark applies to many subse- quent examples. {125-126] PHYSICAL APPLICATIONS. 315 Ex. 2. The radius of gyration of a uniform circular wire of adius @ about its axis is evidently a. To find the radius of gyration about a diameter, we have 4 fee a* sin’ @. adé Fk Ge sa Mace a eae II Nie g bo on we) — soe eee eee eee ease Qa Ex. 3. To find the radius of gyration of a circular plate about an axis through its centre, perpendicular to its plane. If we divide the disk into concentric annuli, the area of one of these may be represented by 277ér, and its radius of gyration by 7, Hence _ Ex. 4, To find the radius of gyration of a circular disk about a diameter, we make use of the result of Ex. 2; viz. that the square of the radius of serene of the annulus 2erdr is dy”, Hence the final result will be 3 of that in Ex. 3; or P=10 seceeeeteessceenseseceseea(4), To find the radius of gyration, about the axis of , of the a¥a included between a curve 7] = vo) (x) wintatetuis sre) eateteiclaibisie, o/s 6.u( ajsiais (5), tl® axis of x, and two bounding ordinates, we may divide the alga into elementary strips yd. The square of the radius 0 gyration of a strip is 1y*, by Ex. 1, above. Hence _fxy.yda 3zsydax : k? = Sarde = jyde (6), the integrals being taken between the proper limits of «. 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 YrtsTa, i \ 316 INFINITESIMAL CALCULUS. _ (CH. VII where a is half the base, and h the altitude... The radius : gyration for the whole triangle is evidentiy the same as fo either half, whence 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 epee ves ‘ z jee jnab or (9). If we put x=acos¢, y=6 sin ¢, this becomes 4 am : (Pee ty Ke wa pales 7 k 36 i sint pdd = 40"... 3 eee (10 Similarly, for the radius of gyration about the minor axis, should find 127. Three-Dimensional Problems. The following problems in three dimensions are i portant. Ex. 1. To find the radius of gyration of a uniform th spherical shell of radius a about a diameter. Take the origin at the centre, and the axis of a along th diameter in question. If the shell be divided into nar w zone by planes perpendicular to a, the area of any one of these ti ay . denoted by 2zada, by Art. 111, Ex. 1, and its radius of gyrati by y. Hence RE Ose e 2Qradx =o (a? = Sa (1). an) as y d r-' VS - 126-127] _ PHYSICAL APPLICATIONS, 317 UN anes 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 a i 292 Aaridr 0 pe DE a a 4 PO a ry (2). gra 5 A general formula for the radius of gyration, about its axis, of a uniform solid of revolution, 1s easily obtained. Dividing the solid into circular laminze by planes perpen- dicular to the axis, which is taken as axis of xz, the volume of any one of these lamine may be represented by my*dz, where y is the ordinate of the generating,«urve, and the square of its radius of gyration by 47’ (see Art. 126, Ex. 3). Hence | palsy myde _sfyde (3) fryda fy-dx ONS) ob 6.0 bese & > . Ex. 3. To find the radius of gyration of a right circular cone about its axis, we put where a@ is the radius of the base, and h/ the altitude. Thus Lx. 4. For a solid sphere of radius a, we have a dar ee ee ae ee eet 3 a 4 care pkey (a? a)? dx = 20? ......(5), _ asin (2), above. A similar result can be obtained for an ellipsoid __ of revolution. of Ex. 5. A plane area having a line of symmetry, revolves _ about a parallel axis in the same plane. To find the radius of _ gyration, about the axis of rotation, of the solid generated. Let y be the distance of any point of the area from the axis _ of rotation, and let us write 318 INFINITESIMAL CALCULUS. AOR: WIT - where a is the distance of the line of symmetry of the area from the axis. If 8A be an element of the area, and & the required radius of gyration, we have ,_ aly? /2rySA) 3 (y®SA) ~ &2ydd) —-& (yd) _ a? 3 (8A) + 3a®S (8A) + 303 (734) +S (y834) a@ 3 (8A) + 3 (754) Now, in consequence of the assumed symmetry of the area the sums 3} (5A) and & (7°34) must be unaltered when the sign of is reversed, and must therefore vanish. Also if « be the radius of gyration of the given area about the line of symmetry, we have 3 (P34) Pf ae R= S Gd) (7) ‘Hence the above formula reduces to ke = 6? + 82 eee (8) The same result holds whenever the generating curve has a centre, te. a point such that any chord through it is bisected there*, The proof may be left to the reader. For example, in the case of the solid ring whose sectional area is a circle of radius 6, we have x’ = 40’, and therefore Boa + 80 (9), a result easily verified by a direct calculation. A similar investigation applies to the case of the surface generated by the revolution of a plane arc, having a line of symmetry, about a parallel axis in its own plane. The only difference is that the element of area (8A) is now to be replaced by the element of arc (8s). The result has the same form (8) as before. . Thus the radius of gyration of the surface of the anchor-ring, - about the axis, is given by A= Of + 30 sere sce eran (10}- This, again, is easily verified independently. * Townsend, Quart. Jour. Math., t. x, (1870), and t. xvi. (1879). 127-128] © PHYSICAL APPLICATIONS, 319 128. Mean-Square of the Distances of a System of Particles from a Plane. With a view to further calculations we may introduce the conception of the mean square of the distance of the mass from a given plane, or from a given point. Thus if 1, Lp, X3,... be the distances of the particles m,, m2, m3, ...; respectively, from a given plane we write — Mert mn? + mre+... > (ma?) i a? ——} aT eae S.. oo” owe 4 ale ee oe ee we Clr: My + My + Mg + oo >(m) and, if 7), 72, 73,... be the distances from a given point, — mretmretmre+... > (mr?) eae ae eS es ee, (2). M+ My + Ms +... > (m) Thus, in two. dimensions, if & denote the radius of gyration about a line through the origin perpendicular to the plane of the system, _in(#’@t+y) =,5 i SaaS (Get at) nla ahsloveretev ele, ete eet (3). Ex. 1. In the case of a rectangle, taking the origin at the centre, and the axes of a, y parallel to the edges (2a, 26), we have MPH) g eh ES, Siveds Ficude ces (4), by Art. 126, Ex. 1, and therefore Ue RS | JSON a Rete ete ee De (5). Ex. 2. In the case of a circular hoop, taking the origin at the centre, we have the equality of the first two members being due to the symmetry about the origin. Since & is evidently equal to the radius (a), we infer that Cf, Art. 126, Ex. 2. * The symbol a? must not be confounded with =. For example, in the case of two equal particles we have x? = (ay?+ 2,7), @? = {i (X71 + 2) }%. “oid 320 INFINITESIMAL CALCULUS. [CH. VIII Ex. 3. Similarly, in the case of a uniform circular disk, we infer that of =a = 1g ee Nr ree (8), since 4? = 4a’, by Art. 126, Ex. 3. In three dimensions, if k,, ky, k, be the radii of gyration with respect to rectangular axes Ow, Oy, Oz, respectively, we have keg = y? + 2, ky? = 2 + a, kZ=e+y? eae (9). Also, if r denote the distance of any particle from the origin, we have a Se ). 2m (e+y +2) eee r= > (m) Sin) es Hence, and from (9), kg+ky+k?7= Dri ae (11). Ex. 4. For a rectangular parallelepiped, taking the origin at the centre, and the axes of «, y, z parallel to the edges (2a, 26, 2¢), we have by a, calculation similar to that of Art. 126, Ex. 1, eoeoeoeeeeereerseoe 8, ole bo II Col oO aS i) ll Col fm) bo — — bo — ~~ and therefore k,g{=4(6+0), kR=t(P+a’), kZ=4 (e+ 0)...(13). Ex. 5, In the case of a uniform thin spherical shell of radius a, taking the origin at the centre, we have . ky = ky = key by symmetry, and r?=a?, whence kg =k? =k? = 20 (14), as in Art, 127, Ex. 1. Ex. 6. For a uniform solid sphere, we have a i ” Arrdr = 0 } 7? = Te Rielettsv sreketerehenchetavetenetste (15), and therefore kif = hyp he = 20 voices (16) ; cf. Art. 127, Exs. 2, 4. 128-129] PHYSICAL APPLICATIONS. 321 Hex. 7. Ina uniform solid ellipsoid bounded by the surface since the section by a plane perpendicular to « is an ellipse of area we have ay ee ae) ee eee 18 Srabe : (18), 7m} 2_. 172 2 1,72 and similarly Te Sie Coma ee Hence &f=1(b +0), hy=4(et+a*), k2=1 (a+ bien 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 2, %, #3, ... be the distances of the particles M,, Mz, Mz, ..., respectively, from the first-mentioned plane ; and let % denote the distance of the centre of mass. If we write %=C+8, mM=B+E,, =24+&,,..., ...(1), ve have —_ = (ma*) _ = {m(@+ 6} ~~ &(m) > (m) _ &(m).2 +28 . &(mE) +3 (mE) Now 2(m£) = 0, by Art.116; and the fraction 5 (m£?)/S (m) s denoted, in accordance with our previous notation, by &. dence L, 21 322 INFINITESIMAL CALCULUS. -[CH. VIII 9°, 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 taken as axis of z, and if the coordinates be rectangular, we have, by the preceding case, Of pH=eP+PaPt+ Pt G+P)ccee (3). Hence if & denote the radius of gyration of the system about an axis through the centre of mass, and k’ the radius of gyration about any parallel axis, we have kt = 28 + Nie eee (4), where h is the perpendicular distance between the two axes. This is a very important result in the dynamical appli- cation of the subject. Ex. 1. The radius of gyration of a rectangle about a side is given by Ke? = a? + ha? = FO? o.ccnscsseceecenennes (5), if 2a be the length perpendicular to that side. This may be easily verified by direct integration. Ex. 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 Ki? = (a? +B) + § (a? +B?) = § (@ 4 0)... cee eeeee (6). Ex. 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 Ki? = a? + fat = BaF sr eterna (7). Similarly, the radius of gyration about a tangent line is given by hi’? = a? + Sat = 2a? cg eee eee (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 determining 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 #-coordinate of the centre of pressure is, by Art. 119, E& = lim Seal errs (1). Tn our present notation, we have = (#5A)=a?x (SA), S(wSA)=zExE (5A), ultimately, and therefore Ex, In the case of a circular area, having its centre at a distance 4 from the line in which its plane meets the surface, we have and therefore Sey ere ed Be (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 where y is the distance from a certain line in the plane of the section, called the ‘neutral line,’ R is the radius of the curve Into which the beam is bent, and / is a certain coefficient depending on the material. If 84 be an element of area, the total force across the section is the limit of “2 SHOE ey spelen eel (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 ine parallel to it) is got by multiplying the force on each element 21—2 324 INFINITESIMAT, CALCULUS. (cH. VIII 5A by its distance from the neutral line. In this way we get, as the value of the ‘ flexural couple,’ sp x lim 3 (p84) ieee (6), or BABIR gr (7), where A is the area of the cross-section, and & is its radius of gyration about the neutral line. The ratio of the flexural couple to the curvature (1/2) is called the ‘flexural rigidity’ of the bar. For bars of the same material it varies as Ak’. 131. Homogeneous Strain in T'wo Dimensions.* Taking first the case of two dimensions, let us suppose that, in any plane figure, the rectangular coordinates of a point (a, y) are changed to (w’, y’), where ajar, y = Bye. eee ‘aes aand @ 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 « be the common section of the two planes, we have a=1, B=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 : | | | varying 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 Rankine’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 GaSe sa Aan as Stone obs tse oe ek « 90 (2), ean ee Sia a Ta (BN Re Jb ed ei epee | a where Pe = rn O UNG Ae fee oct (4), 326 INFINITESIMAL CALCULUS. (CH. VIII and it is evident that by a proper choice of the ratios a, 8B 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 a8. For this is evidently true of any rect- angle having its sides parallel to the principal directions of the strain; and any area whatever can be approximated to as closely as we please by the sum of a system of rectangles of this type. Ex.1. Thus, the area of the ellipse (3) is af times that of the circle (2); and so =aPB.7a@=7.aa.Ba=rad'. Again, a chord cutting off a segment of constant area from a circle touches a fixed concentric circle. Hence, a chord cutting off 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 6A, 6A’ be corresponding elements of area, we have 7 lim 4) a x > (6A’) a >a ¥- 6 since dA’ =aBdA. Hence, and by similar reasoning, af = 00, of = BY ce ee (5). Ex. 2. The centre of mass of a semicircular area is on the radius perpendicular to the bounding diameter, at a distance 4/37 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 4/37 of its length from the centre. i 131-132] PHYSICAL APPLICATIONS, 327 Finally, mean squares of distances from the axes of a, y transform into mean squares of distances. Thus MI ESS (af2844) 3 (#84) 2? = lim ———,—- = a? x lim =. =. 3 (54 > (84) Hence, and by similar reasoning, Bo tt ft OU ich cases suis (6). Hx. 3. The mean squares of the distances of points within the circle (2) from the coordinate axes are w= qa, y? = Ha’. Hence, for the ellipse (3), v=lo%g?=10", w=16%a?= tba eb aod (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 P= 4 (a? +67) ....... Peo Wee (3), where a’, b’ are the principal semi-axes. b] 132. Homogeneous Strain in Three Dimensions. There is a similar method of transformation in three dimensions, the formule of transformation being now / U ! Z AD, eA re PY rie ITVS ah elas vob nastte ves (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 ie ert en (freee er. Se Lae (2), transforms into the ellipsoid ae 3 y” Zz a st b2 ie oa 1 Eee OA raga ee (3), where @=aa, WU =Ba, 6 =O ...0.000.00- (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 aBy. For this is 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 ay . Sra = frac. 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 8 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 2 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 PAC Ene es Le ey. Pee, ee fist fs v= =; Yrs; a= sQ e+ eeeeeeecee core (9), it follows that, for the ellipsoid (3), — FOC ees Wp eae Walaa A 2._.1,,/2 er =1a, yP=FO%, BPH ZO? secrecseeesere (6). The radii of gyration about the principal axes of the ellipsoid are therefore given by ; k= (b% +02), k2=2 (c?+a°), kat (a? + 67), (7). 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 go, gu, 5(a°+0?), respectively, where 2a, 2b 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 2d, is given by kK? = 26°, 3. The radius of gyration of the same segment about the tangent at the vertex is given by iat he. 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 (1 - 5) 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 i? ae, 16, where 0 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 sin 2a jo (1-5). 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 me =a? (1-7); Qa 330 INFINITESIMAL CALCULUS. (CH. VIII and the radius of gyration about a parallel axis through the middle point of the arc is given by I? = 2a? (-=*). Q 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 2 2 4 (a +0°), 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, 6, is } (a? +2). 11. The mean square of the distance, from the centre, of points within an ellipsoid of semi-axes a, 6, ¢, is 4 (a" +6? +’). 12. The mean square of the distance, from an equatorial — plane, of the surface of an anchor-ring is 46%, where 6 is the radius of the generating circle. 18. The mean square of the distance, from the same plane, of the volume of the ring, is 40°. 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’CP which bisects the horizontal chords, and is at a distance 10 PCH from the centre C, where H is the point in which PP’ produced meets the surface of the liquid. PHYSICAL APPLICATIONS. perk 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 3:7, 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 2a’. 19. If ds be an element of are of an ellipse, and @ the parallel semi-diameter, the value of the integral ds Be taken round the curve, is 27. CHAPTER IX. SPECIAL CURVES. 133. Algebraic Curves with an Axis of Sym- metry. The method of tracing algebraic curves of the type where f(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 Two points of novelty here present themselves. Since the equation gives two equal, but oppositely-signed, values of y for every value of «, 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 a (if any) for which f(a) is negative. Thus if f(#) contain a simple factor «—a,, so that the equation is of the form y? = (@ — a) b () (3), the right-hand member will change sign as # passes through 133] SPECIAL CURVES. 333 the value 2,. Hence on one side of the point (a, 0) the ordinate is imaginary. | Also, we have, at this point, (x2) fered (i) da) («#-a%) 4-2,’ and therefore, dy/dr=o. The tangent is therefore perpen- dicular to Oz. If, on the other hand, f(«) contain a double factor, say apes (Mi) Ay racecars citeaet (4), the right-hand side does not change sign as w passes through the value x, Hence the ordinate is real on both sides of the point (z,, 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 (x,, 0) is an isolated or ‘conjugate’ point on the locus. The directions of the tangent-lines at the node are given by Ue oe (SY) =lim ay = (@). If f(x) contain a triple factor, say Gf (GE — Bey OD (Ly avai cite reer esas s« (5), the right-hand side changes sign at the point (a, 0); the curve 1s therefore imaginary on one side of this point. Also since dy/dx here = 0, the curve touches the axis of «. We proceed to some examples; beginning with cases where f(z) is integral as well as rational. Ex. 1. In the cases where f(x) is of the first or second degree, say eA Bef Ae Bee OA, (6), the curve is a conic having the axis of # as a principal axis. Ex. 2. The cubical curves er et ACL) Saag Peas ost 4 cone (7), include some interesting varieties. 304 INFINITESIMAL CALCULUS. [CH. Ix (wa) If the linear factors of the right-hand side be real and distinct, we may write ay? = (% — a) (a — B) (2—y). mitecerenessues (8), and there is no loss of generality in supposing that a is positive and aa, but vanishes — for «=f. The point (8, 0) is here a node; it may be regarded as due to the union of the oval in the former case with the infinite branch. (dq) If, however, the two smaller of the quantities a, B, y | coalesce, so that ay? = (x — a)? (@—¥) «2. ee eeeeee clea (11), y will be imaginary for x a. See Fig. 77. The curve is known as the ‘witch’ of Agnesi. a °- Teo ee races Bae Ns (15). There is a node at the origin, and the curve cuts the axis of x again at (—a, 0). For a#>b, and «<-a, y is imaginary. The line «= 6 is an asymptote. See Fig. 78, Ex. 6. y= — +h eee eT et (16). This is obtained by putting a= 0 in ( 15). The loop now shrinks into a cusp; see Fig. 79. The curve is known as the ‘cissoid. L. 22 338 INFINITESIMAL CALCULUS. [cH. IX : : ma isle ak 2 Oo — Fig. 78. Fig. 79 L+a E: . oh a : ———— ee eres everest CSO Pete rere 17 e x Y= es (17) Since y is imaginary for a>ax>-a, except for «=0, the origin is an isolated point. To find the oblique asymptotes we have a\ 4 ee a x a a’\ -% S=+ =+(1+) (i= i) x a x x jar fee x Ha (Lt Se St) cocssessesntnaenneneeeen (18) Hence the lines Yy = +.(GO+4 O):.. oncegeoe deen rer eeane Lae are asymptotes. See Fig. 80. 133] SPECIAL CURVES. 339 ‘ s x ee rr ee ree www wwe ween La ecenuceerscans SSS mene Shwe Stee meme Abas eee ene e Yy’ Fig. 80. EXAMPLES. XL. 1. Trace the curves y=4a(l—-ax), ya=attatl. 2. ‘Trace the curve . ay’ = a (a— x), and shew that it forms a loop of area ,8,a?. Find where the breadth of the loop is greatest. [a = 2a] 3. Trace the curve ary? Aa (a? 7 a), and shew that it forms two loops, each of area 2a?, 22—2 340 INFINITESIMAL CALCULUS. [CH. 1x 4. ‘Trace the curves y=a(e?—l1), yaa (l- aw). 5. Trace the curve aty? = ot (a? — a), and shew that it encloses an area j7a’, 6. Trace the curve ary! = a (a? — 2%), and shew that it encloses an area 3a?. 7. The length of an are of the curve } p= (Fig. 75), from the vertex to the point whose abscissa is 2, 1s 1 Be Eines s #_ 8 oT Ja (9x2 + 4a)? — Sa. 8. The mass centre of the area included between the curve ay? = a and the line «=h is at the point (?h, 0). 9. If the curve ay?=2* 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 il 1 Oe OWENS Gide rs Teg ee x (1 — a)” 11. The area included between the curve y a-2% a? a (Fig. 77) and its asymptote is za’. If the same curve revolve about its asymptote, the volume of the solid generated is $7°a*. 12. ‘Trace the curves SPECIAL CURVES. 341 13. ‘Trace the curves Determine the maximum and minimum ordinates (if any), and the points of inflexion. 15. The area included between the curve 8 ak ae are pest (Fig. 79) and its asymptote is 37a, If the same curve revolve about its asymptote the volume of the solid generated is 177a'. 16. Trace the curve aa? ~ et a? and shew that the area included between its two branches and either asymptote is 2a?. 17. Shew that the area included between the curve aG+e a—2’ (Fig. 78) and its asymptote is } (7 +4) a y= a 18. Trace the curve 2 act es ary and shew that the area included between the curve and either asymptote is d7a*. 19. Trace the curve ox. ee a? + a? and shew that it forms a loop of area } (a — 2) a*. 342 INFINITESIMAL CALCULUS. (CH. Ix 20. Trace the curve x? y= 2 (2a — a) (a — a), and shew that it encloses an area 37a’. 134. Transcendental Curves; Catenary, Tractrix. 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. 54, viz. a= h(t), YY = (Oe eee (1), where ¢ is a variable parameter. The ‘catenary’ is the curve in which a uniform chain hangs freely under gravity. It appears from elementary statical principles that if s be the arc of the curve measured from the lowest point (A) up to any point P, and w the inclination to the horizontal of the tangent at P, then where a is a constant. Hence if w#, y be horizontal and vertical coordinates, we have ip Gap! & dy dyds_. ee dy 7 dsdy 78h Y asec =a tan sco y, Integrating, we find x=alogtan(tr+4), y=asecy...... (4). - The omission of the additive constants merely amounts to a special choice of the origin, which was so far undetermined. Since the formule (4) make, c=0, y=a for ~=0, it appears that the origin is at a distance a vertically beneath A. From (4) the Cartesian equation can be deduced without difficulty. We have © = log tan (Jw + 4) = log (see + tan Wy, 134] SPECIAL CURVES. 343 whence sec yr + tan = e%4, (5) and therefore Bech tani nress Go, jon eae Hence, by addition and subtraction, x y = asec v=acosh fa See at (6). : x s=a tan f =a sinh — @ Fig. 81. Some further properties follow easily from a figure. If PN be the ordinate, P7' the tangent, PG the normal, VZ the perpendicular from the foot of the ordinate-on the tangent, we have N4Z=yosp=a, PZ=atanpe=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 ZV; in other words, ZV is a tangent to the locus of Z. Hence this locus possesses the property that its tangent ZW 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 (1) of which is made to describe a straight line (OX). Fig. 82. 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 dy with one another, the area swept over by the tangent is given by gfady, taken between the proper limits. The whole area between the curve and its asymptote is thus found to be 47a, 135. Lissajous’ 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=boos(mt+e’)......6. (1), and it is further obvious that we may give any convenient value to one of the quantities e, ¢’, since this amounts merely to a special choice of the origin of a” 134-135] SPECIAL CURVES. 345 When the periods 27/n, 27/n’ are commensurable, we can by elimination of ¢ obtain the relation between w# and y in an algebraic form. Ex. 1. In the case n’ =n, we may write = GCOS (Né +), Y=DCOSNE ....cccerceeees (2), whence ~ - cos € =— Sin nt sine, ; sin € = cos né sin «. Squaring, and adding, we find Ce 207 y? sane — — ——~ COS €4+ 5 = SIN? €......ccccecceecns 5). Gon ab b? (3) ec akan | ee e,, wae as pa Hoar ee H hs ea i ' ! ' oN van : H ' PRS of ‘ ' ' i ‘ / ; ‘ i ' ' 1 i t ' H 4 4 i H 1 H ny i ' ' H ih H H | ‘ ' ' ty et | Leathe ie t ! Vt ae en wm ewe ee — Vege --e e Fig. 83. : _ This represents an ellipse. In the special case of «=0 or e=7, _ the ellipse degenerates into a straight line | 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. 1X its form owing to a continuous variation of the relative phase (¢) of the two component motions. When the ellipse (3) is referred to its principal axes, the coordinates of the moving point take the forms e=acos(nt+e) y=bsin (nt+e)..... eee (5). We identify nt+e with the ‘eccentric angle’; and since this increases uniformly with the time it appears “that the point (a, y) moves like the orthogonal projection of a point describing a circle of radius a with a constant velocity na. 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 m. CD, where CD is the semi-diameter conjugate to CP, C being the centre. The type of motion here considered is called ‘elliptic har- monie.’ Hx. 2. If n’=2n, we may write e2=acosnt, y=b cos (2b Pei OP Here y goes through its period twice as fast as a, and the point (0, — bcos ec) is passed through twice as nt increases by 27. The curve therefore consists in general of two loops. For e=+47, the curve is symmetrical with respect to both axes, the algebraic equation being 2 2 ne” = 45(1- 3) see meee ence necccecce (7). When ¢=0, or a, the curve degenerates into an are of the parabola 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 auxiliary circles, mark out 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 A in the figure) where the curve is furthest from the fixed straight line or ‘ base’ (BD) are called ‘vertices’; the points (D) half- way between successive vertices, where the curve meets the base, are the ‘cusps.’ A line (AS) 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 JP’ be any | NIN | A Fig. 84. other position of the rolling circle, J the point of contact with the base, C the centre, 7’ the opposite extremity of the diameter through J, and let P be the position of the tracing- point. Draw PMN parallel to the base, meeting 77 and AB 348 INFINITESIMAL 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 P will be e=NP=BI+MP, y=AN=CT-CM. Let a be the radius of the rolling circle, and @ the angle (PCT) through which it turns as the tracing point travels from A to P. We have, then, Bl=aé, PM=asin8, CM =acos 8, and therefore We y =a(1 —cos 0) From these equations all the properties of the curve can be deduced. Thus if Wy denote the inclination of the tangent to AT, or of the normal to BA, we have dy dy [dc sm Gee tan = Fe ob | Go-To whence = 40 0.060535. (2). Since the angle 77P is one-half of TCP, it follows that JP is the normal, and PT’ the tangent, to the curve at P. Cf. Art. 164, below. Again, to find the are (s) of the curve, we have dia:\* dy oe 2 2 in2z 6@| — 2 21 (5) + (35) = a? {(1 + cos 6)? + sin? 0} = 4a? cos? $8, whence, by Art. 109, s = 2afcos46d6 = 4a sin 30, or, in terms of w, 8 = 40 sinh ....sceesee eens (3), no additive constant being required, if the origin of s be at A. This relation is important in Dynamics. Since 7’'P = TI sin yf, we have arc AP = 27P =2 chord AG (4). In particular, the length of the arc from one cusp to the next is 8a. 349 SPECIAL CURVES. 136] "eg “St ~wewoen ~ — oe, ~_—aeem PO eae 350 INFINITESIMAL CALCULUS. [CH. Ix If we put y =~IM=a(1+cos0)2 (5), the area included between the curve and the base is given by fy'da =a [ (1 + cos 0)?d0@ = 4a? [cost £0d0 = 8a? fcos* yr dy. Taking this between the limits + 47, 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 CP, at a distance & from the centre, its coordinates will be x=a0+k sin 0 y=a—kcosé } When & >a we have loops, which in the particular case (k = a) of the cycloid degenerate into cusps. When k }, in the pericycloids a< 6. 0, x% =(a—b)cos 8 —bcos y =(a—b6)sin 0+ bsin Fig. 87. Similarly, for the locus of P’ we obtain % = (a — b) eos 64} cos ee b eb Pe (6). y = (a—b)sin 6-6 sin 6 b 137] SPECIAL CURVES. 353 To find the tangent at any point of an epicycloid, we have from (1), since dd/d@ = a/b, dy cos 9 + cos (0 + ¢) et and ren Oy 7 OT HP). _ On reference to Fig. 86, we see that 6+ 4¢ is the inclination of JP to OA. Hence /P is normal to the epicycloid at P. A similar result can be deduced, of course, for the peri- eycloids and hypocycloids, from the equations (5). Cf. Art. 164, Again, from (1), (a) + (34) (a+ 6)'0" oh bs (2 + 2cos ¢)= Age ie g cos’ 4, dd or is we ee eA) COS Ap aes ee dtes caves (8). 4(a+b)b.. _ sod) sin $¢.. Sere no additive constant being necessary, if s=0 for ¢=0. If we denote by y the inclination of the normal JP to OA, we have Hence a+2b vp=0+4¢d= 2a Crease rita fees (10), _4(a+b)b. a and therefore s= arg ll aeop ee fee eae (11) The formula (9) has a simple interpretation. It appears from Fig. 86 that 7’P = 2b sin4¢, whence In particular, the length of the curve from one cusp to the next is 8(a +b) b/a. _ The corresponding results for the pericycloids and hypo- cycloids are easily inferred by changing the sign of b. * Cf. Newton, Principia, lib, i., prop. xlix. 354 iINFINITESIMAL CALCULUS. {CH.EX The curve traced out by a point of the rolling circle which is not on the circumference is called an ‘epitrochoid,’ or a ‘hypo- trochoid,’ as the case may be. If & denote the distance of the tracing point from the centre of the rolling circle, the expressions for the coordinates x, y in the various cases are obtained by writing & for 6 in the coefticients (only) of the second terms. 138. Special Cases. 1°. If the radius of the fixed circle be infinitely great we fall back on the case of the cycloid. The corresponding equations are easily deduced from Art. 137 (1), writing «+a for 2, a0 = bd, and (finally) @ = 0. 2°, Again, making the radius of the rolling circle infinite, we get the path described by a point of a straight line which rolls on a fixed circle. The curve thus defined is called the ‘involute of the circle’; see Art. 161. The equations may be obtained as limiting forms of Art. 187 (4), or they may be written down at once from a figure. We find “2=acos 6+ aé@sin 0, y=asin 0 — af cos 6 Fig. 88. The corresponding trochoidal curve is x=(a +h) oO ee y =(a+h) sin 6 — a6 cos 0 where h=PQ in the figure, @ being the tracing point. The particular case of h=-a gives the ‘spiral of Archimedes,’ see Art, 140. a | 137-138] _ SPECIAL CURVES. 355 3°. If the radu a, b be commensurable, then after some exact number of revolutions the tracing point will have returned to its original position, and its subsequent course will be a repetition of the previous path. In such cases the curve is algebraic, since the trigonometrical functions can be eliminated between the expressions for # and y. Sometimes the equation is more conveniently expressed in polar co- ordinates. Figs. 89, 90, 91 shew the epi- and hypo-cycloids in which the ratio of the radius of the rolling circle to that of the fixed circle has the values 1, $, 1, respectively. Fig. 89, Fig. 90. 356 INFINITESIMAL CALCULUS. (CH. 1x Fig. 91. We proceed to notice in detail one or two of the cases which have specially important properties. Ex. 1. The ‘cardioid.’ If in Art. 137 (3) we put b=a, we get a2=2acos6+acos20, y=2asin#+asin 20, whence x+a=2a(1+cos6)cos6, y=2a(1+cos 6) sin 6...(3). This shews that the radius vector drawn from the point (—a, )) as pole is given by 1 = 2a (1+ COS) ...sccccsrecsoessreeees (4). Fig. 92. 138] SPECIAL CURVES. 357 This is otherwise evident from Fig, 92, where A'P =2A'N =2 (OI + A’M). The corresponding trochoids are given by «= 2acos6+kcos26, y=2asin6+ksin 26. Referred to the point (—/, 0) as pole these formule are equi- valent to T= 2(G4+K COS O) .015-- voce cveciseees (5), which is the polar equation of the ‘limagon’ (Art. 141). This equation, again, is easily obtained geometrically. Ex. 2. = COb @.... 55s cee eee 1), EY cot a (1) whence, integrating, log r = @ cot a+ const., or Y= (69 8" ere ae (2). Fig. 100. As @ ranges from — 0 to +0, 7 ranges from 0 to 0. See Fig. 100. r 140] | SPECIAL CURVES. 367 Since, by Art. 110, we have dr/ds=cosa, it appears that the length of the curve, between the radii 7,, 7, is ra ds i} ie dr = (7, — 1) SOU Goose apistols a (3). 1 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, ash EEO whence TE nat See ee St ae is (4), ifa=u/n. Fig. 101, Fig. 101 shews the curve. The dotted branch corresponds to negative values of 0. Another mode of generation of this curve has been explained in Art. 138, 368 INFINITESIMAL CALCULUS. [CH. 1X 3°. The ‘reciprocal spiral’ is defined by the equation If y be the ordinate drawn to the initial line, we have : * sin @ y=rsind=a——. As 6 approaches the value zero, r becomes infinite, but y approaches the finite limit a Hence the line y=a@ is an asymptote. | Fig. 102. The dotted part of the curve in Fig. 102 corresponds to negative values of 0. 141. The Limagon, and Cardioid. If a point O on the circumference of a fixed circle of radius 4a be taken as pole, and the diameter through O as initial line, the radius vector of any point Q on the cir- cumference is given by r= 6.6080... csi (1). If on this radius we take two points P, P’ at equal constant distances c from Q, the locus of these points is called a ‘limacon.’ Its equation is evidently | T= 0 COS O 4.6...) Sie eee (2). This includes the paths both of P and of P’, if @ range from 0 to 2zr. 140-141] SPECIAL CURVES. 369 If ¢ a, r cannot vanish ; see the curve traced by P;, P, in the figure, Fig. 103, In the critical case of c= a, the loop shrinks into a cusp. The locus is now called a ‘cardioid’ or heart-shaped curve. Its equation is Pe CIP COS Oi. stoke cceocc. (3). See the curve traced by P,, P; in the figure. Also Fig. 89, p. 355. L. 24 370 INFINITESIMAL CALCULUS. [CH. IX 142. The curves r”=a" cos 70. A number of important curves are included in the type 7” = a” COS NO. a ee (2): Thus if n= +1, we have the circle Y= 0 008 0... ca eee (2) and the straight line . f COS: 0: = & ..so0 ete (3). If n= + 2 we have the ‘lemniscate of Bernoulli’ 7 = a? C0820 1.5) (4), and the rectangular hyperbola 7? COS.20 = 07. eee (5). The equation (4) makes r real for values of @ between + lr, imaginary for values between 47 and #7, and so on. Also r? is a maximum for 6=0, 0=7, ete. It follows that the lemniscate consists of two loops, with a node at the origin. See Fig. 113, p. 389. If n= + 4, we have the cardioid v= atcos40, or r=ha(1+cos@)......... (6), and the parabola 2a ricos40=ai, or r= Dicos Qc (7). The curves corresponding to equal, but oppositely-signed, values of », 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, ldr cot p=— 7, = tan 10 3. tee ot eee (8), or b= hr +005 ee (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 6 and ¢ between the formulze ldr p=rsin q, do Ct CSTE TEES (1) (for which see Art. 55) and the given equation. From (1) we obtain ital Tee fare —=— 0) ee nt a re 2). waz jal + cot 6) = 45 (B) eeneeeee (2) It is occasionally convenient to employ the reciprocal of the radius vector instead of the radius itself. If we write = : we have aS UE (3) = Y’ dé = 7 dé cose esccesee 2”), and the formula (2) takes the shape 1 (du? eee Ee Aap owes tulle Sos CF dain dies 4). p Ue (a) (4) Exe. 1. In the equiangular spiral, we have | | WP eTOCs keeU ett eh Et Cte asc eu e: (5) Peoamem the circle, t= Lo sin 0. ilis ch sysus Ainild es sae se (6) we have, as in Art. 55, #=8, and therefore p/r =1/2a, or 5 1 pel SO Nag ah a eg (7). Hx. 3. In the parabola Hise BG BOG hE ie Ma. hee cafe de fae, (8), where the focus is the pole, we find $6=47- 430, p=7 cos 18, whence 5 VE et PROTEC CERT ER RE TS (9) This is a well-known property of the curve. 372 INFINITESIMAL CALCULUS. [CH. IX This example, like the preceding, is included in a general result embracing all curves of the type 7” = a" COS 10 ...c2 et (10). By Art. 142 (9) we have p=rsin 6=% COS 10 fanaa eee (11), whence, eliminating 0, pala” 3s (12). Thus in the case of the cardioid (n= 4), we have ; PaO... (13). Hix. 4. The tangential-polar equations of the central conics may be given here, as they are sometimes employed in Dynamics, although the proofs will not require the use of the Calculus. First, let the origin be at the centre. The Cartesian equation of the conic being — +=! MP (14), a we have, if 8 be the semi-conjugate diameter, pB=ab, and f+ =0' 20 (15), by known properties of central conics. Hence 22 < = 6? © G3 17. ste (16). In the particular case of the rectangular hyperbola we have DY = 0 0. i (17), since 8=r. This is also obtained by making m =— 2 in (12) above. Ex. 5. Again, taking a focus as pole, let us denote the per- pendicular and radius vector corresponding to the other focus by p and r’. Since the tangent makes equal angles with the two focal radii, we have p/r = p'/r’, and therefore BUY, rr Now pp’=0", and, in the ellipse, r+7’=2a. Hence, for this curve, p'/r = 8'/(2a—r), or, if / denote the semi-latus rectum (b?/a), Late | 143] SPECIAL CURVES. 373 In the hyperbola, we find y Ze} ++ the upper sign relating to the branch nearest to the origin, the lower to the further branch, It is important, with a view to some applications in Dynamics, to notice that if the tangential-polar equation be given, say aes e (To) ee ee ce ee (20), (Sag tan oan eae (21), ig gia ad aia whence O—a =| 2 Seo ony (22). A variation of the additive constant a has merely the effect of turning the curve bodily through an angle about 0. Ex. 6. To find the curve in which Substituting in (21), and integrating, we find rdr r 0 — cae i i in-t = 2 eee —7) 2 CE, or foes BIN OG =a) 3.01), aos, Megeediet (24), a lemniscate. EXAMPLES. XLII. 1. Prove that all equiangular spirals of the same angle are identically equal. - 2. Prove that in an equiangular spiral of angle a the area Swept over by the radius vector (7) is 4 (r?— 7,7) tan a, where 7,, 7, are the extreme values of +. 374 INFINITESIMAL CALCULUS. [CH. Ix 3. Prove that in the spiral of Archimedes the angle (¢) between the tangent and the radius vector is given by a a FP)’ cos ¢ = Weer) 4. Prove that in the reciprocal spiral the area swept over by the radius increases proportionally to the radius. 5. Shew that all chords drawn through the pole of a cardioid are of the same length. Does the same hold of the limacgon ? 6. The area of the cardioid 7 =a(1+cos 6) is 37a. 7. Prove that, in the cardioid, d. = = 2a cos 40, and thence that the whole perimeter is 8a. 8. The volume generated by the revolution of the cardioid about its axis is $7a’. 9. Prove that, in the cardioid, the maximum breadth (per- pendicular to the axis) is 3,/3a, and that the double tangent cuts the axis at a distance 4a from the pole. 10. Find the maximum ordinate, and the minimum abscissa, in the limacon r=acosO+e¢, 11. The area of the limagon r=acos6 +e, when ¢>4a, is am (c? + 30°). 12. Prove geometrically that if two straight lines, touching two fixed circles, make a constant angle with one another, then intersection traces out a limagon. 13. The whole area of the lemniscate 7? = a? cos 20 J) ~I Or SPECIAL CURVES. 14. The perimeter of either loop of the same curve is hr do 0 vi ( (1 — 2 sin? 6) ° Prove that, in the notation of elliptic integrals (Art. 109), this is equal to J2aF, (= 5 ae 15. The mass-centre of the area of either loop of the lemniscate 2a - is at a distance 4,/27a from the pole. 16. Shew that the area included by one loop of the epicyclic *=asin mO is ra?/4m. 17. Trace the curve 7*?=a’? cos 8. 18. Prove the following properties of the ‘solid of greatest attraction’ oz. the figure “generated by the eavolaien: of the curve 77= a’ cos 0 Biot the inisal line): (1) The volume is pera’ ; (2) The greatest breadth is 1:2408a, at a distance °4389a from the pole; (3) The mass-centre of the volume is at a distance 13a 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 d6/dr. 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 para, the centre being pole. 21. Shew that in the spiral of Archimedes (Fig. 101) 9 og ee ae 376 INFINITESIMAL CALCULUS. [cH. 22. Shew that in the reciprocal spiral (Fig. 102) 1 Sal St Pe = 7 aie we ° 23. Shew that in the curve a ~ cos m6’ 1 l= 24. Shew that in the curves . Laas: eae = coshm0 eo Senne ae 1 14m? _m!? respectively. 25. Prove that, in the epicycloid (Art. 137), 4(a+b)b ary? What is the corresponding formula for the hypocycloid ? = @2 + 2 26. Prove the formula rdr V(r? =p") for the are of a curve whose tangential-polar equation is given. — 27, Prove the formula pds = r°dg, 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 ee | tangent to the path. : 144. Associated Curves. Similarity. . There are several methods of associating with a the curve another curve connected with it by a definite rots } 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 poimt 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 o straight lines in the other, and angles in the one figure to qual angles in the other. If we take rectangular axes OX, OY in the one figure, nd the corresponding axes 0’X’, O’Y’ in the other, the oordinates x, y of any point P in the one figure will be onnected with the coordinates a’, y’ of the corresponding oint P’ of the other by the relations here m is a constant ratio. If by a displacement of either figure in its own plane ‘X’ and O’Y”’ can be made to coincide with OX and OY, spectively, the two figures are said to be ‘directly’ similar. the new position, any two corresponding points P, P’ are a straight line with 0; moreover OP’=m.OP, and any 0 corresponding straight lines are parallel. The same atements hold if either figure be turned about O through o right angles, until OX’ coincides with XO produced, and Y" with YO produced. In either of these two relative sitions the figures are said to be ‘similarly situated,’ and e origin is called the ‘centre of similitude. If Meath) ere eh ey. ee ma ees ys (3) the polar equation of any curve in the one figure, the uation of the corresponding curve in the other will be eG Ua Dar yes ete re (3). It may happen, however, that when O’X’ is brought into incidence with OX, the line O’Y’ coincides, not with OY 378 INFINITESIMAL 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 (— 0)... (4). We shall not have occasion to consider this kind of relation. Ex. 1. Conics of the same excentricity are similar. Let the two conics 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 ? LENE eV aecee folate (5), Lr r whence it follows that radii drawn in the same direction are in the constant ratio 7: 0’. Ex. 2. ‘All catenaries are Similar curves. For the equation y = Cosh a/0* i205. as den eee (6) is unaltered when 2, y, ¢ are all altered in any the same ratio. 145. Inversion. If from a fixed origin O we draw a radius vector OP to any given curve, and in OP take a point P’ such that OP.0P'=4#. i (1), where i is a given constant, the locus of P’ is said to be the ‘inverse’ of that of P. The point O is called the ‘centre, and + is called the ‘constant, of inversion. Q’ Pp? Fig. 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 CURVES. 379 curve, and P’, Q’ the corresponding points on the inverse curve, we have OP.OP’=OQ.0Q’, and therefore Nien OG Whete Gwenn sscntaves Fen (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 augles 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 TV; = CONSE., 175 = CONSE: «i. edecesscses (3), we have els == CONS. Vays ses os aches ee, (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 O from the given straight line. If P be any point on the line, and if OP meet the circle on OA as diameter in P’, we have (ine tales (1 A8 cairn ees clek cose. (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 _ fx. 2. More generally, the inverse of any circle is a circle. pr P p' Fig. 106. Let O be the centre of inversion, C’ the centre of the given circle, a its radius ; and let P=00_8 (6). If, then, we draw any chord OPP’ through 0, it is known from Geometry that OP. OP =00 —@ak ae (7) Hence P’ traces out the inverse of the locus of P; 7¢. 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, O be within the given circle, the constant in (7) is negative. This means that P’ and P are now on opposite sides of O. 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 O. 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 O. If W be the intersection of the diagonals of the rhombus, we have OP.0Q = ON? ~ PN? = 0A? ~ AP? =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=SP, the locus of P is a circle through O, 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’ ABCD formed 382 INFINITESIMAL CALCULUS. [cH. 1x of four rods jointed at their extremities, the alternate sides being equal. A point O in one side AB is made a fixed pivot, and P, Q are points in AD and BC such that AP: PD=(CQ: QB=A0: 0B =i ay. Evidently O, P, Q will lie in a straight line parallel to AC and BD. If H, K be the orthogonal projections of A, C on BD, and if NV be the middle point of BD, we have AC. BD=2NH .2NB = DH’ —- BH? =AD?— AB’. Now OP: BD=A0:AB=™ Gee and 0Q:AC=B0O :AB=n :m+n. Hence OP.O0Q= a a AD? AB?) = const. ...... (2). Hence P and Q describe inverse curves with respect to O. As before, by connecting P to a fixed pivot S 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 O 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 O. 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 be the angle which OZ makes with any fixed straight line, then p, ~w may be taken to be the polar coordinates of Z with respect to O as pole. Hence if the relation between p and wW can be found, the polar equation of the pedal can be at once written down, Hx. 1. If the origin be at the centre of the conic and w be the angle which p makes with Oz, it is shewn in books on Conic Sections that p= oF costay + OF sin” aes eee (2). 146-1 47] SPECIAL CURVES. 383 Hence the polar equation of the pedal is Fp Fe SR aR eS Vad cb thane alee ia (3). In the case of the rectangular hyperbola PO at A eek i yackes sew cl Ces Se (4) the pedal is the lemniscate Wigs, COSAD. tenet e asta cetees (5). Ex. 2. In the case of a circle of radius a, the pole O being at a distance c from the centre C, and the line OC being the origin of w, we have at once from a figure HR ORO CON Wc ein tanee stents 1b eos (6). Hence the pedal is the limagon TEAC COR-U es tes vent ve rens 155 C58 (7). If O be on the circumference, we have ¢ =a, and the pedal is the cardioid Pre RREGETCOS 2) A ch cooks ce kcr cued o (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 O on two consecutive tangents PZ, PZ’, and let OU 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 original curve, respectively. Also, by similar triangles, we have OU OF O70 Lae 22 te sth. cers (9). Hence if r be the radius vector of the original curve, p the 384 INFINITESIMAL CALCULUS. [CH. Ix perpendicular from O on the tangent, and p’ the per- pendicular from O on the tangent to the pedal, we have, ultimately, plo=olr, or Pp =p (10). Again, if OZ’ meet PZ in NV, we may write, OZ=p, OZ’=p+8p, 2240Z'= ZZPLZ’=5. Neglecting small quantities of the second order, we have 6p = NZ’ = PLZ'dyp. 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. This result enables us easily to solve the problem of ‘negative pedals, viz. to find the curve having a given pedal. Taking O as origin, and the initial line of W as axis of a, the coordinates of the poimt of contact P are given by x=OZcosw—ZPsiny, y=O0Zsin y+ ZP cosy, or o=poosy— FP sin yp, d ft crea CE2), Ny ) Ee ae, cos x. 3. To find the curve whose pedal is the cardioid ¢=a(1 + cos 0)... pence ee (13). Writing p=a (1 4.008 W)=:..- ee (14), the formule (12) make x=acosy+a, y=asiny, whence (2 —~ 0)" + =O) verse. eee (15), a circle through the origin. 148. Reciprocal Polars. The locus of the pole of the tangent to a curve S, with respect to a fixed conic , is called the ‘reciprocal polar’ of 147-148] SPECIAL CURVES. 385 S. It is proved in books on Conics that if S’ be the locus of the poles of the tangents to S 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 %isacircle. If O be the centre of this circle, and k denote its radius, the pole P’ of any tangent to the curve S is found Fig. 110. by drawing OZ perpendicular to this tangent, and by taking in OZ a point P’ such that O78 O PEA eng re (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 S02. OP =e ob... (2). Hence P describes the inverse of the locus of 7’. se 25 386 INFINITESIMAL CALCULUS. [CH. Ix Hx. 1. The reciprocal polar of a circle with respect to ha origin is a conic having the origin as focus. As in Art. 147, Ex. 2, the formula for the pedal of the ake is P=H=G4+C COSY swe eee ee (3). Writing @ for y, and #/r for p, we get the equation of the reciprocal polar in the form k2 = =a +008 8 » cnpasiealeaeeneee cs ate (4), which represents a conic, having its focus at the origin, of eccentricity c/a. 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 = 02 y” ae = % = 1 cevecssvcreresseseecertoecs (5), with respect to the centre, is given by p) =a? cos’ yt 0? sin? W aac eee (6). Hence the reciprocal polar is 4 L = a* cos? 0+ 6? sin? 0 (ieee Ga or aig? + by? = I os ines ane (8), a concentric conic. 149. Bipolar Coordinates. A curve may be defined by a relation between the distances (7, r’) of any point P on it from two fixed points, or foci, S, S’; thus If we denote the angles PSS’, Ps 'S by 0, 0, respectively, and the angles which the radii 7, 7’ make with the tangent by ¢, ¢’, we have, as in Art. 110, dr dr’ ; rhe rr (2) dd + / de’ ae ot ee Sra . ee ee ? Wp SNe 1 48-$49] SPECIAL CURVES. 387 We have, in addition, the relations rsind=r'sin@, rcos@+r' cos @’=2c...... (3); where c=4SS’, Fig. 111. Ex. 1. In the ellipse we have ihe wearer pe operrres (4), dr dr’ and therefore Tag that is, cos¢+cos¢’=0, or gf =r7—............ (5). The focal distances therefore make supplementary angles with the curve. Similarly, in the hyperbola WIE Ab sens ctascesedine gaat ss ts.<8 (6), we find COB = COR ie a cote ehiy es fia : (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 8’. 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. [CH. Ix In the case of refraction, we have, if » and p’ be the refractive indices of the two media, painy—plsin Yr (8), where. X= (4r=—$9), ¥=t Gro (9). Hence pcos b+ pt cos hb =O) ee ee (10), or | = (ur + po) = O35 ee 17). Integrating, we have | pr tr’ = const ee (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 circle r/o! c pel ph cic ee (13). See Fig. 112. =" 149] SPECIAL CURVES. 389 Hx. 3. The ‘ovals of Cassini’ are defined by k being a given constant. Since for a point P in SS’ the greatest value of 77" is c’, it follows that the curve will consist of two detached ovals surrounding S, 8’, respectively, or of a single oval embracing both points, according as k& < ¢. fe) ees DRG Fig. 113. In the critical case of k=c the curve is known as the ‘lemniscate of Bernoulli’; this presents itself in various mathe- matical problems. If O, the middle point of S’S, be taken as pole, and OS as initial line, of a system of coordinates 7,, 0,, we have 7=7,2+c?—2cr,cos0,, r?2=717,? + c? + 2er, cos 6, ; the equation of the lemniscate is therefore (72 + 07)? — 4071? cos? 6, = c4, which reduces to Pig 20 COMA A ais Sead se (15). Cf. Art. 142. Ex. 4. The magnetic curves. If S, S’ be the N. and 8. poles of a magnet, the forces at any point P may be represented by p/7? along SP, and p/r? along PS’. A ‘line of force’ is a line drawn from point to point always in the direction of the resultant force. Expressing that the total force at right angles to the line is zero, we have B Esin } + 4, sin ¢' =0, 390 INFINITESIMAL CALCULUS, [CH. Ix 1d0 1d0’ or PE EAP = Opn. as i eee (16). Hence, since 7 sin 6 =7" sin 6’, we have dO _ de’ sin 6 — EF + sin 6’ — ae =) or cos 6+ cos 6 = const. s. css. peers (17). An ‘equipotential line’ is a line such that no work is done on a magnetic pole describing it. Expressing that the total force in the direction of the line is zero, we find ldr 1dr ce = Pci reek eee (18), whence bas = == CONSE. . 22ehs,sausen eee (19). r Lr The equipotential lines will necessarily cut the lines of force at right angles. EXAMPLES. XLIII. 1. The inverse of an equiangular spiral with respect to the pole is an equal spiral. 2. The inverse of a hyperbola with respect: to the centre has a node at the centre. 3. The inverse of a rectangular hyperbola with respect to the centre is a lemniscate of Bernoulli. 4. Prove by means of the polar equations that the inverse of a straight line is a circle through the pole of inversion, and conversely. 5. Prove by means of the ee equation that the inverse of a circle is a circle. 6. The inverse of a parabola with respect to the focus is a cardioid. The inverse of any conic with respect to a focus is a limagon. pias 149] SPECIAL CURVES. 391 7. Prove that the inverse of the ellipse cree Sy a! Bt 2 l with respect to the centre is the curve a 2 (ac? + y*)? = k* (+k): Also shew that the curve, where it cuts the axis of y, will be concave or convex to the origin according as 6? s 2a’. 8. From the fact that the cardioid is the inverse of a parabola with respect to the focus, or otherwise, prove that the normals at the extremities of any chord through the cusp are at right angles, and that the line joining their intersection to the cusp is perpendicular to the chord. 9. Prove by inversion, or otherwise, that the cardioids r=a(1+cos@), r=b(1—cos 6) cut one another at right angles. 10. If ds, ds’ be corresponding elements of a curve and its inverse, eR S Rea Vie Hike —' Kar eace, where 7, 7’ are the radii. 11. The pedal of a parabola with respect to its vertex is the cissoid (Art. 133 (16)). 12. If two tangents to a curve make a constant angle with one another, the locus of their intersection (P) touches the circle through P and the two points of contact. 13. Prove that the area of a pedal curve is given by the formula 4 [p'dw. 14. Prove that the arc of a pedal curve is expressed by frdy. 15. The area of the pedal of an ellipse, the centre being pole, is dar (a? + 6°), where a, b are the semi-axes. 392 INFINITESIMAL CALCULUS. | [CH. IX 16. The pedal of the hyperbola x a? apo? with respect to the centre consists of two loops, each of area pea! dab + 4 (a — 6°) tan™ ae 17. If p,, p, be perpendiculars on the tangent to a curve from the origin of (rectangular) coordinates, and from the point (x1, Y¥,) respectively, prove that Pi=Pp cosy — Y sin YW; where y is the inclination of the perpendiculars to the axis of a. 18. If A,, A, be the areas of the pedals of a closed oval curve with respect to the origin O and with respect to the point (x,, ¥,), both these points being within the curve, prove that lr Qa A,=A)- x, | py cos wd — yy | py sin wd + da (a? + yy"). 0 0 19. Prove that the locus of a point such that the pedal of a given closed oval curve with respect to it as pole has a given constant area is a circle; and that the circles corresponding to different values of the constant are concentric. Also that, if O be the common centre, the area of the pedal with respect to any other point P exceeds the area of the pedal with respect to O by the area of the circle whose radius is OP. 20. The negative pedal of the parabola y? =4ax with respect to the vertex is the curve 27 ay? = (x — 4a), 21. In what case is | p=acosy} 22. Prove that the curve for whieh p=asin y cosy is the astroid ei+ ys = as, SPECIAL CURVES. 393 23. State what property follows by differentiation with respect to the arc (s) from the equation rP+rt!=k, and verify the result geometrically. 24. Prove the following construction for the normal at any point P of a Cassini’s oval: In PS, PS’ take points Q,.Q’, re- spectively, such that PY = PS’, and PQ’= PS; the line joining P to the middle point of YQ’ is the required normal. 25. A system of parallel rays is to be reflected so as to pass through a fixed point ; prove that the reflecting curve must be a parabola, 26. A system of parallel rays is to be refracted so that their directions pass through a fixed point; prove that the refracting curve must be a conic, and that the eccentricity of the conic will be equal to the ratio of the refractive indices. 27. Prove that the equation of a Cartesian oval, referred to either focus as pole, is of the form rv —2(a+ bcos 6)r+c=0. 28. Prove that a Cartesian oval is necessarily closed, if we except the case where the curve is a branch of a hyperbola. CHAPTER X. CURVATURE. 150. Measure of Curvature. As regards the applications of the Calculus to the theory of plane curves we have so far been concerned chiefly with the direction of the tangent at various points. We have not considered specially the manner in which this direction varies from point to point. The subject of curvature, to which we now proceed, can be treated from several independent stand-points, and although all the methods lead to identically the same formule, it is important for the student to observe that they are in their foundations logically distinct. In the first of these methods*, we begin by defining the ‘total’ or ‘integral’ curvature of an arc of a curve as the angle (dy) through which the tangent turns as the point of contact travels from one end of the arc to the other. The ‘mean curvature’ of the arc is defined as the ratio of the total curvature to the length (és) of the arc; it is there- fore equal to ov os * The ‘curvature at a point’ P of a curve is defined as the mean curvature of an infinitely small arc terminated by that * Other methods are explained in Arts. 153, 154. * 150] CURVATURE. 395 point. In conformity with the previous notation it is denoted by dy > (1) In a circle of radius R we ae 6s = Row, and therefore ayy _ ds == 2.e. the curvature of a circle is measured by the reciprocal of its radius. Hence, if p be the radius of the circle which has the same curvature as the given curve at the point P, we have P= dap Coc ccc crore teeccccesecceces (2), A circle of this radius, having the same tangent at P, and its concavity turned the same way, as in the given curve, is called the ‘circle of curvature, its radius is called the ‘radius of curvature, and its centre the ‘centre of curvature.’ The length intercepted by this circle on a straight line drawn through P in any specified direction is called the ‘chord of curvature’ in that direction. If @ be the angle which the direction makes with the normal, the length (q) of the chord is given by If &, » be the rectangular coordinates of the centre of curvature, we have by orthogonal projections E=a—-psny, n=y+pcosw............ (4), provided the zero of yy be when the tangent is parallel to the axis of a. The centre of curvature is the intersection of two con- secutive normals to the given curve. Forif PC, P’C be the normals at two consecutive points, including an angle dy, and if és be the arc PP’, then drawing the chord PP’ we have (see Fig. 114) (GE men Chae: PP’ sindy ’ a : ae oy os or CP=sin CP’P .—— Serioal oe 396 INFINITESIMAL CALCULUS. [CHyex When P’ is taken infinitely near to P, the limiting value of each factor on the right hand, except the last, is unity. Hence, ultimately, CP =ds/dv = p. A. Pp Fig. 114. In modern geometry a curve is regarded as generated in a two-fold way, first as the locus of a point, and secondly as the envelope of a straight line (see Art. 158). Considering any continuous succession of these associated elements, the straight line is at any instant rotating about the point, and the point is travelling along the straight line; and the curvature dy/ds expresses the relation between these two motions. If at any point the curvature is zero, the rotation of the tangent is momentarily arrested, and we have what is called a ‘stationary tangent.’ The simplest instance of this is at a point of inflexion (Art. 68), where the direction of the rotation of the tangent is reversed after the stoppage. If at any point the radius of curvature (ds/d) vanishes, the motion of the point along the line is momentarily arrested, and we have a ‘stationary point.’ The simplest instance of this is at a ‘cusp’ such as we have met with in Figs. 75, 79, 84, 88, etc. The direction of motion of the point is in such cases reversed after the stoppage. In the examples of Art. 133 a cusp was regarded as due to the evanescence of a loop: this shews in another way why the radius of curvature should vanish there. The consideration of curvature is of importance in numerous dynamical and physical problems. For example, in Dynamics, if the force acting on a moving particle be resolved into two 150-151] CURVATURE. 397 components, along the tangent and normal to the path, re- spectively, the former component affects the velocity, and the latter the direction of motion. If from a fixed origin we draw a vector OV to represent the velocity at any instant, the polar coordinates of V may be taken to be v, y, where v= ds/dt. Hence the radial and transverse velocities of V will (see Art. 110 (8)) be dv dw oF and On ae (5), respectively. These are the rates of change of the velocity estimated in the direction of the tangent and normal to the path of the particle. Since a= yt ds_ wv 6) is ae ioe pe at), the latter component is equal to the product of the curvature into the square of the velocity. 151. Intrinsic Equation of a Curve. The formula is of course most immediately applicable when the relation between s and w for the curve in question is given in the form This is called the ‘intrinsic’ equation of the curve, for the reason that its form does not depend materially on space- elements extraneous to the curve. The only arbitrary elements are the origin of s and the origin of w, and a change in either of these origins merely adds a constant to the corresponding variable. Hx. 1. In the catenary we have EAE PAS Sty hove ow Mae mee | (3), whence Pe POE Ai ee WSO Urea crc ness teen (4), the notation being as in Art. 134. On reference to the figure there given it appears that the radius of curvature is equal to the normal PG. 398 INFINITESIMAL CALCULUS. [cH. x Hx. 2. In the cycloid (Art. 136) we have §=4.Sin Wiiesccseseeee aie ore notes (5), and therefore p =a COS Us :. ss .e neces eee (6). Hence in Fig. 84, p. 347, we have p=2PJ, or the radius of curvature is double the normal. Ex. 3. Again, in the epicycloid we have (Art. 137 (11)) 2 (Gb) 0 asa bra aera Uf acco cete west tigas (1, and therefore _4(a+b)b a 4(a4+d)b- | = on O88 gop cos £...... (8) Hence, on reference to Fig. 86, p. 351, it appears that 2 (a + b) = non PL. i. casciep eee (9), where PJ is the length of the normal between the tracing point and the fixed circle. If the intrinsic equation be not known, we may employ one or other, of the formule of Art. 152; or we may, in particular cases, have recourse to special artifices. Ex. 4. In the parabola y? = 4ax we have, by Art. 53 (9), y = 20 Coby...) 2 veaseeeen eee (10), ne Ree 2a dw whence sin y= det ae ae 2a or ie rn fob uhed oo bette ae Seer (11), the negative sign representing the fact that y diminishes as s - increases. Ex. 5. Tf the ellipse 2 = COS hb, f= 0 SIND sates Re awaeee (12) be supposed derived by homogeneous strain (or by orthogonal projection) from the circle e=a4C0S h, Y=QsiD Ge ee (13), 151] CURVATURE. 399 we have _ Sars ate rita sss ager’ Dees (14), do where £ is the semi-conjugate diameter. For the element of arc is altered from ad¢ to ds, and the parallel radius from a to £. Also since $6°6y and 4a73¢ represent corresponding elements of area, we have Boy = 2 x wd, dp _ & or Fy RSA Pa ot Oe (15). _ds_dsdd_f? Hence Eee edd dl cab Seneaepinre ere nerd (16). If p be the perpendicular from the centre on the tangent-line, we have p = ab, so that our result may also be written 2 ab? 1 =—, or oo eee eT usashv rcs ceeab 7). Pp D p (17) Since pp? = a cos? yy + 6 sin? y = a? (1 — e sin? W), the last form is equivalent to 1-é pe OTe sin yh eialara' dials mleis ticiisteveicl eral cvera (18). 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 y, is, if we neglect e, Gna —@ + $e" sin? Wy) =a (1 — de — Se cos 2) ... (19), where «= (a—b)/a=e?; 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 y, s=a(1—te) W— faesin QW ............ ee (20). Ex. 6. In the equiangular spiral (Art. 140), we have gee ie MG 4 Peta ie LA Ree EA a (21), whence diy/ds = d6/ds = (sin a)/r, : = 22 or Sg tates (22). Hence the radius of curvature subtends a right angle at the origin. 400 INFINITESIMAL CALCULUS. [CH x 152. Formule for the Radius of Curvature. The expression d/ds for the curvature is easily translated into a variety of other forms. 1°. In rectangular Cartesian coordinates, we have and therefore Dae ee d (3%) _ ey da _ : poe ‘ds ds\d«)~ da ds dx?’ d’y whence es ee «ice pega dns ak One (2). he dae This form shews, again, that the curvature vanishes at a point of inflexion. fx. 1. In the catenary Y = @ COSN 2/0 vee, on keene (3) we have AY i) ey ty aa “2 dy\2= it Jy 7 nhs daa = OO Gg? 1+ (3) = cosh a3 whence p= weosh’ /a=77/a 2 (4). Since y = asec y, this agrees with Art. 151, Ex. 1. When dy/dz is a small quantity the formula (2) gives, approximately, the proportional error being of the second order. The form (5) is an obvious transcript of dy/ds, since when w is small we may write dy/dx (= tan ~) for y, and d/dx for d/ds. 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] CURVATURE. 401 deflection at any point, the ‘bending moment,’ or ‘flexural couple,’ is 2°. It was proved in Art. 147 that the projection (¢) of the radius on the tangent is given by If OU, OU" be the perpendiculars trom the origin on two consecutive normals PC, P’C, and if OU’ meet PC in N, we have, ultimately, OU’—OU=U'N=CNby, or dSt=CNSw. The limiting value of CU or CN is therefore dt/dy, whence rns TE VASO sg yd oh amy Fig. 115. 3°, With the notation of Arts. 110, 147 we have ds _dsdrdp _,dsdr dy dr dp dy ‘drdp’ L. 26 Since 4,02 INFINITESIMAL CALCULUS. (CB x. this gives POET steerer (10), a form which is very convenient of application when the taugential-polar equation (Art. 143) is given. Hix. 2. In the parabola P= P/O ood (11) adr? 2p? 29 we have Pe Es ee Gb ess (12). Ex. 3, In the central conics we have (Art. 143, Ex. 4) yp a 6? oe ae + had see ree ese reso essere (13), 2,2 and therefore pat ws oe ves ihe eee (14). Cf. Art. 151, Ex. 5. 153. INNewton’s Method. In another method of treating curvature, employed by Newton*, a circle is described touching the given curve at P, and passing through a neighbouring point Q on it, and we Q’ Q ee Pp T Fig. 116. investigate the limiting value of the radius of this circle when Q is taken infinitely near to P. * Principia, lib. i., prop. vi., cor. 3. - 152-153] CURVATURE. 403 We can easily shew that in the limit the circle becomes identical with the ‘circle of curvature’ at P, as defined in Art. 150. For if C be the centre, then, since CP = CQ, there will be some point (P’) on the curve, between P and Q, such that its distance from C is a maximum or minimum, and therefore* such that CP’ is normal to the curve. In the limit P’ approaches P indefinitely, and C, being the inter- section of consecutive normals, will coincide with the ‘ centre of curvature ’ (Art. 150). Newton’s method leads to a very simple formula for the radius of curvature. Let Q’Q7 be drawn perpendicular to the tangent at P, meeting the circle again in Q’, and the tangent in 7. Since eee ; [idee we have Zo = dQ t= ie oes (1). If Q’QT be drawn at a definite inclination to the normal at P, instead of parallel to this normal, the limiting value of the same fraction gives the chord of curvature in the corre- sponding direction. It occasionally happens that the chord of curvature in some particular direction can be found with special facility; the radius of curvature can then be inferred by the formula (3) of Art. 150. Hx. 1. In the parabola, let QR be a chord drawn parallel to the tangent at P, to meet the diameter through P in V; see ise lis.) We have, then, from the geometry of the curve, QV2=4SP. PY, where S is the focus. Hence, for the chord of curvature (q) parallel to the axis, 2 gq =lim oe If 6 be the angle which the normal at P makes with the axis, we have cos @= SZ/SP, where SZ is the perpendicular from the focus on the tangent at P. Hence roe ge SP4 p= 39 see 0 = 2 Ge = 2 ah DOOR OCR ICOKD IOC (3), since SZ?=SA.SP, A being the vertex. * See Art. 55, Hx, 2. 26—2 404 INFINITESIMAL CALCULUS. [cH. xX Fig. 117. Ex, 2. In the ellipse (or hyperbola), if Q&, drawn parallel to the tangent at either extremity of the diameter PCP’, meet this diameter in V, we have QV?: PV. VP’ =CD?: CP, Fig. 118. where C'D is the semi-diameter conjugate to CP. Hence, for the chord of curvature (q) through the centre, QV? CD? 2 am py = im Gp: VP =2 CP ecer ccc vcce (4). g=lim 153] CURVATURE. 405 If CZ be the perpendicular from the centre on the tangent at P, and @ the angle which CP makes with the normal, we have cos 6 = ('Z/CP, and therefore 2 CD p = 39 sec 0 =p G cmaiodeaaes Fu tpwe Pict (5), in agreement with Art. 151 (17). Again, if 6’ be the inclination of either focal distance to the normal at P, it is known that cos 6’ = CZ/C'A, where A is an ex- tremity of the major axis. The chord of curvature (q’) through either focus is therefore given by CD: q' = 2p cos 6 = 2 GA ee (6). Ex. 3. To find the radius of curvature (p,) at the vertex of the cycloid *x=a(8+sin0), y=a(1—cos6) ............ (7). We have : in 16\2 gama (6 +sin 6+ d4sin? Joa (145°) 2 (S08) 2 2y 6 ; x whence Py = limo=o By maa ee eee. ct (8). Newton’s method, combined with the result of Art. 66, 2°, leads to a general formula for the chord of curvature parallel to the axis of y, and thence to the Cartesian expression for the radius of curvature. Denoting the chord in question by g, we have, in Fig. 43, p. 155, 1 . V ° V LA 2 7 obit au = lim en. cos? yr = $4” (a) cos? ...(9), where yf is the inclination of the tangent at P to the axis of az. Since q=2pcosy, tanw=¢'(a), it follows that Le MC) -= a) Cos? af = ——_>* —__ ........, 10). pene Ms treta aye EO) This is identical, except as to notation, with the formula (2) of Art. 152. 406 INFINITESIMAL CALCULUS, [cH. x 154. Osculating Circle. A slightly different way of treating the matter is based on the notion of the ‘osculating circle. If Q and R be two neighbouring points of the curve, one on each side of P, we consider the limiting value of the radius of the circle PQR, when Q and & are taken infinitely close to P. We can shew that if the curvature of the given curve be continuous at P, this circle coincides in the limit with the ‘circle of curvature. for if C be the centre of the circle PQR, there will be a point P’, between P and Q, such that CP’ is normal to the given curve, and a point P”, between P and &, such that CP” is normal to the curve. Let P’O and P’C meet the normal at P in the points C’ and C”, respectively. Under the condition stated, C’ and C” will ultimately coincide with the centre of curvature at P, and, since CC" < C'O”, C will @ fortiori ultimately coincide with the same point. Fig. 119. Since, before the limit, the circle PQR crosses the given curve three times in the neighbourhood of P, it appears that the osculating circle will in general cross the curve at the point of contact. See Fig. 123, p. 422. 154] CURVATURE. 407 Ee. Ifin Fig. 117, p. 404, the circle PQR meet PV in W, we have QV.VR=PV.VW, and therefore VW=4SP. Hence the chord of curvature parallel to the axis of the parabola is 48P. A similar argument may be used to find the chord of curvature through the centre, in the case of the ellipse (Fig. 118, p- 404). If in Fig. 42, p. 153, QV meet the circle through P, Q, P’ again in W, we have VV PVsOV, and therefore, for the chord of curvature of the curve y=(«), parallel to the axis of y, 1 3 OV ; V A wt 2 7a lim pro lim AF A008 v= 3h" (a) cos? wp, as in Art. 153 (9). EXAMPLES. XLIV. 1. Prove that the circle is the only curve whose curvature is constant. 2. Prove that the intrinsic equation of an equiangular spiral is of the form 5 = aev cota, 8. Prove that the intrinsic equation of the tractrix may be writtcn s=« log cosec w. Prove that in the tractrix the curvature varies as the normal. 4, By ditferentiation of the formule dx biel oe ae cos y, ate sin wy, 1 dx [dy d’y |dx prove that Soe ae ease 1 du\2 /d*y\2 and a (53) + (=) 2 408 INFINITESIMAL CALCULUS. [en. x 5. Ifa curve be defined by the equations a= I(t), y =f (é), 1 _ ay" ox ya p a (a? + y’\t 9 where the accents denote differentiations with respect to 7. prove that 6. Apply the preceding formula to the cases of the ellipse x= a Cos d, y=bsin d, and the hyperbola x=acoshu, y=bsinhw. 7. Prove that the curve whose intrinsic equation is s=ksiny is a cycloid. (Use the method of Art. 134 (3).) 8. Given that in the ‘catenary of equal strength’ p=ksecy, where wy is the inclination to the horizontal, prove that if the origin be at the lowest point c=kb, y=klogsecy, the axes of x and y being horizontal and vertical. 9. Given that the intrinsic equation of a curve is s=ksin’ yf, deduce the Cartesian equation wt + yt = (Zh)A. 10. If the coordinates x, y of a point on a curve be given functions of ¢, prove that die Oe iV ee dt dt (3) sin yy, d*y d's , 1 /ds\?2 gen gen +-(F) cos yy, and give the kinematical interpretation of these results. Hence shew that dnt (ae ey, Zr _ fds 1 {(S5)+($0)-(B)-(Q). ae Se = eee Fees a 1ew how to express the coordinates «, y of a point on a artesian equation is given in terms of the inclina- ‘ta pean, Ann BENE: that ‘15. Also that, in the hyperbola 16. Also that, in the parabola y? = 4aa, 17. Also that, in the semi-cubical parabola ay? =x’, I — 18. Also that, in the cubical parabola a’y = se 410 INFINITESIMAL CALCULUS, 19. Also that, in the astroid a8 + yz aS: a’, a ~ \ aE aie Prove that ‘the chord of curvature through the pole of a cardioid is 13 times the radius vector. me CURVATURE. 411 27. Prove that the chord of curvature, through the pole, at any point of the curve r”= a” cos mé is 2r/(m + 1). «28. Prove that the curvature of the pedal of a curve r=/(p) with respect to the origin is , where 7, p, p refer to.the original curve. 29. Prove that the curvature at any point of the pedal of an ellipse of semi-axes a, b with respect to the centre is equal to aa +O ’ nd where 7 is the radius vector of the pe ponding point of the ellipse. 30. Prove the formula ae (e-=(Z) ~aefef e () fs p tr r\ds ds hap ds ' and apply it to deduce the conclusions of Ex. 20. 31. Prove that in polar coordinates the condition for a stationary tangent is where u=1/r. 32. From the formula w=0+6=6 + cot- as r do deduce the formula for curvature in polar coordinates : 1 a eae dr\? dr\*) # AL pee pe ee ee a 1 ponte: p f "de (a) | . \r + (a) f _ (du \= {1 1 du\*)# =(aet#)+ 1+ Ga) } where w= 1/r. 33. With the same notation, prove that the chord of curvature through the origin is 2 {1+ Ga) } Gar»): 412 INFINITESIMAL CALCULUS. (CH. x 34. The radius of curvature of the curve ay? = (% — a) (# — B)* at the point (a, 0) is (a — 8)’/2a. 85. Prove by Newton’s method that the radius of curvature at the vertex of the catenary y =acosh «/a { is equal to a. \ 36. The radius of curvature of the curve | y? = a" (a + a) |x at the point (-a, 0) is da. 37. The radius of curvature of the ‘ witch’ y =a (a—x)/x | : at its vertex is da. 88. The radii of curvature of the trochoid c=a0+ksind, y=a—kcosé. at the points where it is nearest to and furthest from the base are (a+ k)?/k. 39. Prove that in the meridian-curve (r?=a?cos 6) of the ‘solid of greatest attraction’ (see Ex. 18, p. 375) the radii of curvature at the extremities of the axis are o.and 2a, re- spectively. 40. Prove that by Newton’s method the radius of curvature at either vertex of the lemniscate 7?= a? cos 26 is 3a. 41. Apply Newton’s method to shew that the radii of curvature of the epicyclic U=A, COS NE +a, COS Nt, Y=A,Sin n,t + a,sin nf, at the points nearest to and furthest from the centre, are (72101 + N22)? 1°, + Ne7Ay Infer the condition that an epicyclic, at the points of nearest approach to the centre, should be concave to the centre (as in the case of the orbit of the Moon relative to the Sun). aa 155] CURVATURE. 413 42. If a curve be referred to polar coordinates 7, 0, and if the pole be on the curve, and the initial line be the tangent at the pole, prove that the diameter of curvature at the pole = lm 1/6. Find the radius of curvature at the pole of the curve r=acosm#, 43. If P bea point of a curve where the.curvature, but not the direction of the tangent, is discontinuous, and if Y, & be neighbouring points on opposite sides of P, prove that the curvature of the circle PYF is ultimately equal to my Ma Pr Pa where p,, p, are the radii of curvature of the given curve on the two sides of P, and m,, m, are the limiting values of the ratios PQ/QH and Pk/QAR, 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 46s/p, where ds 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 7’, then 7’P and 7'Q are ultimately in a ratio of equality. Why does it not follow that the line joining 7’ to the middle point of P@ will be ultimately perpendicular to PQ? 46. Assuming that the radius of the circumcircle of a triangle ABC is equal to 4a/sin A, 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. *S OR. 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 imtersection’ 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 Pee 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 C'C’ 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. ang 120. Hx, 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 P, the triangles APA’, BPB' will be equal, whence PA PA ePR aE 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=2a, OB=2y, and therefore . 2xy sin w = k?, The envelope is therefore a hyperbola having the coordinate axes as asymptotes. Fig. 121 illustrates the case of w= 4dr. 155-156] CURVATURE, 415 Fig. 121. 156. General Method of finding Envelopes. The equation of any curve of the system being MCRAE) a Orta wa car cna ate ts (1), where a is the parameter, then at the intersection with another curve ALCO, i EN Ee eee ae ra, (2), we have, evidently, POG Ys SV PM GPA) yg a’ —a eG) When the variation «’—a of the parameter is infinitely small, this last equation takes the form 6 (0, Y, A) =O vssesscsrsrneseee (A) where 0/da 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. KCH? < Ex. 1. The circles considered in Art. 155, Ex. 1 may be represented by (e—a)?+ =a? iiss scceuneee ee (5) Differentiating with respect to a, we find e~a=Q 2... noe eee (6). Eliminating a between (5) and (6) we get Y = b Os ovine sane i! (7), the envelope required. Fx. 2. Tf a particle be projected from the origin at an elevation 0, with the velocity ‘due to’ a height /, the equation of the parabolic path is 0 y=atan 9-4 sec? O MOE SEA ae henge (8), where the axes of x, y are respectively horizontal and vertical. Writing a for tan 6, we get 2 y=an—-1i— (eae re (9). To find the envelope of the paths for different elevations, and therefore for different values of a, we differentiate (9) with respect to a, and find This is satisfied either by x=0, or by aw=2h. 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 v= 4h (h— J). ; eee are CeO), If the origin of y correspond to a cusp instead of to a vertex, the cosine of the angle must be replaced by the sine. Hence, for the evolute, we have iv er: p=—asin ——3¥- Sw Peay (19), which can be brought to the same form as (18) by an adjustment of the origin of y. The evolute is therefore a similar epicycloid in which the dimensions are reduced in the ratio a/(a + 26). For a hypocycloid we have merely to change the sign of b*. 160. Arc of an Evolute. The difference of the radii of curvature at any two points of a curve is equal to the arc between the corresponding points of the evolute. To prove this, let the normals at two neighbouring points P,, P, of the curve meet in C’; and let C,, C, be the corre- sponding centres of curvature. Asin Art. 158, C,CC, is in general an obtuse-angled triangle; and when P,, P, are taken infinitely close to one another, C,0+4.CC, is ultimately in a ratio of equality to C,\Q). Also, since the triangle P,CP, has the angles at P, and P, infinitely nearly equal to right angles, the difference between CP, and CP, is ultimately * It appears on examination that the equation p=ccosmy, or p=csinmy, represents an epi- or a hypo-cycloid according as m1, provided we include the pericycloids among the epicycloids, in accordance with the definition of Art. 137. The pedal of an epi- or a hypo-cycloid with respect to its centre is therefore an epicyclic of the special type referred to in Art. 139, Ex. 2. Thus Fig. 97 represents the pedal of a four-cusped epicycloid, and Fig. 99 that of a four-cusped hypocycloid. 426 INFINITESIMAL CALCULUS. [CH. x of the second order of small quantities. Hence, neglecting C) Fig. 126. this difference, we have C.P,—0,P,=C,C + Cae ce It follows that if p be the radius of curvature of the original curve, and o the arc of the evolute, we have dp=6c, ulti- mately, or Hence, integrating, where C is an arbitrary constant depending on the origin of measurement of o. Otherwise: by differentiation of the equations E=n-psinW, y=y+pcosw.......... po) of Art. 150, we find, since dx/ds=cosy, dy/ds=siny, dy/ds=1/p, dé L ddp U8 pine a dee Ae de ole eintelanerateneysiereters (4). dn Hence es COU Yh ive ti ete eee eee (5), which shews that the tangent to the evolute is normal to the original curve, and Bm M(B) +(Q) aR ee (5), which gives on integration the result (2). 160-161] CURVATURE. | 427 For the case of the cycloid, this property has been already obtained in Art, 159 (15). Ex. The radii of curvature of an ellipse of semi-axes a, 6, at the extremities of these axes, are b?/a and a’/b, respectively. Hence the length of any one of the four portions into which the evolute is divided (see Fig. 124) by its cusps is a/b—b?/a or (a?—6*)/ad, 161. Involutes, and Parallel Curves. If a curve A be the evolute of a curve B, then B is said to be an ‘involute’ of A. We say an involute because any given curve has an infinity of involutes. To obtain an involute we take any fixed point O on the curve, and along the tangent at a variable point P measure off a length PQ in the direction from O, so that arc OF. + PQ) = const, cass. er bees ab) It is easily shewn, by an inversion of the argument of Art. 160, that the tangents to the given curve are normals to the locus of Q, so that this locus fulfils the above defini- tion of an involute. And, by varying the ‘constant, we obtain a series of involutes of the same curve. As a concrete example we may imagine a string to be wound on a material arc of the given shape, being attached to a fixed point on it. The curve traced out by any point on the free portion of the string will be an involute. This is in fact the origin of the term. Hx. 1. The tractrix is an involute of the catenary; see Art. 134. Ex. 2. In an involute of a circle of radius a we have, evidently, ds rn a ee 2 AN = eat LI be a aaa iaed ree ea Oy (2), if the origin of be properly chosen. Hence, integrating, Be RMN ras rats Kistatle Sev ee Cais (3), no additive constant being required, if s be measured from the cusp (=0). 428 INFINITESIMAL CALCULUS. [CH. X In this particular case (of the circle) it is evident that all the involutes are identically equal. It is therefore customary to speak of the involute of a circle. The curve is shewn in Fig. 127. Fig. 127, If a constant length be measured along the normal to a ziven curve, from the curve, the locus of the point thus determined is called a ‘parallel’ to the given curve. If CP, CP’ be two consecutive normals to the given Q’ Ones P Q Fig. 128.- curve, and Q, Q’ the corresponding points of a parallel curve, we have PO ae Since the difference between CP and CP’ is of the second 161] CURVATURE. 429 order of small quantities, it follows that the same holds of the difference between CQ and CQ’, and thence that the angles at @ and Q in the triangle CQ are ultimately right angles. Hence CQ, CQ are normals to the parallel curve. Hence two parallel curves have the same normals, and therefore the same evolute; in other words, parallel curves are involutes of the same curve. Conversely, it is evident that the various involutes of any curve form a system of parallel curves. EXAMPLES. XLV. 1. The envelope of the parabolas y°? = 4a (x— a), where a is the parameter, is a pair of straight lines. 2. The envelope of the parabolas ay” =a" (x - a), where a is the parameter, is the curve Aap = Be 3) Sar « 3. Circles are described on the radii vectores of a curve as diameters ; prove geometrically that their envelope is the pedal of the given curve with respect to the origin. 4, Find the envelope of the circles described on the focal radii of a conic as diameters. 5. Chords of a circle are drawn through a fixed point on the circumference; prove that the envelope of the circles de- scribed on these chords as diameters is a cardioid. 6. The envelope of the circles described on the central radii of a rectangular hyperbola as diameters is a lemniscate of Bernoulli. 7. Prove that the envelope of the curves Pcosa+Qsina=k, where P, Y, & are given functions of #, y, and a is a variable parameter, is D moa a) 8 430 INFINITESIMAL CALCULUS. [CH. X 8. Find the envelope of the circles a” + y° — 2am cos a — ay sin a = 0, and interpret the result. 9. A system of ellipses of constant area have the same centre and their axes coincident in direction; prove that the envelope consists of two conjugate rectangular hyperbolas. 10. A straight line moves so that the product of the per- pendiculars on it from the fixed points (+ ¢, 0) is constant (= 6?) ; prove that the envelope is the ellipse red y” Prieta or the hyperbola Se l fa bey Bee according as the two perpendiculars are on the same or on opposite sides of the variable line. 11. Circles are described on the double ordinates of the parabola y’=4ax as diameters; prove that the envelope is the parabola y? = 4a (x + a). 12. Circles are described on the double ordinates of the ellipse s + v= 1 as diameters: prove that the envelope is the ellipse ie y* be Bae 13. n, If n’ 2. We note that if m= 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 ao, da M, N are homogeneous functions of # and y, of the same degree. In this case the fraction M/N is a function of y/z only, and we may write dy _ (y F(%) Maeda ts hate (1) If we put y= av this becomes dv ‘Zep Uae (i) acre eee Wee ceo. (2) The variables x, v are now separable, viz. we have da dv a ~ F(v) —0 © sie’e.e\s 60/0) 01+) 6 eae 01's (3), dv whence log @ = | >———_ 4. O oa eeeeccececec es (4). g ee (4) After the integration has been effected we must write y= y/x. U Ex. (2° y?) — 2ary = 0 ie SSS (5). y 97 dy ea Here Se NY oy So re eS Sve ook fon cece, 6 dx 1_¥ (6), oe? dv 2v dv_v(1+*) h pe eRe Lge Sa a vhence a7 te igesar, wr ae ae dx 1 -v? I 2v q _ = a Ca ence rma) v (5 =) Les Se tis 2 (7). ntegrating, we have log «= log v — log (1 + v*) + const., 470 INFINITESIMAL CALCULUS. [CH. XI which is equivalent to x(1+v*)=Cy, or 22 4:9) = Oy 1 ee cr (8). In the geometrical interpretation, the general solution of a homogeneous differential equation must 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 (y/«=m) through the origin, dy/dx 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 « at the origin. If in (4) we put C= lore y/« or v is determined as a function of a/c. In other words, the primitive is homogeneous in respect to a, y, and ¢, and is therefore of the type 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. XLVIII. d. ohio 1. Integrate = =2 : [y=Cx.] yp ey a rane 2. Integrate ee ly ae Fee dy : 3. Integrate aE: cot « cot y. [sin « cos y = C.] Yan | » dy . Integrate itera ne k= es [y=1+ Ce] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 47] 5. Solve m(y+b)dx+n («+ a)dy=0. [(w+ a)" (y+6)"=C.] dy 1+¥ e+e 6. Solve Bre a = Ton" | 7. Solve (1 + y?) da — avy (1 + x”) dy =0. [((1 +.@) (1 +47) =Ca’.] 8. Find the curves in which the angle between the tangent and the radius is one-half the vectorial angle (6). (The cardioids 7 =a (1 —cos 6).] 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 += 2a cos 6.] 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 ay = C.] 11. Find the curves in which the subtangent varies as the abscissa, by Circe 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 (a— a)/a. | 14, Find the curve in which the polar subtangent is constant (=0). [r= a/( -2),] 15. Find the curve in which the polar subnormal is constant (=a). [r=a(6—a).] 16. Find the curves such that the area included between any two ordinates is proportional to the intercepted are. [The catenaries y=a cosh (#— «)/a. | 17. Find the curves such that the area included between any ordinate, the axis of #, and the curve may be 1/nth of the rectangle contained by the ordinate and the corresponding abscissa. | Ly 2 Ca", | 472 INFINITESIMAL CALCULUS. [CH. XI 18. Find the form of a solid of revolution in order that the volume cut off by any right section may be 1/nth of the product of the area of this section into the length of the axis. [The equation of the generating curve must be ¥?= Ax”—",] 19. In a suspension-rod of uniform strength the area of the cross-section (S) varies as the total stress across it ; prove that if x be measured vertically downwards the relation between S and x must be of the form S=A-B | Sade. 0 Hence shew that the form of the rod must be that generated by the revolution of a curve of the type y=berrs about the axis of a. 20. Find the form of a cuive, symmetrical with respect to the axis of a, such that the centre of mass of the area cut off by any double ordinate may be at a distance from this ordinate equal to 1/nth of the length of the axis. y= Cat, | 21. Solve (a? + day”) dx + (y? + 32°) dy =0. xdy — yda eta E + 4° =.2a? tan“! Lacs ¢. | a 22. Solve eda + ydy =a? dy x 23. Solve ore J (x? + y’). | y=esinh ==" | 24. Solve ee —y=,/(x?+y") Sipe! y): Give the geometrical interpretations of the differential equation and of its primitive. [a? = 2C'y + C?.] da dy 25. Solve a — Ixy = yp — Quy ° [ay (x = Yy) = C.] dy nt 26. Solve aw 7 ty= ay. [xy = Ce*.] 176] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 473 27. Solve a au —y=ny. [y= Cae”. | 28. Shew that the equation dy _axt+by+e dx wa+b'y+e’ is rendered homogeneous by the substitutions axt+by+c=&, aur+byt+c=n. 29. Shew that an equation of the type dy ) Wi =f (ax oe by) may be solved by the substitution ax + by=2%. 30. Shew how to solve any equation of the type dy aw + by +e ) ee +by+c]° 176. Linear Equation of the First Order, with Constant Coefficients. A ‘linear’ equation is one which involves y and its derivatives only in the first degree. Thus the linear equation of the first order is of the type where P, @ are given functions of a. We will first take the case where P is a constant, the equation being as this will be of special use to us later. If Q=0, we have, by separation of the variables, dy o4% are is erred (3), 474, INFINITESIMAL CALCULUS. whence logy—aw=A, ye*=C, or y= CO cies fs eeesaweeeeeeeeaes (4), where C, = e4, is arbitrary. It appears from the above process that the factor e~” renders the left-hand side of (2) an exact differential co- efficient. This gives the key to the solution in the genera! case where Q+0. Thus (2) is equivalent to — d oF (e27H) = Qe"). ee (5), whence eg wty = f Qe “da + C, ae y= | Qe da + Cee ae (6). In accordance with a general usage (see Art. 185), the first term on the right-hand of (6) may be called the ‘particular integral,’ and the second the ‘complementary function.’ The following cases are important : 1oet Q=He® Gt); wehave [Qe “de=H fe? %%dx= yaa Ne Bhs and ees et + Ce ee (8). a—a That the first term on the right-hand is a particular integral of the proposed equation is verified at once by inspection. 2°, The result (8) needs correction when a=a, or In this case we have [Qe “de =H lda= ae and y= Hae + Ce. (10). re 3 O= Hate. eee (11), Soe a a Harn we have (Oe. =e On = age 176] =‘ DIFFERENTIAL EQUATIONS OF Flo. ura Ha n-+1 = UH Ax “ey y= patos GE et. teen - and Kx.1. Ifa particle be subject to a resistance varying as . velocity, and to some other force which is a given function of the time, its equation of motion is of the type dv at eat Reena toa Men esses dae (13) The integral of this is Wier ee | Ged (L) Ob tice. Deda tees oss (14). For example, if Leo; a constant, we have v=Ce-# + ; Pee Scat uraccta esate aie. (15). This might have been obtained more simply by writing the differential equation in the form d g OMe G(v-Z) +#(0-$)=0 Peake (16), whence ) -{= Rea: SO et dee Liste, Se a (17). As ¢t increases, v tends asymptotically to the constant value g/h. Hx. 2. Tf an electric current of strength « be flowing in a circuit of self-induction Z and resistance A, and if # be the extraneous electromotive force in the circuit, we have the equation dx L Wi WED pe aa eee ae (18) If # be a constant, the solution of this is y R waa + Oe! a Rey (19); where C is arbitrary. The current therefore tends to the constant value L/h. If, for example, we suppose that the circuit is completed at time ¢=0, we have to determine C’ so that 2=0 for ¢=0; this gives egy le teat ta Ps cate Poe Lip. LER Rp Pee te (20). The second term represents the ‘extra current at make.’ «MAL CALCULUS. [CH. XI Ei = Hy cos (pi + €)...2; eee nen (21), Be ee | (pel \ eel: oT 7 (cer) ZT, ¢” 60s (pt +), R Ce, Clad and dividing by ek’, we find R R C= Ca L Bao € H fe cos (pt +) dt Ei, R?4 Rs PL see Art. 79 (14). Hence as ¢ increases, the current settles down into the steady oscillation May pest Ff cos (pt + €) + pL sin (pt + )}...(22);. = e+ pL) COS (pt + €— €) ..-ceceeees- (23), where e, = tan Be wae eg oad pea eg tae ane oa ates The effect of the self-induction (L) is therefore to cual the amplitude of the current in the ratio R] (E+ pL’), and to retard its phase by «. 177. General Linear Equation of the First Order. We return to the general linear equation of the first order, d me te lady = Q 010 6.6 © 616) eb ehalolelmipuanenete ate (1) If Q =0, we have ldy y dx + P = 0 eo 610 @ 6) 0:a\ejeceiereleierstaterceels (2), whence logy +fPda=A, ie. yor Pit — Ol (3). This shews that e/?? is an integrating factor of (1), viz. we have ef Pda (3! ct Py) iz - (ye SPae), 176-177] DIFFERENTIAL EQUATIONS OF FIRST ORL Hence (1) may be written 2 EEA OLS Ree pee ted ett (4). Integrating, we find ners ae — fei hee Cine Cis seteteeehe wees (5); dy Ee. 1. dg TY Cob w= PEON rede AU rey gee (6). Here P=cotz, fPdx=logsina, efPé—sing, Hence, multiplying by sina, fy Sin a AAT oe COM Ds ost, reas s t9 ce (7), y sinx=sin?« + C, Y = SING pve seeeeeeeecteeeeceeees (8) dy THIS (1 ca eee ake Soe ee eae (9). Dividing- by 1 — a, we have dy x ST ar es Cee rere sec cesses (10). Here Pk — [Pde =}log(1—a%), e!P4® = (1 — 23, Multiplying (10) by the integrating factor, we get J(1 - #’) abe Tia qa d 1 pr da (V1 — 2°) 9} = Fea vos eee es (11). Hence, integrating, JQ — x) y=sin-!x+C, sin-) x C Y= Ti — a) * Jd —2%) Mclatslare eeiarers titisn (12). or INFINITESIMAL CALCULUS. [cH. XI y Aegrating factor will often suggest itself on inspec- ne equation, without recourse to the above rule. hi 3, YJ (13). The steps are a” es ae nay = mie gittns+t xy =——__—__ + C dimen 178. Orthogonal Trajectories. Suppose that we have a singly-infinite family of curves h(a, y, C) = O05. eee (1), where C 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 #, and elimination of C. See Art. 170. If two curves cut at right angles, and if wy,’ be the angles which the tangents at the intersection make with the axis of 7, we have — wr’ = + 47, and therefore tan r= — cot y’. Hence the differential equation of one family is obtained from that of the other by writing dy ¢. Wy . —1 rF for Ae Otherwise: if dz, dy be the projections of an element of one of the curves (1) we have 177-178] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 479 Hence, if dx, dy be the projections of an element of the The differential equation of the trajectories is then obtained by elimination of C between (1) and (3). Ex. 1. To find the orthogonal trajectories of the rectangular hyperbolas GPU eee eee ferent haere Tat ager te faz (4) Differentiating, we find ils Oe asd ig nO be ere ee rea ares (5), and therefore, for the trajectories, Ee Ye winger ety chooks arty oie (6), whence | nln Tele Oy hig Sent TGA ee ae (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 sed els eT Pe gl RS ee a (8), where pw is the variable parameter. Differentiating, we have ada + (y +p) dy=0, and therefore, for the trajectory, xdy — (y +p) dx=0. Eliminating » between this and (8), we find ny | 2ay ant op ies Os ok, Sohn vas: (9), d (y”) 2 or Poy eee Sia saghe et eletateta b/ el e(elel sia (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 1/z* Introducing this we have d /y’ o k aie) ase 480 INFINITESIMAL CALCULUS. [CH. XI whence a eee w 1 or oe? + a? — DAse +? = Oe (11), Xd being arbitrary. The original equation represents a system of coaxial circles, cutting the axis of # in the points (+4, 0). The trajectories (11) consist of a second system of coaxial circles having these points as ‘limiting points’; viz. if we put A=+é we get the point- circles (ct kP +4? =O 5 pee teeete (lay; see Fig. 148. roe xe SALT TR RRS RIS SEER Fig. 148. If the equation of the given family of curves be in polar coordinates, thus 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 ¢ =— cot ¢’. : Hence the differential equation of one system is obtained from that of the other by writing SE aoa Se dd dr Or, differentiating (13) we have Ftv -2F rd9 =0 A AQ LO =O vrevreerenesees (14), and therefore, for the trajectory, OF neice Uae a. rdd —— aadr=0 RE gee S| (15). The elimination of C between (18) and (15) leads to the differential equation of the required system. Ex. 3. In the circles ee CUNO er peer eee aston thie aces (16), which pass through the origin, and have their centres on the initial line, we have = tan LETS ae ee Sein (17), and therefore, for the trajectory, rd@=tan 6dr, or ae GOUT ros ee he (18). Integrating, we find log r= log sin 6 + const., or Pere CMB Stacy een evan be Teeth as sie (19), which represents another system of circles, passing through the origin, and touching the initial line. EXAMPLES, XLIX. 1. Solve = +ytanv=secx, [y=sina+C cosa] ay Ly 2. Solve (1 — act) T+ wy = a, [y=at+C /(1—2*).] L. 31 482 INFINITESIMAL CALCULUS. [CH. XI 3. Solve ot rary=0 [w? + Qay =C.] 4. Solve ARIE [a?— 2Qay = C.] di & dy é 2 —x? 5. Solve Fp t Pay = 1 + 2a. [y=u+Ce*.] 6. Shew that the equation dy a Hee Py = Qy is made linear by the substitution ye) =o (Bernoulli’s equation.) 7. Solve oY ay =x loge. [j= 1 + log e+ Cx | 8. Solve COS &@ a —ysin“x+y=0. E sin « + C cos x. | 9. If the two plates of a condenser of capacity C’ are connected by a wire of resistance & (and zero self-induction), the equation connecting the charge (7) with the electromotive force (£) is . Jo ee Lh a a Integrate this in the cases #=0, H=const., H= EL, cos (pt + «). 10. Find the orthogonal trajectories. of the straight lines y = Cx. [The circles x? +7’ =C.] 11. Find the orthogonal trajectories of the curves aly =o", . [The conics a7 +z =U. | 12. Find the orthogonal trajectories of the circles e+ y? = 2cy. [The circles ~ + y?= 2c’x. | 13. Find the orthogonal trajectories of the cardioids r=a(1—cos 6). s {The cardioids r= 6 (1 + cos @).] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 483 14. Prove that the orthogonal trajectories of the curves 7e™ = a™ cos mO are the curves 7 =) sin m0; Interpret the cases of m=1, —1, 2, —2, 4, —4, respectively. 15. Prove that the orthogonal trajectories of the curves 7” = A cos 0 are the curves ry = Bsin? 6. 16. Prove that the differential equation of the confocal parabolas y’ = 4a (x+a), is yp’ + 2ap —y =0, where p = dy/dzx. Shew that this coincides with the differential equation of the orthogonal curves; and interpret the result. 17. Prove that the differential equation of the confocal conics 2 oc y Gad” PEN” is ay p? +(x? —y? —a? +b) p—ay=0. Shew that this coincides with the differential equation of the orthogonal curves, and interpret the result. 1, 18. A system of rectangular hyperbolas pass through the fixed points (+a, 0) and have the origin as centre; prove that their orthogonal trajectories are the Cassini’s ovals (a? + y?)? = 2a? (a? — y*) + C. 19. If in bipolar coordinates (Art. 149) the equation of a family of curves be TCT y=; the differential equation of the orthogonal trajectories is é "or Hence shew that the orthogonal trajectories of the circles rin, are the circles 64+07=C. = 31—2 484 INFINITESIMAL CALCULUS. Kol: ape ae 20. Also that the orthogonal trajectories of the Cassini’s ovals 17 =O, are the rectangular hyperbolas 6-V=C. 21. Also that the orthogonal trajectories of the equipotential curves beged. creat) . Rig are the magnetic curves cos 8+ cos 6’ =C. 179. Equations of Degree higher than the First. The general type of an equation of the first order and nth degree is p + Pip" + Pip’ + ... + Paap eee), where P= fF oc (2), and P,, P,,... Pn are given functions of # and y. It is usually implied that these functions are algebraic, and | rational. The equation (1), being of the nth degree in p, indicates that | » branches of the primitive curves go through any assigned point in the plane wy. Some of these branches may of course be imaginary, and for some ranges of « and y all may be imaginary. There may also be a real locus of points at which two of the values of p coincide; this locus is of special importance in the higher development of the subject. For example, in the equation of the second degree, p?+Pp+Q=0 a 0 lene etpiecereceiaienelelatetels state nie te (3), the values of p will be real and distinct, coincident, or imaginary, =. according as P?=4Q. And the locus of points at which the two | <= values of p coincide is the curve P?=4Q. If the left-hand side of (1), considered as a function of p, | can be resolved into linear factors, thus (p — 1) (Pp — Pra) +++ (P — Pn) =O ceseceserese (4), 179-180] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 485 where pj, 2, --- Pn are known functions of w and y, the . complete solution will consist of the aggregate of the solutions of the several equations dy da da dg = Py Fao Bases ge = Pn ee eae (5). Ex. Cy (oe a) py = Oy net cnet rate (6). This is equivalent to (ap +.y) (yp — 2) =0 0... ececeeeeeceeees (7); and the solutions of xp+y=0, yp-«x=0, are, respectively, A hel OE Ses aE ee pdt Om Piso ee ae (8). The product of the two values of p given by (6) is—1. This shews @ priori that the two branches of the primitive curves which pass through any point («, y) will be at right angles to one another. Cf. Art. 178, Ex. 1. 180. Clairaut’s 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 dy/dz, the form in question is 1) Ey iG ey AG 1) OAD trae (1). It was proved in Art. 53 that the intercepts (a, 8) made _ by the tangent to a curve on the axes of w and y are given by a=(ap—y)/p, B=Y — UD cecrvcscrves. (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 * Viz. the equation is equivalent to B=f(-B/a), or $(a, B)=0. 486 | INFINITESIMAL CALCULUS. (CH. XI evident that this relation is satisfied by any straight line whose intercepts have the given relation. Along any such straight line we have p=0 ee (3), and we thus get the solution y= Ca +f (C) ee (4), involving an arbitrary constant C. But the equation will also be satisfied by the curve which has the family (4) of straight lines as its tangents; in other words, by the envelope of this family. This envelope is found by expressing that (4), considered as an equation in C, has a pair of equal roots, ze. by eliminating C between (4) and otf! (C)= 0) cere (5); see Art. 156. Ex. To find the curve whose pedal with respect to the point (a, 0) as pole is the straight line «= 0, The expression of this property is a= pB, where 8 is the intercept on the axis of y, or and also by their envelope: .y?7=4am ...:.0:sasseeenae eee (8) ; see Art. 157, Ex. 2. . The usual method of deducing the above solutions is to differentiate (1) with respect to 2; thus d ve p=apt+ {ath (pe, whence {o+f’ (p)} = oP =) ee (9). 180] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 487 This requires, either that “ SA Bh ar Ue EL Oe ey A (10), or that Bee fGC Dy — Osan ees My cee cnr (11). The former result makes p = C, and ee tae ae ears tore a tie (12). The alternative result (11), combined with (1), leads, on elimination of p, to a particular relation between # and y. Since the result of eliminating p between (1) and (11) must be the same as that of eliminating C between (4) and (5), we identify this second solution with the envelope aforesaid. The solution (12), involving an arbitrary constant C, is called the ‘complete primitive. The second, or envelope- solution, is not included in the complete primitive, 7.e. it cannot be derived from it by giving a particular value to C. It is therefore called a ‘singular solution*,’ EXAMPLES. L. f 2 I. Solve ($) ~ (a+ p) + af =0. [y=ax+C, y= Pux+C.] 2 2. Solve (<2) = aint ar [y=C + cos x. | 2 3. Solve (4) = my, [y = Cetmea, ] 2 4, Solve y? (3 ) ier [y?=C + 4az. | dy\* 5 5. Solve 2 (<“) gh [y= C+ 2/(ae).] * 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- extensive with, the theory of envelopes. 488 INFINITESIMAL CALCULUS. (CH. XI 6. Solve (1 — 2?) 4) = [y=C +sin7 x. | dx : = dy (dy \ _, 7. Solve a & + y) = 2 (e+ ¥). [y=d4er+ OC, yada Ce | dy (dy \ _ 8. Solve ae & + x) =(“+yY) y. [y=Ce*, y=1l-—a“+Ce] x dy\° dy ss aors D 9. Solve a (2) —2y = —#=0. [a? = 2Cy + C®.] ‘dy\? dy ye: 10. Solve y (“) +20 Uy =0, [V@+y)=C+2.] 11. Find the curve such that the product of the intercepts made by the tangent on the coordinate axes is constant (= 4’). The hyperbola 4ay = k2. yP Yy 12. Find the curve such that the perpendicular from the origin on any tangent is equal to a. [The circle a? +y?=a?.] 13. Solve y = ap + ,/(b? + ap’). 002 y? | Singular solution : tat 1 14. Find the curve such that the product of the perpendiculars from the points (+ ¢, 0) on any tangent is equal to 67. ’ u? Yy” hha y” | The conics Poe 1 i OB 5-8 | 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 0”, 2? Oe a OE a 1. | | The conics a 16. Solve y=xp + ap (1 —>p). — [Singular solution: (@ + a)? = 4ay.| DIFFERENTIAL EQUATIONS OF FIRST ORDER. 489 17. Solve (c—a) p?+(x—y)p—y=0. [Singular solution: (# + y)? = 4ay.] 18. Find the curve such that the sum of the intercepts made by the tangent on the coordmate_axes is equal to a. [The parabola (#— y)?— 2a (a+ y) +a=0.] 19. Shew that any differential equation of the type dy _ (dy BY ie (2) represents a system of parallel curves, 20. Shew that any differential equation of the type : ] L,Y, P—-— }= 0 f( YP y represents two systems of orthogonal curves. CHAPTER XII. DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 181. Equations of the Type d’y/dxz?=/ (a). This chapter is devoted principally to differential equa- tions of the second order, and especially to such types 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 This requires merely two ordinary integrations with respect to #; thus w= [f@) de + A, y= |{[f(o) da} de + Awt B... Pek (2), where the constants A, B are arbitrary. 181] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 491 Ex. 1. The dynamical equation 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 ax 7d (4), whence 3 =gt+ A, OS EQE + ALP Bie occ cesccnncvas ces (5) Again, if a pS ene ey re. (6), the force varying as a simple-harmonic function of the time, we have ieee nt+A, dt n es Ja eee lip (7) =— 2, sin t+ At+ Bo... 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 subject to the conditions that y=0 and dy/dz=0 for x=0. This is the problem of determining the flexure of a bar which is clamped in a horizontal position at one end (a=0) and supports a given weight (W) at the other end (x =). Two successive integrations of (8) give dy _ B= W (la — ha?) + A, By = W (phe? — 408) + Awe Bo .....00.. (9), 492 INFINITESIMAL CALCULUS. ; (OH, Xt - where A, B are arbitrary. The terminal conditions require that A=0, B=0, whence e | =4 yt (l- 3) one eee (10). 182. Equations of the Type d’y/dx? = f(y). If the equation be of the type a first integral may be obtained in two ways. In one of these we multiply both sides by dy/dz, and then integrate with respect to #; thus dy dy, . dy di di +(5 sy =[F) oh de +A=[F(y) dy +A Ve (2). The second method is to introduce a special symbol (p) for dy/dx. Since this makes dy _dp_dpdy_ dp Ae SHE ~ dy de = Px dy + aieirelie 9° eecanelatare tele (3), we have, in place of (1), 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 p= lf y) dy +A rere (5), which is equivalent to (2). To complete the solution, we write (2) in the form dy Jif) dy + 24} = + dt (6). 181-182] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 493 The variables are here separated (Art. 178); 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 ay Sele eal sane Dae Cp emt Ae (7). ax By a change of dependent variable, writing y,+6/a for y, and afterwards omitting the suffix, this is reduced to the somewhat simpler form 2 WRG Lie aie tee te (8). a The first integral of this is Gy ae (3) Shp = EL Boe re (9) If a be positive, we may write Oe Seat Oe Cxe dulce ae PEPE ee (10), it being evident that, if we are concerned solely with real quantities, C must be positive. Thus dy SE ae We ea 38 mda a alee eletateiestis 2 3 sre ll - V(r — y’) a whence cost =+(me« +e), or P= OCOS(MI T €) ci icescesivcrcstes (12). This is the complete solution of (8), and involves the two arbitrary constants a,e«. If we put At=2 COs ef) 10 = — OSI 6 2.4.6. .75 (13), we obtain the equivalent form y=A cosma+ Bsin M&........cs0000 (14). These results are exceedingly important, and should be remembered. 494 INFINITESIMAL CALCULUS. [CH. XII 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 + Bsinh ma .........00. (15), where m=,/(— a). 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 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 dx/dt, thus da dcx da Gi ap aH and integrating both sides with respect to ¢ In this way we obtain jk (G) = [F@) Sat C= [re du +C.....(17), which is the ‘equation of energy.’ The second method consists in writing v for dz/dt, and therefore vdv/dx« for d’«u/dt; cf. Art. 39. Thus Hence, integrating with respect to x, we have 40° = { f (x) de +2, ey a ee (18), in agreement with (17). Ex. 1. If a particle be attracted to the origin with a force varying as the distance, the equation of motion is dx de WWE ~ cleo ai0.0:9;0)6) isin stars elerereieters tie (1 9). This is of the special type (8), and the solution is 0 = 4 COS (/ pt + €)..-.1c eee (20). 182] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 495 This represents a ‘simple-harmonic’ motion. The values of x and da/dt both recur whenever ,/ué increases by 27; the period of oscillation is therefore 27/,/u. The arbitrary constants a and « 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 where g is the acceleration of gravity, and / is a certain length depending on the structure of the pendulum. In the case of a ‘simple’ pendulum / 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 The solution of this equation is g=acos(, /4.t+«) AG ear? oe (23), and the period is therefore 27,/(J/9). The accurate equation (21) can be integrated once by the method above explained; we thus find dé\2 \ (=) Eg CORD ACE cs est (24), but the second integration cannot be effected (except in the particular case of C=0) without the introduction of elliptic functions. Hiv, 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 Ge pe | (25), whence, as in Art. 173, Ex. 3, ENA eS i Gee eee 0 re eee (26) 4,96 INFINITESIMAL CALCULUS. [cH. XIt If the particle start from rest at the distance a, we have C=— p/a, dx Qu\t (ab — De Z and a-(*) (7) ‘side tucbesod le Ore (27), the minus sign being taken since the velocity is towards the origin. The second integration is facilitated by the substitution = @ COS" O scene eee eee ee (28). Separating the variables, we find (1 + cos 26) d0 = (2) a Pettetereree (29); 2u\% 6 +4sin 26 = (=) (iClear: (30). As « diminishes from a to 0, 6 increases from 0 to $a. Hence the time (¢,) of falling from rest at the distance a into the centre of force is given by 7 a t, Eo nip pec te dee- cece nceceene (31). The time (é) in which a particle would describe a circular orbit of radius a about the same centre of force is rai t, = Api See 32 ae (82) fj eee | Hence hee 177-3 ee (33). 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 *177 of its period of revolution. 183. Equations involving only the First and Second Derivatives. If the equation be of the type ze. the variables w, y do not appear (explicitly), then, writing p for dy/dx, we have 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 dy EO = reeer ETE Ltt Oka (3), the ae) dx or se + Aae Meigen e. (4). (1 + p*)3 ae Integrating this we have (Art. 76 (13)) Dine yer al ph Gots (5), where a is an arbitrary constant. This gives dy x— a ing ies Ja (a — a)* BieWeieisa wisis ares ofe).6 (6), whence Y¥— B= + 4a" — (e— a)" eo. sweet fe eh (7), if B be the arbitrary constant introduced by this last integration. The result may be written tere ye aol een D> aes KE van asia dv oe dango & (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. Hx. 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 dx sd ae £ (=) 2s Ae anaes (9), which evidently comes under the type (1). Writing v for dx/de, we have dv dv dv ah oI oe bape rca (10). b. 32 498 INFINITESIMAL CALCULUS. [CH. XII For example, if the particle be subject solely to a resistance varying as the velocity, we have dt = —k eee ce cesar seeeescerers (41), whence, by Art. 176, OD yee — ie ato : waz Ae Att ee es (12). Hence, whatever the circumstances of projection, # will approach asymptotically, as ¢ increases, to a constant value (B). Again, if the resistance vary as the square of the velocity, we have dv dv _ Le , Fee — Gr ahd, aot cee ante (13). dx 1 Hence Bud = rlog (kt+ A)+ B...... (14). Bhs eee We see that, although v tends asymptotically to zero, there is now no limit to the space described. The equation (1) may also be reduced to an equation of the first order, with y as independent variable, by writing as in Art, 182 dp g GY. Dee for Ta? thus b(n. p)=0 Reh ete (15). For example, in the dynamical example just given, the equa- tion (9) may be replaced by | dv Da Sf (O)orereerreieis tetteen (16) | Thus, in the case of resistance varying as the velocity, we get dv ae o=— kt yes (17) dix Hence aaa ka= Cae (18), and therefore, by Art. 176, where C, D are arbitrary constants. This agrees with (12). 183-184] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 499 Again, if the resistance vary as the square of the velocity, we have pe pate er eee (20) Hence eeweyy FB = Ob + Deseessvesseees-(21) Ue SLi or We log (hGG + KDY- i assy: ncsvds vie (22), a form not really distinct from (14), as may be verified by putting A=kD/C, kB=log C. 184. Equations with one Variable absent. 1°, If the dependent variable do not appear explicitly, the equation being of the type Bendy NS o(S, ee ees (1), then, writing p for dy/d«, we have an equation of the first order in p; viz. | (0, B v) =0 ais senses (2). If the solution of this be put in the form Wizet UDR ies ete fer es (3), where A is the arbitrary constant, a second integration gives CTP 8 Aa Ree 2) be (4). That one of the arbitrary constants would occur as an addition to y might have been anticipated @ priort, since the equation (1) is unaltered when we write y + © for y. dy dy S97 —_ = ce 1, (1 ) die ede eae ery seee ieee (5). Writing this in the form ldp. a pdx 1—2’ we find log p =— #4 log (1 — 2”) + const., dy A or mF = P = Jd =2) eid eOW.m (6 s16iCelsiMl ieleie 66: se (6) Hence Me Aaing at Bataicgh hou. 2h CEy 32—2 ia | | 500 INFINITESIMAL CALCULUS. [CH. XIT Ev. 2. In the theory of the potential we meet with the equation : CV 2dV a dr Regarding dV/dr as the dependent variable, we have dV dr dV dr 0 ee (8). whence log a + 2 log r=const., av _d 9 adr. 33% We (10). or Integrating again, we find 9° If the independent variable do not appear explicitly, the type being : 4 (ty dy = (ae 7) y)=0 (12), we write as in Art. 182 (3) and obtain dh (oe , Dp, y) =e): eu Sapeeeeeee (14), : an equation of the first order between p and y. If the solution of this can be put in the form | | pal (y, A) eee eee (15), the next integration gives d i s laa Bee (16) form of the given equation (12) that one of the arbitrary constants would consist in an addition to 2. 184-185] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 501 Ex, 3. To find the curves in which the radius of curvature is equal to the normal, but lies on the opposite side of the curve. Referring to Arts. 53, 152, we see that the expression of the above condition is u(Z)} dex dy\2)4 = = (2)} eh (17). das? Simplifying, and making the substitutions (13), we find es Didge gin aera (18) 2 Hence 3 log (1 + p?) = log y + const., 1l+p?= - bee hely: where c’ is written for the arbitrary constant, which must evidently be positive. This gives TYR te NY we ©) 9 pe =pH=+t Pic Jee ce P oisteleletelora isis (20). Separating the variables, we have tf ee ae ay, #2) ‘ Jee) =e? cosh aise rer ee (21), where a is the second arbitrary constant. Hence, finally, y =c cosh ——* Se ee Foe ere (22), a family of catenaries. Cf. Art. 151, Ex. 1. 185. Linear Equation of the Second Order. A linear equation of the nth order is one which involves the dependent variable and its first n derivatives in the first degree only, without products. Thus, the general linear equation of the second order may be written dy d Pa o ot Wy = Tiela’ rele ave'ae'e:@ si alalsters by where P, Q, V are given functions of «. There are several important properties common to all linear equations. We give the proofs for the equation of the second order, but the generalization will be evident. 502 INFINITESIMAL CALCULUS. [CH. XII 1°. The complete solution of (1) may be written where w is any function whatever which satisfies (1) as it stands, and wu is the general solution of the equation ay da? which differs from (1) by the absence of the right-hand member. +P y= (3), For, assuming that y=u-+w, where w satisfies (1), and u is to be determined, we find, on substitution in (1), d?u ait gee + Qu ve at Po — + Qw=.V, d?w dw or, since ack P a +Qu=V 2 (4), i d?u du . by hypothesis, ae dB Tae Qu =0. oe ee (5); a.¢., the function wu must satisfy (3). The two parts which make up the general solution of (1), viz. w and wu, are called the ‘particular integral, and the ‘complementar y function,’ respectively. It is to be observed that the particular integral may be any solution whatever of the original equation; the simpler it is, the better. The complementary function must be the most ‘general solution of (3), and will involve two arbitrary constants. 2°. If uw, uw be any two solutions of (3), the equation will also be satisfied by y= Cu, + Cue 2 (6), where C{, C, are arbitrary constants. This is easily verified by substitution. Hence if the functions w,, uw, are ‘independent,’ ze. one is not merely a constant multiple of the other, the formula (6) gives a solution of (3) involving two arbitrary constants. 185] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 503 3°. If a particular integral (v) of the equation (3) be known, the complete solution of (1) is reduced by the substitution — JENA ee pay eles peer Cree Cry. to the integration of an equation of the first order in dz/dx. For (1) becomes dz dv dz (dv dv van (25, + Po) ak (Gate St Qu) z= V, which reduces, in virtue of the hypothesis, to d?z dv dz vga (257+ Po) = ale tere. elare-s ate 5 (8). This is linear, of the first order, with dz/dx as the dependent variable, In particular, if V=0, we have whence log = + 2loguv+ fPdx = const., or —fPdz Fresco z= A’ SM eT ramn ee (11), VU The complete solution of (3) is therefore e—SPdx y= Ay | We add a few examples of the integration of linear equations, by various artifices. The method of integration by series will be noticed in the next Chapter. JE iy aT i ae 4 L2,): y? fx. 1. In the theory of Sound, and in other branches of Mathematical Physics, we meet with the equation 504 INFINITESIMAL CALCULUS. [Gre xir If we multiply by 7, this is seen to be equivalent to FUP) 5 42 (rp) =0 ce eee (14). Hence, by Art. 182, rp=Acoskr+ Bsin kr, A cos kr + Bsin kr or ie Senn (15). | dy, dy a2. (l= ol) a to ee ae ne ee (16). A particular solution is obviously y=x. We therefore put ! = US ses ene ee ee tb); : dz which leadsto «#(1— ee Te ee (2 — 32") TG ON ee iliays (18). Separating the variables, we have Che dx 2 by de 5 = jaw =... 4 (19); da dz A oe a eee ey whence ae jl) (20), ee AN!) (21). The complete solution of (16) is therefore ysaA JL —0) 4+ Bbe eee (22). Ex. 3. (1 + 27) CY. 30 W 4 y= O ocd cox etane (23). This happens to be an ‘exact equation,’ 2.e. the left-hand side is the exact differential coefficient of a function of a, y, and dy/dzx, for it may be written ee {oe i} = {1 +08) TY + Be SY} oH a yl =0. The integral is (1 + a?) wu + y= Aye ae es (24). 185] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 505 This is linear, of the first order, and the integrating factor is seen to be 1/,/(1 pe! We thus find Z W040) =a Hie eee Dane e a eee ae (25). EXAMPLES. LI. 2, bs oS a1. [y=xloga+ Ax+ B.] d’y x 2. dat = te [y =(w—- 2) e*+ Ax + B.] 2, 3. o oY oa, [ y=alog® + Aw B. | 4. The differential 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 y=0, dy/dx?=0 both for «=0 and forw=/. (This is the case of a uniform beam of length / supported at its ends.) [By = prva (I — x) (P + le —2°).J 5. Solve the same equation subject to the conditions that y =0, dy/dx =0 for x=0 anda=J. (This is the case of a beam clamped at both ends.) [By = gypwa? (1 — x). 6. Solve the equation of Ex. 4, subject to the conditions that =0, dy/dx=0 for w=0, and dy |da? = 0, dy/deé=0 for x=1. (This j is the case of a beam clamped at one end and free at the other.) [By = = gpwa? (61? — dla + 2”). | 7. Solve the equation dia and interpret the result. [aw =f] +a cos (,/pt + €).] 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 woes of repulsion varying as the distance, viz. We pa, is of one or other of the types: x=acosh(,/pi+e), ec=asinh(,/ut+e), aw =ae*Nette ; and interpret these results. 9. z., D—dz, coe where Aj, Ag, Az,-.. are the real roots of fd)=20.... (7). If ® be a simple root of (7), one of the equations (6) is of the form cal (D a r) Y= O we yee (8), the integral of which is known to be y= OO (9). And if the roots of (7) be all real, say they are Ay; Agu. A, a solution of (1) involving n arbitrary constants is y = Ce? 4+ Oe +... + Cee (10); chart 186-415); 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 \ be an r-fold root of (7), J (D) contains a factor (D—2)". To solve we assume Y =O" 2 csccc (12), which makes (D—d)"y=(D—-))".e% 2 =e Dz, by Art. 188, 2°., and the solution of Dr z=0 is obviously 2=B,+ Bye+ By +...+B, a, whence y =(B,4+ B+ By? +... 4 Bie eee (13). 189] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 521 We have here r arbitrary constants, corresponding to the r-fold factor of f(D), Cf. Art. 186 (17). If f(D) contains, once only, an irreducible quadratic factor D’ + aD + 6, where 4a? < b, then part of the solution of (1) consists of the solution of Cer tes, ces ietane ss (14). If we put Nats ok ea ePhet fae 0 ea BP EE (15). we have, by Art. 188 (5), (D+ aD +b) y= {(D + $a) + B?} eM? z = ¢242( DP)’ + 6") z. And the solution of (D*?+’)z=0, is z=KHcos6a+ F'sin Ba. We thus find y=e%(HcosSv+ F'sin P2)......... (16), as in Art. 186 (22). Hence for every distinct quadratic factor of f(D) we obtain a solution involving two arbitrary constants. Finally, if f(D) contains an irreducible quadratic factor which occurs r times, we have to solve CEL) rr atl I nae eae CLT): or, making the substitutions (15), as before, Carey er IEA) Ma (18). To solve this, we assume 2=UCOS B+ USIN Bw .....ccccccesee (19). Now, by actual differentiation, we find (D* + B").ucos Bx = 28. Du.cos(Bx+4)+..., (D* + 2"). wcos Bx = (28). Deu. cos (Ba + 17) + ..., and, generally, (D* + 8)". u cos Ba =(28)". Dru.cos (Ba+ihrr)+...... (20), where only the terms of lowest order in the derivatives of u are expressed. Similarly (D’ + 2’). usin Ba =(28)". Dv. sin (Be+4rr)t+...... (21). 522 INFINITESIMAL CALCULUS. [CH. XII Hence the equation (18) is satisfied, provided y=), D9 =O ee (22), 1.e. w= y+ hye + Ko +... +b, ae) v= FPF, + Fat Fet...4+ Fa The complete solution of (17) is therefore y= (b+ Bye t+ f+... + #2") e ™ cos Ba + (Fy + Pie + Fy +... + Fa") e4@ sin Ba...(24), involving 2r arbitrary constants. dy @& Ex. 1. 5 = a4 (25). This may be written D?(D-—1)y=0, and the complete solution is accordingly made up of the solutions . of D¥y=0, (D-1)y=0, thus y= A+ Ba + Ce oe eee (26). dt Ex. 2. a = MY .s\ i Gias (27). This may be written (D —m) (D +m) (D? + m*) y =0. Adding together the solutions of (D—-m)y=0, (D+m)y=0, (D?+m’)y=0, we obtain y=Ae™+ Be-™+ Hcosmut+ F sin mez ...... (28). d'y dy Ex. 3. Te ae + ae +y =>. 2 (29). This is equivalent to (D?+ D+1)(D?-D+1)y=0. Hence y =e (4 ene sina) + e2% (4 ae a+ B' sin V3 2 2 2 2 Fase © Payee (30). dty ay Ka. 4. Tat + 2 Tat My= O ss eee eee ee (31). Writing this in the form (D? + my = 0, we find y = (L, + £0) cos ma + (+ Fx) sin me 1... (32). 189-190] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 523 190. Particular Integrals. We proceed to the determination of a particular integral of the equation in the two most important cases. 1°. If V contain a term for this makes f(D)y= a CL) oeart a by Art. 188 (4). | The rule fails when a is a root of ae ee ace gat excess ote knee (4). If it be a simple root, we may write GES ATOM GAD YA ELIE a ycqulaet Mp cee (5), where #(D) does not contain the factor D—a. The equation BCDC G) eat eres UO a: (6), is satisfied provided (D—a)y= Ha) Cave and we have seen, in Art. 176, 2°, that a particular integral of ae Le this is y= b(a) Oe Ee te Ry ey ORE EE (7): If a be an r-fold root of (4), we may write ECD yb CD) (CD) Say ees voces (8), where ¢ (D) does not contain D— «a as a factor. The equation CNG OU CRESS ATi hd einen tad (9), is satisfied, provided H D—a) y= —~e*. Ee ERTO) Now if we put y = ez, 524 INFINITESIMAL CALCULUS. [Cu XII we have, by Art. 188 (5), , (D—a)"' y=e* Dz, HT hence Vee ei | (4) This is evidently satisfied by Hf 7? ~ rig(a)” A particular integral of (9) is therefore yal UN ea marae ce eeeweeccenratess (10). 2°. Let V contain terms of the type H cos aa + K sin a7 ee (11). Since the operation f(D), performed on y=A cosar+ Bsin am ...1. 1052.0. +s (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 yin the equation FO) y=4i cos a2 + K sin Ga (13), and determining A and B by equating coefficients of cos ax and sin aw. 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 } (D") y= Hf cos aa + K sina (14), i.e. f(D) contains only even, powers of D. We have, then, by Art. 188, 3° y= ic aA a+ oa aye CB eee (15). This result fails if ¢(—a?)=0, 2.e if ¢(D*) contain D* + a’ as a factor, in which case terms of the type (11) occur in the complementary function. If the factor D’+ a? occur once only, we write pb (D) =x (D) (D4 ae (16). Now the equation x (D*) (D’ + «) y = H cos avx+ K sin aa...... (th); 190] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 525 will be satisfied if ak K D? + a?) y =——— cos av + ; eer Ge eu YG) The problem is thus reduced to one already solved under Art. 187, 2°, viz. we have the particular integral sin av...(18). K Tore gyn Ls pamper aot debe Ress (19). If the factor D’+ a occur r times in ¢ (D’), we write (D*) = y (D*) (D? + a?) oc. cece (20), and the problem is reduced to finding a particular integral of (D’ + a)" y Saar, cos aa” + a sin az...(21). If we assume y =u cos (aw —4rr)+ usin (ax —4$rm)...... 2)s, , (Di+a'yry = (2a)". D”u.cos ax + (2a)". D'v.sin ar+... by Art. 189 (20), (21). Hence (21) is satisfied provided H K it pp eg eine OL) Gay Geet eas Gav (Go) Hat Kur ““rearx a) Tri Gayy ay © particular integral is therefore | iar ah {H cos (aw —4rar) + K sin (ax —4r7)} In the general case, where f(D) contains both odd and even powers of D, the assumption (12) fails in like manner if, and only if, f(D) contains the factor D’+a’. Writing ACD Y= VOD) CDitinay eee ae be (26), * The assumption y=ucosax+vsinax 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 y+ (D) does not contain the factor D’?+.’, we first obtain a particular integral of the equation vy (D) y= H cosax+ K sin at ...... aie (OA » in the form y = Hi, cos av+ K, sin a@......... (28). It then remains only to solve | (D* + a’) y = H, cos aw + K, sin az.........(29). This has been treated above. . Another case where a particular integral can be obtained is that of where X is a rational integral function, of (say) degree r. We put y=x2"v, where m is the lowest index of D which occurs in f(D), and v is a rational integral function of x, of degree . The coefficients in v are then determined by substitution. 191. Homogeneous Linear Equation. An equation of the type dy Gray w da” 10 + a es a ee + An ay Agia (1) | nat Ft Any =Vereeree. : | is sometimes called a ‘homogeneous linear equation. The | complementary function in this case consists in general of a series of terms of the form Ca™, 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 Hz?, the corresponding term in the particular integral will in general be Ba?, 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 2 PLS php Lica 3 ) Bec: ax 7 + by =0 re EBON (2), we assume y= OL" (8). This will satisfy the equation provided {m(m —1)+ am + bd} Ca™ = 0.....c.ceeeee (4). 190-191] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 527 Equating to zero the expression in { }, we have a quadratic inm. If m,, m, be the roots of this, the solution is 1) Sd OLE SAUER Ts tame ee OED Opt a Yee (5). Again, a particular integral of the equation dy dy a O Tg t yt Oy = He Kae eee (6), will be SM ih bok © icra adlles Bere te: tf), provided {p(p—1)+apt+b} B= H....... ee. (8). Oa ach Ew. 1. Soa ieee (9). If we multiply by 7° this comes under the form (1); thus Sail: aV eyes ar ON Pehiataes cvet's es UD, (10). Assuming Ke Gr, we find m(m—1)+2m=0, or m(m+1)=0. The admissible values of m are 0 and —1; and the solution is therefore : V=A+ 2 Feith eRe ee ose sheds os (11) Cf. Art. 184, Ex. 2. PY 5, YY mt Dt Dy HO Lo ec cac cence ees Ea. 2. oat Lier, 2y =x (12) To find the complementary function, we assume y =C'x”, and obtain m(m—1)+2m—2=0, or (m—1)(m+2)=0, whence m=1 or —2. Again, a particular integral is y= C2, provided (2-1) (2+2)C=1, or C=}. Hence y= Ant 54 yd SE Ny ey (13). Complications arise when the equation in m has imaginary roots, or when it has coincident roots; there are, further, difficulties in connection with the particular integral when V contains terms of the type #?, where p is a root of the equation in m. ‘To avoid special investigation, we shew how 528 INFINITESIMAL CALCULUS. PCH il the equation (1) can always, by a change of independent variable, be transformed into a linear equation with constant coefficients. 7 If we put EM MMO Oy ne (14), then, w being any function whatever of x, we have du _ du, do_, ,du da dé. G0 Stags du du 5 or sd led (15) We will denote the operator d/d@, which is seen to be equivalent to ad/dx, by 3. Then, D standing as usual for d/dx, we have £D (a D™ y) = a DOH y + ma Dy, or amt! Pmt y = (aD —m) (a D”™ u) = —m) (a Du) Putting m=0, 1, 2,... in this formula we can express eDu, «Du, «2D u,... in succession in terms of Su, S’u, S°u,.... Thus, since the operator 3, = d/d0, is commutative, eDu=Su, 2° D’'u=3 (98 —1) u, «Du =3 (8 —1)(8—2)4, and so on, the general formula being a’ Du =3 (3-1) (9-2)... 8-741)... 17). If we substitute for the several terms of (1) their values as given by (17), we get a linear equation with constant coefficients, of the form d f@)y=V, or f(g) y= 7 eet (18). a? d. Ex. 3. oie tyne se dale altttgad aaa (19). This becomes {9 (J —1)—94+ lb y=e9, or (3-1 y =e 191-192] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 529 Allowing for the difference of notation, the solution of this is, by Art. 186, y=(A + BO) e + £679, or, in terms of 2, G=A+ Blogn)a+da (log)? ....0..0.4 (20) dy , dy a eh aE = Ex. 4. watts ty ia Pete ae Perret (21). This gives (+1) y=e%, y=A cos 6+ Bsin 6 + ye? = A cos (log x) + B sin (log x) + 42° ......... (22). 192. Simultaneous Differential Equations. In dynamical and other problems we often meet with systems of simultaneous differential equations, involving two or more functions of a single independent variable, and their derivatives, the number of equations being always equal to that of the dependent variables.) We may denote the dependent variables by the letters a, y,..., and the inde- pendent variable by t. Without entering into questions of general theory, it will be sufficient here to give a few examples exhibiting the methods which are most generally useful. In the first place, it may happen that each of the given equations involves only one of the dependent variables, and so can be treated separately. Ex. 1. In the case of a projectile under gravity, if the axes of x and y be horizontal and vertical, we have ately ick gee g (1) pao gpa cee Hence ie. We MAE Bead oa ed oh eee aoe ee Oe ee (2). The arbitrary constants A, A’, B, B’ enable us to satisfy the initial conditions as to position and velocity. Hx. 2. In the case of a particle attracted to a fixed centre (the origin) with a force varying as the distance, we have dx dy Amen? SF aa wg Bee ceecccccccscees (3), 530 INFINITESIMAL CALCULUS. [cH. XIL whence a= A,cos ,/pt+A,sin /ut, y=B,cos ,/pt+ B, sin ,/pt. If we eliminate ¢, we find (Bye — A,y)? + (Bow — Any) = (A,B, — A,B,)......... (4), which shews that the path is an ellipse. If the given equations, which we will suppose to be n in number, are not of this simple type, then by differentiations, and algebraical combinations, we may eliminate all the depen- dent variables a, y, z...,save one (say 2). If we substitute the general value of x, hence derived, in the original system, we shall find that this now reduces to n — 1 equations involving the n—1 dependent variables y, z,.... The process can be repeated until each dependent variable in turn has been determined as a function of ¢ and of arbitrary constants. In particular cases various modifications of the above method may suggest themselves. We shall content ourselves with a few illustrations taken from physical problems. ix. 3, Tf «, y be the coordinates of any point in a rigid plane which is rotating with angular velocity n about the origin, we have the equations da dy Te) Gp Mee (5). Eliminating y, we have da y : , ae =—7 aE =—-NWX, whence a = 0 COS (nt + €)..2.).c0aee ee (6), the constants a and e being arbitrary. Substituting in the first of the equations (5), we find Y = @ Sin. (1b + 6) | ieee ee (7). The results (6) and (7) shew that each point describes a circle about the origin with angular velocity n. fx. 4. In the theory of electro-magnetic induction we meet with the equations da dy qt lg + Rea=4, *§ dx dy coo oreeeeccrececne Mas +N 7 +Sy=F 192] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 531 Here x and y denote the currents in two circuits subject to mutual influence; # and S are the resistances of the circuits, Z and WV the coefficients of self-induction, WM that of mutual in- duction, and #, F are the extraneous electro-motive forces. Let us first suppose that H=0, F=0. The equations are then satisfied by ECCI eT yee ee ee (9), provided _ (LX+ Rf) A+ Byer an MrA + (NA+ 8) B=0 A ret Ch ‘ Eliminating the ratio A : B, we have (LA + &) (NX +8) — Md? = 0, or (LN — M*) + (LS + VR)A+ RS =0......... 59} Since (LS + NVR) -— 4RS (LN — M*) =(LS—- NVR)? +4M?RS, a positive quantity, it appears that the roots of the quadratic (11) are always real. Again, for physical reasons, ZW is necessarily greater than J”. Hence (11) shews that the two values of X must have the same sign (since their product is positive), and further that this sign must be negative (since the sum is negative). Hence, denoting the roots by —),, —A,, we have the solutions ee ie te ptf Ber A, and Bien Arent, y= Be Ast ‘where the relation between A, and &,, or A, and B,, is given by either of equations (10) with —A, or —A, written for A. On ‘account of their linear character, the equations (8), with # = F—0, ‘are satisfied by the sums of the above values of x and y, respec- tively. If # and F are not zero, but given constants, a particular integral of (8) is evidently F jostle Wry ice EAE OE, and the complete solution is E waa + Aje—*tb + Aye - Act, ce Sy te an oe eran (13), i ot Bye—Mt + Bye Act 582 INFINITESIMAL CALCULUS. eee PO eX tl where the relations between A, and B,, and between 4A, and 72, are as above indicated. , | | The first terms in these values of # and y represent the steady currents due to the given electro-motive forces; the remaining terms represent the effects 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 H# is a simple-harmonic function of the time, and Fis zero, Putting, then, Ef = Ecos pt, = Oa eeee (14), a particular integral of the equations (8) may be found ‘by assuming «=A cos pt + A’ sin pt, (15) y= Bes p+ Fin : On substitution we find, equating separately the coefficients of cos pt and sin pt, pLA'+pMB'+ RA=EL,, —pLA—pMB+RA'=0, 16) pA’ +pNB'+S8B-0, (0 (16). —pMA—pNB+SB' =0 These formule 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 ¢ increases. Hx, 5. Asa final example, we take the equations 2, 2, A e+ HEY ax thy=&, Pa iy “0, A, + Bot he + by= ¥ 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, Y=0, and assume c= fe, yo Gs ee mgs) as ep > = ij r - ’ 192] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 533 Ve thus get (Ad? + a) P+ (Hi? +h) G= “t bites (19). (Hd? +h) + (BN +6) G=0 iminating the ratio /: G', we have ; (Ad? +a) (BN +6) — (HAP +h)2=0 wees. (20), 4 Ba=2Hh) 2 + (ab —h?) =0...(21). The expressions’ | : de \« | al. sks — —- = {4 G et dt dt nd 4 (aa + Dhaest OY*) ...cccssnoes sores (23), epresent the kinetic and potential energies of the system. The former is essentially positive; hence A, B are positive and AB>H?. Tt follows that the left-hand side of (20) or (21) will be positive both for A2=+0 and for X®»=—o, whilst for *=0 the sign is that of ab~—A?. Also, from (20), it appears that the left-hand member is negative for A7=—a/A and for *7=—)/B. Hence if the expression (23) be essentially negative, so that a, b are negative, whilst ab —h? is positive, the equation (21) is satisfied by two positive values of A*, one of which is greater, and the other less, than either of the quantities —a/A, —B/d. Denoting these roots by A,, 4.2, we have the solutions ea Pett + Beat + Biers + Peat, y = Gert + Get + Gert + ee Of the eight coefficients, only four are arbitrary. The ratio F,,: G,, which is the same as f' : G,', is determined by (19), with d? written for X» Similarly, the ratio /,:G, or Ff, : G, is determined by the same equations, with A,’ written for *». The four arbitrary constants which remain may be utilized to give any prescribed initial values to 2, y, du/dt, dy/dt. It appears that « and y will increase indefinitely with ¢, unless the initial ‘circumstances be specially adjusted to make /, and /, 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, 6, ab—/h? are positive, the roots of the quadratic in A? will both be negative, viz. one will lie between 7 534 | INFINITESIMAL CALCULUS. [cH. XII 0 and the numerically smaller of the two quantities — a/A, — b/B, and the other will lie between the numerically greater of these quantities and —co. This indicates that in place of (18) the proper assumption now is a= F' cos pt + P' siny G cos pt + G sin pt...(25). (Uf, and (21), with — p* e twe roots of the quadnasie in sane them br p,? and p,”, we get This leads to equatigas written for aS Tt fol : cos pt + Fy’ sin 5 t+F,cos pt + F,' sin pot, -..(26) y = G7 L008 Apes + G,/ sin Soe + Gzcos pot + Gel cos pot} ~ ” where the ratios /,/G,, F/G, F./G., F/G. are determined in the same manner as before. Since ,'/G,'=/,/G,, and F/G, = F,/G,, the results may also be written x = F, cos (p,t + €,) + F, cos (pet + €,), y = G, cos (pit +) + Gy cos (pot +e,) Jf where F/G, and F,/G, 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 \”’ (or in p*) have been assumed to be distinct. It may be proved that they cannot be equal unless a/A = b/B=h/H; and that if these conditions be fulfilled the solution is of one or other of the two types: c= fe+ We, y= Get Gere eee (28), x= cos pt+ IF’ sin pt, y=G cos pt + G" cos pt...(29), 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 ab — h? is negative, and one root of the quadratic in )? is positive, the other negative. The complete — solution is now of the type c= Let + Fe + F" cos pt + LF” sin pt, y = Gert + Glet + G" cos pt + 4” sin pt 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 ORDER. 535 A slightly different method of treating the question is to assume UPS fie 2 ie toe eR ecreys SoeEe p (31). The equations to be solved now take the form 2, (A+ pH) oe (a+ ph) x =0, oe RCS a eda aeeeree peer (32), (1+ pB) a5 +(h+pb)x=0 These are both satisfied by ed Nou) ae OE eS Fe oc Tey ie eee (33), provided patie ameeoee Ghee ay 2 (34) atph ht mM ao, Hence p is determined by the quadratic (Hb — Bh) p? + (Ab — Ba) w+ (Ah— Ha)=0...... (35). If 4, #, be the roots of this, the corresponding values of )? are given by (34). In this way we obtain two solutions, which, on account of the linearity of the differential equations, we may superpose. It may be shewn that if we eliminate » from (34), we get the same quadratic to determine A? as before. Hence the condition for the reality of the roots of (35) must be the same as in the case of the quadratic (21). This is easily verified. If ’ be negative, the solutions are of the types a= FP, cos(pt+e), Yy=ml, cos (pit+ 4), a= PCOS (pot + &), Y=pgl", COS (Pot + €2) where Ff, F,, &, & are arbitrary. Hither of these by itself represents what is called a ‘normal mode’ of vibration of the system. To find the forced oscillations when the extraneous forces A, Y are of the type X=acos(nét+e), Y= sin (nt+6)......... (37), we may assume =F cos(nt+e), Y=Gsin(nt+e)......... (38), and determine the coefficients /, G by substitution in (17). A case of failure may arise, when the expression (23) is essentially positive, owing to »* coinciding with one of the roots of the quadratic in p”. 536 [y=( 12. INFINITESIMAL CALCULUS. (CH. XII EXAMPLES. LII. dy d a4 + 7 yyy [y= A+ Be-*.] a? d =4—3 10y = e Ly = Ac + Be-*.| dl? oT Hy =0. [y = Ae? + Bel] 2 Sys ie ald Tey ee [y =Ae” + Be* + Ce] d’y dy 2 a — 2m 2 + (mn? +n*)y=0. [y =e" (A cos nx + B sin nz). ] 2, Ta * dp t 18Y =: [y = e* (A cos 3a + B sin 3z).] et Oe =O [y=e-* (A cos 24+ Bsin 2x).] os 6 foe l3y=0. [y=e*(Acos 2x + Bsin 2x). ] + my = 0. A cosh — B ~ + Bsinh wp 3) 28 + (A’cosh NE + B’ sinh /2 ma) sin ac dy dy dy FE Ba tae moe is 0. [y=Ae®+(B+ Cx) e~*.] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 537 13. aa sda ae YO [y=(A + Ba) && + Ce-*.] d® d?y = Fr 14. 7a 8 gat ty =0. [y = (A + Ba) e” + Ce-*.] dy dy : ¥ dy 3Fy 4dy w 16. a qat3——y=0. [y=(4+ Bu+ Cn") e] 4, 2 17. CY 5 (mt +n!) 2 + m'n?y =0. [y=A cos (ma + a) + B cos (na + B).] 18. Shew that the solution of the equation iad hiahe eek adi aie is of the form x= Ae-at + Beft, where a, 8 are both positive (if & and p are positive) and a> f. dy dy = 19. Tako dg TY = sin na. | __ (m* — n*) sin nx + 2mn cos nx Ba: e EE dy ay dy 7 20. ia ~ ae + Y= cosh w, [y = pa’e” + dae-* + &e. ] dy | dy : - 21. aa =l+cosha, [y=a% + }e*—4haxe-*+ &e,] d*y 9 pee Sy i hn fe ‘ “ye 29. a ee kx + cosh kx ly = a (cos ka + cosh ka) + to. | ody dy dy * B . 23, Hed 4 Yoo, [y+ 5 + Ce | PV ldvV 24. Solve dr? a ae as a homogeneous linear equation. [V=A logr+ B.] 538 . INFINITESIMAL CALCULUS. - (OH. XII dy dy oe 25. 3 ae [y=A+ Blogx+ Ca'.| d’y B 26. oO 3 AY =e. [y= dat + = — 4x. | d’y dy 1 AB log a 2 mera ees an aa pre 27. wat te + y=. ly Sa | 2, 28. Oe Lynas [y= des = + 20% | 2FY 9 dy 29. o—* — 3a +4y=0% [y=(A+ Blog a) a*+a°.| di’ dx qe Ne A 30. (3-3) I(r) =9. f(r) == + Br+ Or? + Dr4,| 31. Prove that 32. Prove that d ) f (x=) a log a =a" {f(m) log « +f’ (m)}. 33. ete =); OY 4 2+ By =0. We 4 {(4 + B) sint + (4 —B) cost} e™, -(A cos t + B sin ¢) e~™ | d. 34. tose Y aiy, e=(A+ Bt en. = A—B+ Bt ott. g 35. Za Se+y=e, ee Fe [w=(A + Bt)e-#4+ Ae — de, y=—(A+B+ Bithe“*+ Ae + ee" | 36. oe 4y + 3=0, PY 1p 45 =0, [a= (A + Bt) cost + (A’ + Bt) sin ¢— 17, y=4(A+B' + Bt) cost+4(A’-B+ Bt) sint—12.] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 539 37. Determine the constants in the solution of the simul- taneous equations dx dy | miele. aa he so that, for ¢ =0, da dy ee tye (), Fray ao Vs [w=acosh /ut, y=V/.J/u.sinh /pt.] 38. The equations of motion of electricity in a circuit of self- induction Z, and resistance &, which is interrupted by a condenser of capacity 0, are dq G > dt where x is the current, and qg the charge of the condenser. Find the condition that the discharge should be oscillatory. [L>42°C.] da dy pe 39. Solve ap = 9; Boe and shew that the solution represents a conic symmetrical with respect to the axis of a. aa 40. Solve aa 0, ae = p17", and prove that the curves represented by the solution include a family of hyperbolas. dx nt dy 41. Solve de: = hare +f, LS + Re = Ab =X, it ease Gee Ais (nt + €), y=B+Lt+asin (nt +0). | : d*x dy 4 42. Solve 1p > Qn a + ma =0, ay aa + On + my = <1) [w= 1, and Uy (xe) = 8, (xr) ooo sac cee sveee, (14). 2nx Ex. it Tf ae (x) = Lecaiae GOROSTCOO Gao aaG lOsmeotor (15) we have eG eA ANG RAE 7) COLI erat ein Cae 2 (16), N=0 for all values of « without exception. The series is however not uniformly convergent over any range which includes x=0. For Rala¥=8(a\— 8, (a) tae 1+ n°a? and for any given value of 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 x>a, has been proved to be uniformly convergent for values of « ranging from 0 to any fixed value short of a. As a matter of fact it will be uniformly convergent up to z=a, inclusively, but this cannot be established by the above method. t By this notation it is meant that x may range from a to b inclusively. T. 35 546 INFINITESIMAL CALCULUS, [CH. X11 The question may be illustrated graphically * by drawing the curve y= (a), and also the ‘approximation-curves’ y=S,, (&), for n=1, 2, 3,...7.. 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 curves Y= (ec) + O°. ieee (18). In the above Ex. the approximation-curve y = S,, (a) is derived from Fig. 17, p. 31, by contracting all the abscissae in the ratio 1/n The curves tend ultimately to lie between the limits y=+o for the greater part of their course, but (if <1) they will always transgress these boundaries in the neighbourhood of the origin. See Fig. 149. Y Fig. 149. Hae 2s te S,,(#) = tanh 12... ee (19), we have S (x) =lim tanh nv =—1, 0/41 (20), N=0 according as x is negative, zero, or positive; see Fig. 22, p. 43. The annexed Fig. 150 shews that no approximation-curve lies — * The author is indebted, here and in Arts. 196, 198, toa paper by Prof. W. F. Osgood, ‘‘A Geometrical Method for the Treatment of Uniform ae, and Certain Double Limits,” Bull. Amer. Math. Soc., 2nd ser., t. 3 (1896). + See for example Fig. 152, Art. 203. 194-195] INFINITE SERIES. 547 wholly within the required limits, in the immediate neighbour- hood of the origin. We have here an instance of a series whose sum is a dis- continuous function of w, although the individual terms are themselves continuous. Fig. 150. 195. Continuity of the Sum of a Power-Series. We can now shew that the sum of a convergent power- series is a continuous function of « for all values of x within the range of convergence, 35—2 548 INFINITESIMAL CALCULUS. [CH. XIII If x, x denote any two values of the variable within this range, we have, with the notation of Art. 1938, S (a’) — S(«) = Sp (a’) — Sn (@) + Bn (2) — Rn (a)...(1), and the condition of continuity is that, o being any assigned magnitude, however small, a value of 6a shall exist such that | S(a@’) —S(a)| 0, Y = we ew em ee KS - Sm me em ee Fig. 151. 197. Derivation of the Logarithmic Series, and of Gregory’s Series. The following are important applications of the theorem just proved : 1°. If|z|<1, we have 1 97] _ INFINITE SERIES. 553 Hence, integrating between the limits 0 and a, log (1 + w) = a— $a? + fa8— dat oo. .. (2), which is the ‘ logarithmic series*.’ The proof applies only for | «| <1 ; and we cannot assert without examination that the result is valid for # = + 1. For a=1, the terms on the right-hand of (2) are alternately positive and negative, and diminish continually, and without limit, in absolute value. Hence, by a known theorem (Art. 6), the sum is finite, and the remainder after nm terms is in absolute value less than 1/(n+1). Hence this remainder can be made less than any assigned magni- tude by taking n sufficiently great. The same result holds a fortiory when # is positive and <1. Hence the series on the right-hand of (2) is uniformly convergent up to the point x=1 inclusive; it follows by the method of Art. 195 that the sum is continuous for values of w# ranging up to unity, inclusive. The two sides of (2) are therefore finite and continuous up to «=1, and since they have been proved equal for values of « which may be taken as nearly equal to unity as we please, it follows that they are rigorously equal for «=1. Hence log 2=1-—$ +4 —4F4+... voc... ei.e (3). This formula, though exact, is not suited for actual calcu- lation, on account of the very slow convergence of the series. It may be 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 #, we obtain 1 ~ = 2 (e+ ha? +1054...) Sree (4), log a: l—@ which might also have been obtained directly by integration from the identity 1 1 2 ae os = 2 4 ~ wee a 1— 2 2(1 + 2+ a+...) sralotelets (5). * Apparently first obtained by N. Mercator in 1668, 554 INFINITESIMAL oe. [CH. XII If in (4) we put «=1/(2m+1), we obtain m+1 log (m + 1) —log m=log 1 1 1 = Qype Fo cen cecee 6). sacl son ee | } 0, This series is very convergent, even for m=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 yp by which logarithms to base e must be multiplied in order to convert them into logarithms to base 10*. 9° If «<1 we have 1 Se aa eer ers 2 us od 6 e@ eeeoeeoeeeeens ° ieee l—a+at—at+.. oe eilt Hence, by integration between the limits 0 and a, tan“ # = # — 445 + tao — tat (8). This is known as ‘ Gregory’s Series.’ The function tan x here appears as the equivalent of edt 0 1 ot 7 3 and must therefore be taken to denote that value of the ‘multiform’ function tan w which lies between + 41. : For reasons similar to those given at length in connection with the formula (2), the result (8) holds up to both the limits +1 of a, inclusively. Putting «=1, we find tr =1-4 t—F+ see cecccccverseees (9). * The most rapid way of determining « is by means of the 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 has been superseded for the calculation of z by others. Euler used the identity ja = tan} + tan}... ren ont (10), which gives igo.) 1 ee 1 1 eee poh ay a oft a in=(5 3 9+ 59 .)+(5 333+ 53s Athy Machin had previously employed the formula dar = 4 tan *$ — tan shy ...cceceeeee eee (12), which, like (10), is proved in most elementary books on Trigono- metry. This leads to fee 1 ) 1 1 1 ras cars cite Artie) On account of the importance of the matter, it is worth while to give the details of the calculation of 7 from Machin’s formula. To calculate tan™11, we first draw up the following table: 1 1 + 5a aaane 1 200 000 000 0 | +-200 000 000 0 . 8 000 0000 | — 2 666 666 7 5 320 0000 | + 64 000 0 7 12 8000 | —- 1 828 6 9 5120 | + 56 9 11 S02 19 i 8 f 1 The 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“4= "197 395 559 8. Again to calculate tan-1,1, we have the table: 096 INFINITESIMAL CALCULUS. [CH. XIII 1 F 1 239” nN. 239” 004 184 100 4 +004 184 100 4 (EoVo- = 24 4 This makes tan-1,3, ='004 184 076 0. Hence a7 =4 tan™1—-tan,4, =+°789 582 239 2 —-004 184 076 0 =-785 398 163 2, r= 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-1,3,. The errors, therefore, in the inferred value of z, even if cumulative, cannot ‘exceed Ax (4x5xh4+2xh), =44, times the unit of the last place. The first seven decimal places cannot therefore be affected, 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 a7 to ten places is w= 3'141 592 653 6. 3°. If|«|<1, we have, by the Binomial Theorem (see Art. 200), 1 A eevee lL 3 Tease tga gt oF yaar eee coe (14). Hence, integrating term by term between the limits 0 and a, Sippy ee sin t= a+ 53 toa 5 oe a series due to Newton. 197-198] INFINITE SERIES. 557 If we put «= 4, we get Be eee ee tS TOS SSS Ra Ta a from which 7g can be calculated without much trouble. 198. Differentiation of a Power-Series. If the series A, +A,@ + Agu? +... FA RL +# o0. wc eee. (1) be convergent for any one value (a) of w, the series A, +2Age + dAga? +... +A nv + oo. eereeee (2), composed of the derivatives of the successive terms of (1), will be essentially con yerReny for all values of x such that |e| <]al. For the hypothesis requires that |.A,2”| shall tend with increasing n to the limiting value 0. Hence if M denote the greatest of the quantities ACO ea laeen | ne |, wearer ans. (8), the terms of (2) will be in absolute value respectively less than the ao terms of the series a + 26+ 3t7 +... + ni? 3+...) eee. (4), where t=|«/a\. The ratio of successive terms in (4) is of the form BtAL or (1+=)¢ n n and the limiting value of this, for increasing n, is ¢, which 1s 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 t,, where ¢, may be any _ quantity between ¢ and 1. The series (4), and @ fortiort the series (2), is therefore essentially convergent. 558 INFINITESIMAL CALCULUS. [CH. XIII We can now shew that if S(#) denote the sum of the series (1), and f(#) that of the series (2), then JQ =8 (€) eee (5), z.e. the sum of (2) is the derivative of the sum of (1), for all values of « within the range of convergence of (1). Since f (a@) =A, + 24,7 + 8A? +... + Age *+ 2... (6), we have, by the theorem of Art. 196, [4 = A,f+4,64 4,84... = 8 (BE) = Apis stasis (7), provided & lie within the range of convergence of (6). Hence, differentiating both sides with respect to & and afterwards writing # for &, we obtain the result (5). We have thus ascertained that, for values of # within the range of convergence, the sum of a power-series is a differen- tiable, as well as a continuous, function of w, and that its derivative is obtained by differentiating the series term by term ™*. A more general theorem is that if the series S (av) = U, (x) + Uy (00) + Ug (0) +... + Uy (@) +... (8) be convergent for values of « extending over a certain range, and if the series Up. (x) + Uy’ (@) + Ug’ () + 20. + Uy! (0) + ce ceeeee (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 S’ (a). Ex. 1. If|«|<1, we know that 1 ye a 78 Tog 7 itete + OP Utes (10). * 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. biel 198-199] INFINITE SERIES. 559 Differentiating both sides, we obtain 1 a 9 02 3 I ae tae . (I—a) 1 + 2a + 3a? + 40% + (11) A second differentiation leads to 1 ack en at f- Dee ae Ag? Ay i et pee kG A (ay £11.24 2.3043. 40744. 507 + ) (12) Ex, 2. On the other hand, consider the series (considered already in Art. 194) in which a 2nx a 1 + n2x? This makes S (x) = 0, and therefore S' (x) =0 for all values of « ; but, if we differentiate term by term, the sum of the resulting series is : . 2 lb—nie Me ura) elie ee oR Mah (14). nN=0 nao (lL + nix’) This vanishes if «+0, but it becomes infinite for «= 0; the series of derivatives is in fact divergent for x= 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), 1t is often possible to obtain a solution in the form of an ascending power-series, thus y=Ay t+ Aat Ap? t+... + Ane + 00 oece. (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 #, 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 A,, A,, 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 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 (A, — Ay) + (24, — A) e+(3A; — A.) v?+... + (nA, — Ay 3) Oo tee (3), which will be satisfied identically provided A,= Ay, A,=3$4,=4A), 4,=44,= 5-540 and, generally, 1 1 ois = 7 Ag = — 7a A,, ee ccerereees (4) We thus obtain Y = Ay (&). on csddecs steer (5), where, as in Art. 17, E 1 Hp a” (f)= 2a ee (6), and 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 where v is to be determined, and substituting in (2), we find © B (a) +0{B (a) - E(@)} =0 ae (8), ia 2 199] . INFINITE SERIES. 561 or, since £'(#) satisfies (2), 2°, Let the equation be Assuming the form (1), and substituting, we find (1.2A,+ A.) +(2.3A,4+ A,)a@+(8.4A,+ Ay)v?+... + {(n—1)nA, + An_.}a”?4+...=0...(11), which is satisfied identically, provided 4c ip eae siger Gag kal Ay=— 7 Asap Ao Aya— pp A= Av and, generally, Am = 37 (Gai 4” We thus obtain the solution Gre ic Hic Wg = (1-5 R-)+4 Cor Wades hoa (bS)s Sack an Le (e554 Fyn) (08) The series in brackets are easily seen to be convergent, and their sums therefore continuous, for all values of a. It has been shewn in Art. 182 that the complete solution of (10) is y=Acos#+ Bsin@ ......... heh (14). Hence, given A, and A,, it must be possible to determine A and B so that the expressions (13) and (14) shall be identical. L. 36 562 INFINITESIMAL CALCULUS. & XIll For example, putting 4,=1, A,=0, we must have | 2 : 1+ ...=Acosz+Bsin se, | and, changing the sign of a, 12 1-F 4+ - ...=Acosa—Bsinz. Hence we must have B=0, and putting «=0, we find A=1. We thus obtain the formula aren cosw@=1—s5 + 447 cei Pein = (15). In the same way, if we put 4,=0, A,=1, we find A =0, B=1, and therefore é as Set SIN B= & — Bt FH see eet eeteeees (16). 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 coefficients, which the function satisfies*. For example, let. y=(1+@)™ 0.5) (1), where m may be integral or fractional, positive or negative. Taking logarithms of both sides, and then differentiating, we have ldy m yd 1l+a’ * This method was first employed by Newton, to whom the series for cosx and sinw are also due. The manner of obtaining these series was, however, different. e200). INFINITE SERIES. 563 or (142) my=0 Ho ERO Ree (2). Assuming Y= Ayt Avot Ang? + ... + Anat... (3), and substituting, we have (1 + 2) (A, + 24+... + nA,e"1+...) —m(A,+ Aya + Apa? +...4+ Ape" +...) =0, or — (A,—mA,) + {2A,—(m—1) A,} @ + {[3.4;—(m— 2) A,} a? +... + {nA,—(m—n+1) Ana} am 4+...=0...... (4), which is satisfied identically provided A, = zo Ay aM Am TG Ae a ae a eee ae and, generally, Aa hi aw AO) - We thus obtain m m(m—1) =A, }145 2+ Sry, at oe 5 EA Se i ot (6), as a solution of (2); and it is easily verified that the series is convergent so long as |w|< 1. Now if we retrace the steps by which the differential _ equation (2) was formed, we see that its complete solution is Meee Canes Eee tits can cemeee totes Cf), 564 INFINITESIMAL CALCULUS, [CH. XIIl where C is arbitrary. Hence (6) must be equivalent to (7), and putting «=0 in both, we see that C=A,. Hence m(m— 1) 2 | 21 4 mm 1)...dm—n+ 1) on n! (1 +a)"=14+7 04 for all values of w such that |~|<1. This is the well-known ‘Binomial Expansion*.’ As a further example, we take the function sin-!& Y cS J —2) eee ere eeeresvecres sevens (9). Multiplying up by ,/(1 —#?), and then differentiating, we find dy x 1 op?) 4 oe ee NO) de se Ja=2)’ or (1- a) @ Fe 1. eee eee (10). Assuming y= A, +A,ot+ Age’ + ..:+ 4,0" ee (11), we find (1 — a”) (A, + 24,0 4+ 3A,n? +... +A,0" 7+...) —2(A,+A,o+ Ayn? +...+ Ape" +...)=1...(12), or (A,—1)+(2A,—A)) «+ (3A; — 2A) a? +... + {nA,,—(n—1) A,_,a™ 1+... =0...(13), which is satisfied identically, provided | 1 A, =1, A,=5 Ay, a7. 3 ie ieee 44> 749 (14). Lets ys 5 des 45-5405 5.0?) “eT GA ee * Newton (1676), The cases of = +1 would require special investigation. — 200) . INFINITE SERIES. 565 We thus obtain the solution i-th a ee tak a +A (1452 to 4e sed fea +...) (15). Now if we retrace the steps by which the linear equation (10) was formed, we see that its general solution is FEN ad EAST eget es ren) pr CP (16)*, sin-'x A Y= Jd = 2) =f ea) sere er ere rereee and, by the nature of the case, (15) must be included in this form. If we put «=0 in (15) and (17), we see that A= A,. The identity of the two expressions for.y then requires that or sin’ 2 I Jd —#) — 2) Re ae aaron Ee Teg fs abiae.oniee (18), and ial a Sha ea Vist ies Se (19). J(1 — «*) 2 2.4 These series are both of them convergent for | «|<1. The result (19) is a mere reproduction of the binomial ex- pansion of (1 =~«’)-4. If we put «=sin 6, the former ae may be written 2 4 @=sin @ cos (1 + = sin? + 5 3 psintet.. | ven (aU)s Again, if we put tan 6 =z, we obtain the form Ee wi Zia Thr f tan ) tan *=T,2 {1+ 31lae' 305 Ae EBs hc es (21). This series has been made the basis of several ingenious methods of calculating 7. It may be shewn, for example, that 4x7=5 tan" 7+ 2 tan? 3, - whence 28, 2 Ea) 2.4/2? D a1 +3 (700 +355 (imo) =} 30336 (2 (_144 ies 144 \3 on * [00000 { +3 (qoo000 3.5 (i000) + uf... (28), * See also Art. 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. a0 2 ob Oe 3 (sin™ a a" 3 mm +35 6 as a ana boo as (23). EXAMPLES. LIII. 1. Prove by repeated differentiation of the identity 1 l-« where | «| <1, that, if m be a positive integer m(m+1) _ m(m+1)(m + 2) 1% 1.2 1.2.3 =l+wvta?+ar+..., (1—a#)-™=1+mae+ 2. If|x|<1, prove that tanh! a =a%+taP+40°+..., 38. If|e|< , prove that Oe a act git Os —... =(14+2) log (1+) —«a. Does the result hold for «=+1? 4, If|x|<1, prove that a at a8 Ke ny OA Uassiek pF ghoees e. OR Sh =) 2 loca eee So x tan™’ «— 4d log (1+ 2’). Hence shew that 5. Prove that [raw x a 0 x 3.3! Se Ditmar *sinh & 4 i a8 x : if 1 See Sse * For the history of these series, see Glaisher, Mess. of Math., t. ii., p. 119 (1873). INFINITE SERIES. 567 6. Prove that b — aé "E da=log® + (b- rf feats 3-37 + a 2.2! 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 4=1:386 294 361 log 5=1:609 437 912 logo = 1°491- 759.469 8. Prove that log 8=2:079 441 542 log 9=2:197 224 577 log 10 = 2:302 585 093 b= '434 294 482. log 2= Ta—2b+ 3c, log 3=1lla—36+ 5e, log 5=16a—4b+ Te, and thence log 10 = 23a— 66 + 10e, where a= i = 7 eee, +... = 1053605157, b= a + ae 7 eee .. ='0408219945, aoe . iat ‘i Boat = 0124225200. Apply this to find log 10. (Adams. ) 9, Prove that log 2= 7P+ 5Q+ 3R, log 3=11P+ 8Q+ 5R, log 5=16P+12Q0+ TR, and thence log 10=23P+17Q+10R, where P=2 (7 4 ae: es i) = 0645385211, Q=2 (Gt ant Pec a ...) = 0408219945, 19 * 3.493 * 5.493 Te a tee Bel ie 8s) +...) = 0124225200. 161 * 3. 161° * 5.1619 : Apply this to find log 10. (Glaisher. ) 568 INFINITESIMAL CALCULUS. [CH. XII 10. Shew graphically that the series in which Un (x) = (1 — x) a” is not uniformly convergent in the neighbourhood of « = 1. 11. In the series for which ght go” 42 Ue) a 1 n+? we have S'(a) =a for values of « ranging from 0 to | inclusively, whilst the series of derivatives of the several terms has the sum | for 1>a>0, but the sum 0 fora=1. Account for this. 12. Examine the character of the convergency of the series for which S,, (x) has the forms 1 ele . sin 2a” ta? gl: nate, 6 slime : + x NX respectively. 1s Sate) = : log cosh na, prove that S (x) =| «| for all values of «a. 14. If the series Uy + Oy + Ag wee + Ont .s be essentially convergent, the series Ay + A COS % + Ay COS 2H + ... + Ay, COSNL +... will be uniformly convergent for all values of a. 16. If yy My Any ++ ny vee be a descending sequence of positive qualities whose lower limit is 0, prove that the series Ay + Ay COS H + Ay COS 2N+... + Ay, COSNB+... will be uniformly convergent over any range of « which does not include any of the points # = 2s7, where s=0, +1, + 2, +3,... 16. Prove that INFINITE SERIES, 569 17, From the formula a Nee dy “(1—esin® yp) for the radius of curvature of an ellipse, shew that the perimeter is equal to 3 3° .5 Sard ie at sae ee le LF 4 Rin, sak elie 6 27a (1 é)(1+ 56 +o ga +59 ge gee + ); 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 /, oscillating through an angle a (<7) on each side of the vertical, is ar A/; ih J(i—si ~ sia a sin asin? )’ prove that this is equal to ah F ies era am / (1+ jsintda+ ge sint fat, = misin® hat...) 19. The perimeter of an ellipse of small eccentricity e° exceeds that of a circle of the same area in the ratio 1+ ,e%, approximately. 20. The surface of an ellipsoid of revolution (prolate or oblate) of small eethy, e exceeds that of a sphere of equal volume by the fraction ~,e* of itself. 21. Prove that 1 1 ] OI apeessor order 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. 23. Obtain a particular solution of the equation dy dy erat at my = 0 in the form y=A (1-77 + 12.2 2, 9.37 24. Obtain a particular solution of the equation ap ldo, ae is +kh’p=0 22 4, in the form p=A (2 - ay a ) dy. 25. Integrate (1— wo Ta ae 0, [CH. XII by series, and deduce the expansion of sin~'x (Art. 197 (15)). lon 26. Prove that y = sinh satisfies the differential equation Hence shew that, for |#|<1, 3 b) log {e+ te sao 23° 2 toe 27. Obtain a solution of the equation d du Fa (LH) Go} tm + 1) u=0 in the form w= A(1- ae alt (n —2)n nln ae + B(n- tial eee) ae (n — 3) (n—1) (n+ 2) (n +4) ee ay 5! INFINITE SERIES. 571 : 28. Obtain a solution of the equation 2 (1-0) 54+ ty—(a+ f+) a} 2 apy=0 in the form , 2(2+1) 8 (8 +1) Ura aw rag a(t 1) (a+2) 8 (B +1) (B+ 2) eA 7) 1.2.3.y(y+1) (y+ 2) e/a 29. Obtain a solution of the equation a? ld Tats get (@- a) e= 0 in the form 2 ky? ktr* aah (1 ~ F(Qn 42)" F.4(Qn+2)Qnt+4) -) 30. Obtain the solution of the equation 2D aR - 2 (n+ 1) ae +)2R=0 dr? L in the form Jey? k'r4 \ pe oe Ir? kr on 34 ee Se Pe Nae 1 Te chloe Ors) Ae 2(1=2n) * 2.4 (1—2n) (3 2n) ) Sif y =sin (m sin7!«), prove that (1- 2) FY og Hs mty = 0. Hence shew that : 2. a — 3? 2 ops Mise ype UA m sin 6 3! m2 _ 92 cos m0 = 1 — mr sin? 0+ not 32. Obtain a solution, by a series, of 3 (1-04) 4 +-m (n— 1) =0, and give the symbolical expression of the complete solution. CHAPTER XIV. TAYLORS THEOREM. 201. Form of the Expansion. Let f(a) be any function of # which admits of expansion in a convergent power-series for all values of # within certain limits +a. It has been proved, in Art. 198, that the derived function f’(«) will be given by a similar series, obtained by differentiating the original series term by term, for all values of « between +a. By a second application of the theorem cited, the value of /” () will be obtained, for values of a be- tween the above limits, by differentiating the series for f’ (#) term by term. And soon. Hence, writing f(@)=A,+Aet+ Ane? +... 4+ Ana” aparece (1), we have f (@)= A,+2A,e0+...+¢nApx”™ Bvesy f' (2) = 2.1A,+...4n(n—1) Ana”? +..., ee eee wnink )s yf (B)— n(n—1)...2.1Ag+ ecoeceeoeeeoeeeeseoreere eres eovoer-or2*eoooereeeereteoeoeseese soe Heeeeee 8088 Putting «=0 in these pe we find Ay=f(0), Ar =f'(0), 4a = 55 = An= a ") (0) .-.(3), where the symbols /(0), ee (0), f” (0), ... are used to express that w is put =0 after the differentiations have been per- formed. 201] TAYLOR'S THEOREM. 573 The original expansion may now be written F@)=fO) + af O+ GS’ O) +. + ZF" Ot, 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 ro Eat a Nat AU hay ae: tea ee ER (5), we can deduce the form of the expansion of ¢(a@+.) ina power-series, when such expansion is possible. For if we write, for a moment, U=A4+28, we have /(x)=¢(u), F'@)==. 6(== p(u).& = 8 CW, FO) =F 4 u)= 5 $ (u) = 8" (w), and so on (Art. 39, 1°). Hence, putting z=0, w=a, we find F(%)= $ (a), f(0)=$' (a), fF" (0) = $" (a), «fF (0) =$™ (a) so that (4) takes the form P(a+2)=$(a)+2b'(4) + 7-5 “9 (a+. +5 “3 (a)+.. This is known as Taylor’s Theorem}. We have deduced it from Maclaurin’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. a + 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 Cases. 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. Regs Be h(a) = 0" 20.2, see eee (1), we have ¢' (a) =ma"—,. d' (a) =m(m—1)a™,... 6” (a)=m(m—1)...(m—n+ laws (2). Taylor’s formula then gives m — ym m—1 m(m—1) M—2 2 (a+ 2) =a™”+ma BI PRG U7 + ee m(m—1)...(m—n+d) oes Le 1 ose RS ciety 0 | (3), which is the well-known Binomial Expansion. That Taylor’s Theorem cannot hold in all cases, without qualification, 1s shewn by the fact that the series on the right-hand is divergent if |#|>|a|. For |#|<|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. yearn We f(@) Ss @ 0. (4), we have JF (a2) = &, s and therefore fO)=1, f™O)=1. Maclaurin’s expansion is therefore 7 a” f=] ERC oS 1.2.37 ase Te er ieee SOD 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 equal to ¢ (a+). 202] TAYLOR'S THEOREM. 575 b> ® Cf Art. 197, 1°. 3°. Let MLCT =e COR MM Mea tesnaretetertect ser (6). It was shewn in Art. 63 that this makes f™ (#)=cos (a@ + 4n7), so that F(O)=1, f™ (0)=cos gn ....25..5... (7). Hence f™ (0) vanishes when n is odd, and is equal to +1 when 7 is even, according as 4n is even or odd. Substituting in Maclaurin’s formula, we get a a COR ere i tig pot +) Omit (8) 4°, Let Cf Ci) SLL 2 oak SAS § cas ata? (9). This makes J™ (x) =sin (a + $n7z), so that F(0) =0; f™ (0) =sin $nz = sin {4 (n— 1) 7 + dr} ...(10). Hence f™ (0) vanishes when n is even, and is equal to +1 - when n is odd, according as $(n—1) is even or odd. Mac- laurin’s formula then gives LY = Hp a . a 1 sin @=@— a7 + 5 — + +(-) (n+1)i* sees e (11). The results (8) and (11) have been established rigorously in Art. 199, 2°. 5°, Let PCIE For ARE I eh gare (12). This makes —)"-1(n —1)! f'(2)= at and, for n>1, f (2) =! a a oh Hence /(0)=0, f’(0)=1, and, for n>1, : f™ O)=(— )*I1(n- 1)! ... ee. (13). : Substituting in Maclaurin’s formula, we get t Cs tale tl log (1 C2 Need oer rue ah (14). When the general formula for the nth derivative of the _ given function is not known, the only plan is to calculate the 576 INFINITESIMAL CALCULUS. [CH. 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. Hx. ‘To expand tan x as far as a’. Putting J (x) = tan &, we find in succession 7 (2) ]see’ a = 1-+tan’ Zz, J” (x) =2 tan x sec? # = 2 tan «+ 2 tan® a, J" (e) =(2 + 6 tan? x) se?«=2 4+ 8 tan’ x + 6 tan*a, J (#) = (16 tan x + 24 tan? w) sec? x | = 16 tan a + 40 tan? a + 24 tan’ a, JY (x) = (16 + 120 tan? « + 120 tan‘ x) sec? x = 16+ 136 tan? « + 240 tan‘a + 120 tan’ a, J™ (x) = 272 tan x sec? « + &e., I™ (x) = 272 secta +:&c., where, in the last two lines, terms have been omitted which will contribute nothing to the value of f" (0). Hence FoAQ= 9; J’ (0)=1, J” (0) = 9, J” (0) = 2, LA, J” Os FO) 0, JV Oja2eg; and the expansion is aie 200 AG Dia po ROLES pda + ae = 04+ ba + Peat Mea + ce seceeeees (15). That odd powers, only, of x would appear in this expansion might have been anticipated from the fact that tan « changes sign with 2. 203. Proof of Maclaurin’s and Taylor’s Theorems. Let f(«) be a function of x which, together with its: first n —1 derivatives, is continuous for values of # ranging from 0 to h, inclusively ; and let us write J (2) =p (@) + By ()errereeee. Ree vely 202-203] TAYLOR'S THEOREM. 577 where 2 Dy (2) =f(0) + 0f (0) + Ff O) + ot ay SO 0) Riarsat sce (2); ie. D,(#) 1s the sum of the first n terms of Maclaurin’s expansion, and f&,,(#) is at present merely a symbol for the difference, whatever it is, between f(#) and ®,(a#). The object aimed at, in any rigorous investigation of Maclaurin’s Theorem, is to find (if possible) limits to the value of R, (x); in other words, to find limits to the error committed when f(z) is replaced by the sum of the first » terms of Maclaurin’s formula. If we can, in any particular case, shew that, by taking great enough, a point can be reached after which the values of f,,(#) will all be less than any assigned magnitude, however small, then Maclaurin’s series is neces- sarily convergent, and its sum to infinity is f(#). It is evident that the argument cannot be pushed to this con- clusion if f(#) 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 EA et DR he Ae ete A as 5G cicig hance ts (3), has already been utilized in Art. 112. The plan there adopted was to determine the coefficients A,, A,, Ag, ... A,-, so that the function (3) should be equal to f(x) for m assigned values of 2, 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 » — 1 derivatives coincide respectively with f(x) and its first m—1 derivatives for the par- ticular value x=0. The result of this determination is, by Art. 201, the function ®, (x). In the graphical representation, the parabolic curve y = ®, («) is determined so as to have contact of the (n—1)th order (see Art. 206) with a given curve y=/ (x) at the point e=0; and the problem is, to find limits to the possible deviation of one curve from the other, as measured by the difference of the ordinates, for values of x lying within a certain range. This is illustrated by Fig. 149, which shews the curve PEACOCK EL) GEFT Te oe. dade sess a (4), > 578 INFINITESIMAL CALCULUS. [CH. XIV and (by thinner lines) the curves Y= UX, Y= U—ha, y= H%— 4a? + JAP, 0. reveeeeee (5), obtained by taking 1, 2, 3, ... terms of the ‘logarithmic series’ (Art. 202 (14)). The dotted lines correspond to x=+1, and so mark out the range of convergence of the latter series*. y. eeon ee eceeeor co@mes a =~ omen — 4 ee eee ee em eh 88 OSS] se OO wee Bee He Cee eK HH OC OBES He ww oe em wee eeece Al It appears, from the conditions which ®,(«) has been made to satisfy, that R,(a#) and its first n—1 derivatives Fig. 152. * This very instructive example is due to Prof. Felix Klein. / 203] TAYLOR'S THEOREM. 579 will be continuous from 2=0 to «=h, and will all vanish for z=0. Now we can shew that any function which satisfies these conditions, and has a finite nth derivative, must lie between nm 0 As andeR— n! n! where A and B are the lower and upper limits to the values which the nth derivative assumes in the interval from 0 to h. For, let #(#) be such a function. By hypothesis, we have et ee a{()) =O, aut (0) =— 0, (0) = 072,(6), and ANY 0) cB ory Bs So sie dae 8s (7), the latter condition holding from #=0 to c=h. It follows from (7), by Art. 89, 4°, that f "Ade < | F («) da < i ‘Bde*, 0 0 0 or, since #'"-» (0) = 0, CA 6 > 0. This is an accurate form of Taylor’s Theorem. It holds on the assumption that ¢”) (z) exists and is continuous from z2=a to s=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 p(ath)=¢ (a) + hf’ (a+ Oh) ......eee eee (18) ; and, putting n= 2, b(a+h)=¢ (a) + hd’ (a) + Th? oh" (a+ Oh) ...... (19). These agree with Art. 56 (9) and Art. 66 (23), respectively. Another form of remainder may be investigated as follows. If F(x) be subject to the conditions stated in (6), we find by an integration by parts h a\r-1 as | n—2 Aes (2) ea AS ery (n-1) (» | (1 5) 0) (a) dx = if (1 i) FY (a) dee since the integrated term vanishes at both limits. Performing _ this process 2 — 1 times, we obtain [O-Gy mee TR ay | F'(@) de= oo) Fim, % FO=Goy fO- r PO (2) die..e(21), _ Art. 89, 3°, that Since the function under the f sign is continuous, we infer, by F(R) = Gapy h ~ OBO Bl). reseeee (22), “Ty : where 1>6> 0. 582 INFINITESIMAL CALCULUS. [CH. XIV It appears, then, that the last term in (15) may be replaced by h” 7 (n—1)! (1 — 0)" " fC'(Gh) a ee and the last, term in a: ) by rea y= 88 OY (+ OM) oe cecececeee (24). These forms of Le 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 we compare with it the curve y= A, + A\o+ Ago? +...+ A, sa" Age ee), in which the n+ 1 coefficients are determined so as to make the two curves intersect at 2=0 and «=h, and, further, so as to make the values of dy dy = arly da’ da’ dat respectively the same in the two curves at the point #=0. These conditions give 4o=f(0), A=f'O), 4:=5,F7O) » ype a Ff (0) ...(8), as before, and T(h) = Apt Ayht+ Ah? + ... + An ah" + Anh” ...(4), this latter equation determining Ay. Denoting by F(z) the difference of the ordinates of the two curves, 1t appears that F'(0)=0, £’(0)=0, F” (0) =0,... F™) (0) = 0...(5), and Ph) =0 .... \U (6). Since #'(#) vanishes for «= 0 and «=h, it follows, under the usual conditions, that F” (#) vanishes for some value of « j 203-205] TAYLOR'S THEOREM. 583 between 0 and A, say for «= 6,h, where 1>06,>0. Again, since #” (x) vanishes for = 0 and # = 0,h, F” (a) will vanish for some value of « between 0 and 6,h, say for « = 6h, where 0,>6,>0. Proceeding in this way we find that Fe) (x) perishes for #=0 and w=6,_,h, where 1>6,_,>0, and hence that BC OTB 22 Ai se wen Gunes 5 ely Cece ee CE): where 1>@>0. Now, on reference to (1) and (2), we see that mea pee pel (ny eA oo Neveddesis ie (8). It follows from (7) that 1 Fan 18 NSE CRRA ty (9). Hence, substituting from (8) and nee u ey) we obtain Sh) =fO)+Rf (0) +5," (0) +. ae ah" © ge ” f (Oh) ..++--(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(a), or ¢(#); 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 mght to warn the student that the method has a somewhat restricted application, since the general form of the nth 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. Looe BA VEL at aria cthe thd Ata KY (1), z nN a” x we have mide ) (0x) = a Coie. isis det vat (2). * The foregoing proof is substantially that given by Homersham Cox, Camb. and Dub. Math, Journ., 1851. 584 INFINITESIMAL CALCULUS. [CH. XIV Now, whatever the value of 7, we have ” Sep lim my = 0 & va Biviae Peodsletelalaceter eteceanrerer (3), n= om Is since the successive fractions Pine eaten @ 1’ 9” cee Ee Wee diminish indefinitely in absolute value. Hence the expansion (5) of Art. 202 holds for all values of # As already pointed out, this merely amounts to a verification. rn deere le § FG) S008 Bee. ee (4), we have < PUNGLy = — cos (Ox + dnq)............ (5). The limiting value of the fraction #”/n! is zero, and the cosine lies always between +1. Hence the expansion (8) of Art. 202 holds for all values of a. The same reasoning applies in the case of sin @. 3°. If f(@=1+0)" (6), we find | (n) ot m (m — —n+1) ae . This may be regarded as the product of (1+ @”)™ into n factors of the type m—rt+l x ) r Atos 1+ 0x ze If 1>a2>0, the fraction x/(1+ 0x) lies between 0 and ga, and since the first factor in (8) tends with increasing r to the limiting value —1, it appears that by taking n great enough the value of the expression (7) can be made less than or (- 14™**) r any assignable magnitude. Hence, for 1 > #>0,we may write — (14+ #)™=1+4 mae+ ie ee igen ae eres (9) ad infinitum. wy X of 205-206 | | TAYLOR'S THEOREM. 585 We cannot make the same inference when 2 is negative, even if |a|<1. For if we put «=— 4, the fraction «,/(1 — 6x,) is less than 1 only if 6<(1—«,)/a,. And if 2,>4, we have no warrant for assuming that 6 lies below this value. Cauchy’s form of remainder (Art. 203 (23)) is now of service. We have, in place of (7), 1. 2...(n—1) (1 + 6a:)"—™ heh a a This is equal to ma (1+ 6x)" multiplied into n—1 factors of the type and, if |a#|<1, the fraction (a— 0x)/(1 + 6x) is easily seen to be less in absolute value than 2, whether x be positive or negative. Hence the limiting value of the remainder (10) is 0, so long as l>a>-1. Aor 1F F(x) = log a SEE Jaca wtp acces. (12), Be tnd = f° (Ba oe i = (as) st RE (13). The fen value of the ie factor is 0, and, if w be positive and +1, #/(1+0x) +1. Hence the limiting value of (13), for m = #, is zero, and the expansion (14) of Art. 202 is valid from « =0 to #=1, inclusively. Cf. Fig. 149. The above form of remainder does not enable us to determine the case of # negative, even when | «| <1. In Cauchy’s form, we have, in place of (13) Byes tn (Seale ) ear rom (ie Me eer tin)! If |x| <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=O(@), Yaa (1) intersect at the points for which «=a, a, &%, respectively. The function EF (x)= > (@) — @) 6 (2), which represents the difference of the ordinates of the two curves, will vanish for #= a, 2, %. Hence, on the usual assumptions as to the continuity of / (x) and F’ (a), the derived function /” (x) will, by the theorem of Art. 48, vanish for some value of « between a and a, say for «=a, and again for some value of « between a, and a, say for v=ay. Hence, by another application of the theorem referred to, if Ff” («) be continuous in the interval from a to a’, it will vanish for some value of # between a and a’, say for s=a . Hence if, by continuous modification of one of the curves (1), the three points #=4, #,, % be made to coalesce into the one point #=, the values of F(a), F’ (a), Ff” (@) will all be zero; 2.e. we shall have b(a)=V (a), & (a) = (x), $" (a)=p" (2) ...(3), simultaneously, for «= a. In other words, if two curves have contact of the second order at any point, the values of dy dy J, da’ da will at that poimt 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 at a given point. The equation of a circle with centre (, 7) and radius p is (aime Et ( Miers Me pb cca cowesign chr ge wae (5). If we differentiate this twice with respect to x, we find ly x—é+(y- n) = Or ae be Rites (6), 1+ (St) +y- » She Naas (7). In these results, y is regarded as a function of « determined by the equation (5). But if the circle have contact of the second order with the curve (4) at the point (a y), the values of y, dy/dx, and d?y/da* 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, dy/dx, d*y/dx’ refer to the curve (4). The equations then determine the circle uniquely, viz. we find that the coordinates of the centre are dy\*) dy dy\? {! u (2) ee di a (4) g= Dy 1 2= wo (8), dx’ da? cee) Bite 2 x and that the radius is p an REO deri let te Oa Hehe lat (9) ; da? ef. Art. 152. The above considerations may be extended, and we may say that if two curves intersect in n +1 consecutive points, or have contact ‘of the nth order,’ the values of d dy dy d”y D> das? dat? “* dat 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 «= a (say) P(YN=VO) PM=P(),.-. 6” (a)=p™ (a)...(10), and therefore, with F(x) defined by (2), F(a)=0,- F'(a)=0, | F(a) =0, ae ee It follows that, under the usual conditions, hrri eS poate OE FA F(at+h) Ghai (Ce (12), where l1>6@>0. Hence, if h be 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 Fig. 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 (1+) at the origin, whilst the curves numbered 2, 4, 6 do cross it. 207. Maxima and Minima. If ¢(«) be a function of « which with its first and second _ derivatives is finite and continuous for all values of the variable considered, we have d(ath)—d(a)=hd' (a) +59" (a+ Oh) ov Gl) where 1>@>0. By taking / sufficiently small, the second term on the right-hand can in general be made smaller in absolute value than the first, and ¢ (a +h) — ¢(q@) will then have the same sign as h¢’ (a), and will therefore change sign — with h. 206-207 ] TAYLOR'S THEOREM. 589 Now if $(a) is a maximum or a minimum value of ¢ (a), the difference (at h)— (a) must have the same sign for sufficiently 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 ¢ (a) = 0, so that (1) reduces to p(a+h)— (a) -* hb’ (a+Oh) ......... (2): When /h 1s sufficiently small, the sign of the right-hand will be that of ¢”(a). Hence if this be positive we shall have d(a+h)>d(a), whether h be positive or negative; 2. h(a) isa minimum. Similarly, if 6” (a) be negative we have a maximum. If d” (a) vanish simultaneously with ¢’ (a), it is necessary to continue the expansion in (1) further. To take at once the general case, if we have CU, op Ag =O. pad (h)— 0.24... (3), simultaneously, but nN GA ES Na ae Ge eee Ce ere (4), then d(a+h)—- o(a)= a PEC Ol) snc acess (5). If h be small enough, the sign is that of h*6™ (a). If n be odd, this changes sign with h, and we have neither a maximum nor a minimum. But if n be even, we have a maximum or minimum, according as ¢™ (a) is negative or positive. In words, $(x) is either a maximum or a minimum for a given value of # if the first derivative which does not vanish for this value of x be of even order, but not otherwise. And the function is a maximum or a minimum according as this derivative is negative or positive. Ex. MBit = COS aE COS. reek crs ta ss bees (6). This makes f' (x) =sinha—sing, ¢” (%)=cosha—cosa, ¢” («)=sinha+sine, (x) =cosh « + cosa. 590° INFINITESIMAL CALCULUS. [cH. XIV The first derivative which does not vanish for «=0 is $' (2). And since ¢'"(0) is positive, we infer that ¢ © is @ minimum value of ¢ (x). 208. Infinitesimal Geometry of Plane Curves. Let the tangent and normal at any poimt O 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 are OP, =s, say. If, for brevity, we use accents to denote differentiations with respect to s, we have, as in Art. 109, a =cosy, 4 =Ssinth ee. eee (1), and thence ao’ =—-snw.w, 2” =—cosp.v?—sinw.w”.... (2) y = cos yr. af’, y” =—siny.v?+cosw.”,... eee 5) and so on. Now, by Maclaurin’s Theorem, Ly” L=Xyt+ =m + Soe os ae > Yor y+ oi" + where the suffix is used to mark the values which the respective quantities assume for s=0. But, putting y~=0 in (1) and (2), we have D, = 1% Ly exe. epee: eee p ee 4), ‘_() Thesis 1 mt 1 dp Yo = Vs; Yo a Yo eo ee where 1/p has been written for dy/ds. Hence s Stes dp, L=S8— 6p** Ste Op 6p? ae ste (OD), where p and dp/ds refer to the origin. These formule are useful in various questions of ‘infini- tesimal geometry.’ y. 207-208] TAYLOR'S THEOREM. 591 Hx. 1. Thus the second formula in (5) shews that the devia- tion of the curve from the osculating circle at O is ultimately since dp/ds = 0 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 O ina point whose distance from Oisy+acoty. If we neglect terms of the second order in s, this ol Le 3° 2) 8 , s'dp p(1+325 i Hence the distance of the intersection from the centre of curva- ture at s is ultimately When p is a maximum or minimum we have (in general) dp/ds = 0, and the distance is of a higher order, the evolute having then a cusp at the point corresponding to 0. EXAMPLES. LIV. 24 9478 1 cosh x cos @ = 1 — + apm one ae ? 4 9348 910 sin BRT Oe ire aa a es i Qa% =D? 2. cosh a sin @ =a + 3 — Sosy sinh a cos x = ke Nadiad Parca aryiens Helge’ . 3 e*cosx=1+ SE Tal EE eat ap SEER TT Oey Win eh ae aa ik Qa® =D? qe 3g By PADI A Se fT ok RRS EA RESIS 592 INFINITESIMAL CALCULUS. (CH. XIV 4. secx=1 +55 oe 61 2 A 78 4) log sec w= 5 te 6. tanh x=«x—4a3+ PoP — ea’ + .. Oot Pat 7. cos’w= 1 —a¥+ FT — ey + 8. cos” a = eee = at — sin @\" _ oe. nm — 2) pos 9. (ey a1 Fare 5! x Cs Age 10. a ait oe) Le a? Ae sinha 3! 6! x a act 11. at ee tote 30 q] ° 12. tan (47 + x) =1+4 2+ 2x? + Ba? + FPatt..., 13. log tan (4a + @) = 20 + $0? + faP +0... 1 sin? 6 1.3sinté | 1.3.5 sin®6 14. log sec? 16 = = ase m log sect a9 = 5 9 +54 a 15. If D=d/dx, prove that Dex 84 cos (x sin a) = e* S84 cos (x sin a + na). Hence shew that Pe le excos cos (v sin a) = 1 +xcosa+ — 71 008 2a.+ 31 008 3a+. 16. Draw graphs of the functions 1 oe oe Si 3h ae respectively, and compare them with the graph of sin a, TAYLORS THEOREM. 593 17. Draw graphs of the functions a Uae | estes OR: Ga Te a respectively, and compare them with the graph of cos a, 18. SE @(a)=0, w(a)= prove that in general tin p (x) = is (a) : y (x) Ya) 19. Prove that, in the formula (17) of Art. 203, the limiting value of #, when f is indefinitely diminished, is in general 1/(m + 1). 20. Prove that when h is sufficiently small the error in Simpson’s formula (Art. 112 (8)) of approximate integration is equal to d4 - wh 4 nearly. 21. Prove that the mean value of a function ¢ (x) over the range extending from «=a—h oe x=athis $ (a) vue 7 (a) ne a (a) +. Shew that this aa short fe the arithmetic mean of the values at the extremities of the range by Rats ANS 13? (4) +375 ¢ (a) +.... 22. Shew that if, from a given curve, another curve be con- structed whose ordinate, for any value a of «, is the mean of the ordinates of the first curve over the range bounded by x=a +h, where / is a given small constant, the ordinate of the second curve exceeds that of the first by one-third the sagitta of the are whose extremities are x=ath. 23. Prove that if the expansions (4) of Art. 208 be carried to the order s‘, the results are — «594 INFINITESIMAL CALCULUS. [CH. XIV_ 24. If the values of x, y, referred to the tangent and normal at a point O of a curve, be developed in terms of y, the inclina- tion of the tangent to oa axis of #, the results are ds : ee, AG — Fp) ts - = oy 3 SE ye + 25. Prove that, with the same axes, the coordinates of the centre of curvature are d d? f=-b Vb apt ts dp dp 26. If O, P be two adjacent points on a curve, and PQ be drawn perpendicular to the chord OP, to meet the normal at O in Q, then ultimately OQ = 2p. 27. If equal small lengths OP, O7' be measured along the are of a curve, and along the tangent at the point O, and if # be the limiting position of the intersection of PZ’ produced with the normal at O, then OR= 3p. 28. The perpendicular to a chord OP at its middle point meets the normal at O a distance from the centre of curvature ultimately equal to $sdp/ds, where s = OP. 29. The tangents at two adjacent points P, Q of a curve meet in 7’, and V is the middle point of the chord PQ. Prove that TV makes with the normal to the curve an angle tan! (3dp/ds). 30. Prove that if PQ be a small arc (s) of a curve, the are exceeds the chord by yeep", and the sum of the tangents at P and @ exceeds the arc by + L 3/p?, 31. Prove that the form of a curve near a cusp is given by ia. approximately, where a = 2dp/dw. 32. Prove that the form of a curve near a point of inflextan is given by Y=%§ ee a, approximately, where c is the curvature. 209] TAYLORS THEOREM. 595 83. If P be any point on a curve, the form of the evolute of the part near P, referred to the centre of curvature at P as origin, is in general given by ay = x’, where a= 2dp/dy. 34. Prove that if P be a point of maximum or minimum curvature, the form of the evolute is ay” = 2, approximately, where a = 2d?p/dy’. 209. Functions of several Independent Variables. Partial Derivatives of Higher Orders. If uw be a function of two or more independent variables @, Y, ..., the partial derivatives ou du Teas Rieter (1) will themselves in general be functions of 2, y, ..., and be susceptible of differentiation with respect to these several variables. Thus, if Th DA ad) eras aed ass see's os (2), we can form the second derivatives 3 (au), 9H), 8H), 2 aw sl . = (Ga) 7B OF ‘ vs, : or, as they are usually written, Ou 8 ©60*u Ou 8 8©0?u ABC PORE shih cer ep ae ( 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. 3). _ The first derivatives of ¢ (a, y) are sometimes denoted by DalOsy amy (Oey aden es ves torte: (4), 596 INFINITESIMAL CALCULUS. [CH. XIV and the second derivatives (3) by haa (%, Y), Pya(@s Y), Pay(@, ¥)> Py (#, y) --(5). These are often abbreviated into De, — Py: oss nccerpaae eines (6), and Drews Dyn, - Pays Pyy ee (7), respectively. deel Lt b= Any” - onc nene eee (8), h a mAa™-lyr ibis nAgmy-1 (9) we have tee y", ye gfe er eee . ou = mt — 2, Pu as Mp N—2 Bat = Me (m—1) Ax Y; aa ht ee : ou als m-1, 2-1 _ eu ay dat mn Aa tht ety (10). Ex. 2. TE e-aten— 2 (11), 2 02 ay Oz Gx ele eee Seed Bae eee Tee 2 we find an ery? dy wry (12), ee 2axy ez alyr—-x ie Oz es — 2any (13), ~ (a2 + y?)?? you (a? + y?)? axrdy’ ~ (a + 9)? +y’) 210. Proof of the Commutative Property. Let U = (ah, Y).s.0c.scenee eae ee (1), and let us suppose that the functions , oe du Bu Oe ” Ox’ Oy’ Oydxu’ dxdy are continuous over a finite range of the variables, including the values considered. We proceed to shew that, under these _ conditions, ou _ ou : oyox dxdy To this end, we consider the fraction Path, ytk)—o(eth, y-b(@, y+h)+o@y x (h, hb) hie Th he A oe th ee ee 210] TAYLOR'S THEOREM. 597 in which #, y are regarded as fixed, whilst h, k will (finally) be made infinitely small. Let us write, for a moment, EF (a) = (x, y+ hk) — h(a, Y) cecreceseoee (3): By the mean-value theorem of Art. 56 (8), we have F(e+h)—F (a) =hE"’ (a+ Ah)... (6), or, in full, {¢(ath, y+k)—d(ath, y}—-{o(@, yt+k)-$@, yh where 1>0,>0, the value of y not being varied in this process. Hence vy (h, py ee ee Oh, ” (8). If we now write FOS OR CPA A) eee cence ee (9), we have, by a second application of the theorem referred to, Sly th) —f iy) Hh! (y + Ook)... ccceeeee (10), or Hence V(h, k)=hye(at+ Oh, y+ Ook).......0006: (12), where 6,, 0, lie between 0 and 1. By a similar process we could shew that V(h, kh) = hey (e+ Oyh, ytO,k)....002.. (18), where @,’, 8,’ also lie between 0 and 1. | These results are exact, provided +h, y+ le within the range of the variables for which the conditions above postulated hold. If we now diminish h and & indefinitely, it follows from the comparison of (12) and (13), and from the continuity of the derivatives, that yx (x, y) = Dry (a, Yy) ata aonith onions: (14), as was to be proved*. * This proof appears to be due to Ossian Bonnet. An alternative proof is indicated in Art. 211. 598 INFINITESIMAL CALCULUS. [CH, XIV It appears ao (4) that littzey x (8, 8) =; lim, oo path, y) — > (2, Y) any k lim,_, h a (x, ¥ + hk) — hy (x, 2 = Pal ¥ s tO) (15). Hence lim, limjoy x (4, &) = Gye (Gee Similarly, we find limo limpeo x (A, &) = Diy (0,4) eee (17). If, then, we could assume that the limiting value of the fraction (4), when A and & 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 V—kh af (A, k) = h? + h2 + je? we have lim;—» limp f (h, k) = qT; lim;,—) lim;_, ST (h, k) =-+ Le Ex. The condition that Mdau + Ndy ..ii.cte (18) should be an exact differential (Art. 174) is ou aN y mt (19). For if the expression (18) be equal to du, we have dw ou 1 oa: 9 = ay eee cee cer cceccsene (20), and therefore each of the partial derivatives in (19) 4 is equal tc O'u/dy dx or Gu/dxdy. Conversely, we can shew that, if the condition (19) hold, (18) will be an exact differential. Let v denote the function {M/dz, obtained by integrating as if ¥ were constant. We have, then, av ae Moo. ss eee eee a ee (21), 210-211] TAYLOR'S THEOREM. 599 oN ou and therefore aatica By. = aedy ’ 0 Ov Bee —_—)= ay or ms (wv = Ope ea ae. (22) This shews (Art. 56) that the function NV —dv/dy is constant so far as x is concerned. Denoting its value by /’ (y), we have dv ; : Ne a BCD Nt eR eek Pee ee (23). Hence, if we write Mee ET CY Pe wee Die ass g feds xe os (24), we have, by (21) and (23), ? dw dw whet PRN eee ih ta Lees cee 2 ae > by Ne Seah (25), and therefore Moler+- Ny = 0b cing. 28 ses 0e apatsy (26)*. It follows from the above theorem that in the case of a function of any number of independent variables a, y, z, ... the operations Li bela CUP OYR AOS or, as we may denote them for shortness, b DR Dyed Sas are in general commutative, 7.e. the result of any number of them is independent of the order in which they are performed. For example, D,D,Dgte= D, (D,Du) = D,(D,D,) v= D,D, (Du) = D,D,D,u = ete. 211. Extension of Taylor’s Theorem. Let $(2, y) be a function of w and y which, with its de- rivatives up to a certain order, is continuous for all values of the variables considered. It may be required to find the expansion of b (dew, bitte) oan oor, eae. oot (1) * Cf. Murray, Differential Equations (1897), p. 197. 600 INFINITESIMAL CALCULUS. [CH. XIV in ascending powers of h and k. We shall in the first place | give a direct investigation of the expansion as far as the terms of the second degree in fh and &. First expanding in powers of h, we have, by Taylor’s Theorem, p(ath, b+k)=¢(a,b+k)+hd,(a,b+k) + th’dox (a, b+kh)+...... (2). Again, by the same theorem, (a, b+ k) =$ (a, b) + hy (a, b) + $day (a, b) + ha (a,b+k) = bz (a, b) + kbye (a, b) +... (3). hax (a,b + k) = hae (a, 6) +... Substituting in (2), we find d(ath, b+k)=¢ (a,b) + {hdz (a, b) + kd, (a, b)} +4 [hpen (a, b) + Zhkpya (a, -b) + k*hyy (a, b)} +... (4). If we regard the forms of the several ‘remainders’ (Art. 203) in the preliminary expansions, it appears that the re- mainder in (4) will be of the form 1 5] { Rh? + 38hk + 8Thk + UR}... (5), where Rf, S, 7, U are functions of a, b, h, k which remain finite when A, & 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 ¢(#, 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 proof of (4) it was not necessary to assume that bye (ty BY = Bey (0,0). ie (6). If we had begun by expanding (1) in powers of & (instead of h) we should have arrived at a result similar to (4), but with dy (a, 6) in place of d,y,(a, 6). 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 tea ay ap rp na) + 1 | fy? 2 +4 @ aat alles a awe (7), where, on the right-hand, ¢ et for d(a#, y). A more compact form is Path y+tk)=(a,y) + (hbe + kby) +4 (W baw t+ 2hkhey + kh’ dyy) + 0. oe (8). Again, if w be any function of the independent variables x, y, and if, as in Art. 60, dw denote the increment of w due to given increments 6x, dy of these variables, the formula is equivalent to Ou Ou O7u ; "Te Wigs 2 (Ou) + 2 BE Een, Fe An independent investigation, giving the general term of the expansion (7), is as follows. We write h= at, k = Bt, and Fitj=¢o(ath, y+ k)=b(e+at, y+ Bt) ...(10). Regarded as a function of ¢, this can be expanded by Maclaurin’s theorem, and the general term is Now if we put for a moment «+ at =u, y+ Bt=v, we have Op _dpdu_ 0H Aap _Apdv _ AP 12) dc Oudx Ou’ dy ovoy dv where ¢ is written for d(u, v). Hence Fr (4) = 080m 4 2680 _ ee +a ei, 2) ood TARTS Eerie 3, (13). * 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 w and v; hence, by a repetition of the argument, Fr’ =(15 +85) (2+ 6a 0) a (a2 +85.) b (16, 0) ee (14), and, generally, 0 O\" F (t)= (eae b ieee (15), where the operator admits of expansion by the Binomial Theorem, in virtue of the commutative property of the operators 0/da and d/dy. Since t only occurs in the com- binations #+ at, y+ Bt, it is evidently immaterial in (15) whether we put t=0 before or after the differentiations indicated on the right-hand side. The general term of our expansion is therefore ne On mae Ba 5) bay=a (bathe) ey) = (am SF nh-ifp 2? pnaad > rte OE...) da” 0x" “Oy 1.2 Ox" *0y? Staats (16), where ¢ is now written for ¢ (a, ¥). Ex. To prove that if (a, y) be a homogeneous function of x, y, of degree m, we have ah, + Yby= Mp: stone ee (17). Phi + 2Y Pay + Y’Pyy =m (m— 1) d......... (18). The general definition of a homogeneous function of degree m is that if w and y be altered in any ratio p, the function is altered in the ratio pw”, or db (ux, py) =p pb (yy). eee (19). In this equality, let us put »=1+¢. Since p(xtat, yt yt) = (%, y) +t (ey + Y>y) +40 (Wie + 2ey hay + Ypyy) +--+ by (8), and (1+ ¢)" (a, y)= (1 + mt eas y + ) p, by the Binomial Theorem, the results (18) and (19) will follow, 211-212] TAYLOR'S THEOREM. 603 on equating coefficients of ¢ and ¢. More generally, equating coefficients of ¢", and making use of (16), we find o” o” nl(n— on Br ogee ee ac i ¥ a : " aE oe ig =m (m—1)(m—2)...(m—n+1) ¢...... (20). This is ‘Euler’s Theorem of Homogeneous Functions,’ for the case of two independent variables. The extension to three or more independent variables will be obvious. 212. Maxima and Minima of a Function of Two Variables. Geometrical Interpretation. We may utilize the generalized form of Taylor’s Theorem to carry a step further the discussion (see Art. 59) of the maxima and minima of a function (w) of two independent variables (x, ¥). It appears from Art. 211 (9) that when dz, dy are continually diminished in absolute value, preserving any given ratio to one another, the sign of dw is ultimately that of Unless du/da and du/dy both vanish, the sign of (1) is reversed by reversing the signs of 6a and dy. Hence for some varia- tions du will be positive, and for others negative. In other words, u cannot be a maximum or minimum unless we have anaiy Nath On a simultaneously. Let us now suppose the conditions (2) to be fulfilled. We have, then, bu 3 (Oa oP ee ay Baby +o (yy +. setts): When de and dy are sufficiently small, the sign of du will be that of the terms written. Now it is known from Algebra that the sign of a homogeneous quadratic function A+ 2HEn + Br? ...... i Amish (4) is invariable, if (and only if) A Bad is ds Geeta (5), * It is assumed in the investigation of Art. 211 that these derivatives are - continuous and therefore finite. That is, we exclude ab 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 6u will have the same sign for all values of |é62| and |8y| not exceeding certain limits, provided mo dx? Oy?” \dxdy and that the sign will then be that of 0u/da and 0*u/dy? And wu will be a maximum or minimum according as this sign is negative or positive. ty Oe (Ou dx? Oy? ~ \Oxdy then for some values of the ratio dy/é# the increment of u will be positive, for others negative, and the value of 4, though ‘stationary ’ (cf. Art. 50) is neither a maximum nor a minimum, If If Ou Oru ( Ou ) ox oy? \dady the terms which appear on the right-hand side of (3) are equal to + the square of a linear function of dx and dy, and therefore vanish for a particular value of the ratio dy/dz. Since du 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 0°u/0x*, @u/dwdy, o?u/dy? 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 #0, 2.0.40 (9) 0x ay simultaneously. Since these equations ensure that dz shall be of the second order in dx, dy, 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 tangent 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 6z=0, and therefore from (3), if we put dy =méa, and finally make x vanish, the directions of the tangent lines at P to the curve of intersection are determined by Pz 9 Oz Oe Gai Gcoy . dy? This quadratic in m will have imaginary roots if az az Oz \2 we) ay 7 (ea) oe oe (1 yi the surface then, in the immediate neighbourhood of P, will le 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 6°z/dx? and @z/dy? are negative or positive, z.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. a2 oz az \2 But i See tell (reel Deer ut if aa? dy? < Ga (12), 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. Oz de ae \? If ee & ) Dae et ay (13), 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. 0ac" The conditions (9) are satisfied by «=0, y=0, and also by x=a, y=0. The former solution satisfies the inequality (11), Ex. 1. Let gaa — San? — 4ay?+ CO ......ceeeeeee: (14). : Oz dz —= —2 sorties = OGY. © — to) or) D. Table of the Circular Functions at Intervals of One-T'wentieth of the Quadrant. 6/47 | sin 0 cosec 0 sec 0 cos 0 eee [ee ee, EE n,n | 0 0 00 1-000 1:000 | 1:00 05 7078 =| 12°745 1003 “997 ‘95 10 156 6392 1012 988 ‘90 15 233 4°284 1028 "972 ‘85 ‘20 309 3°236 1051 951 ‘80 ‘25 383 2°613 1082 "924 ‘75 30 ‘454 2°203 1122 ‘891 ‘70 "35 522 1914 1173 853 ‘65 "40 588 1°701 1236 ‘809 ‘60 "45 649 1540 1°315 "760 ‘55 DOs *107 1:414 1°414 “707 ‘50 cosec 0 sin 0 6/33 cos 0 sec 0 618 APPENDIX. E. Table of the Exponential and Hyperbolic Func- tions of Numbers from 0 to 2:5, at Intervals of -1. x ex e-& cosh « sinh « tanh x 0 1:000 1:000 1:000 0 0 ‘1 1°105 "905 1:005 °100 "100 9 1:221 *819 1:020 201 "197 3 1°350 ‘741 1:045 *305 "291 4 1°492 670 1:081 “411 380 5 1°649 607 1128 ‘B21 "462 6 S22 549 1:185 637 ae 7 2014 ‘497 1:255 "759 604 8 2226 "449 1°337 "888 664 ‘9 2°460 *407 1:4338 1:027 ‘716 1:0 2°718 368 1°543 1175 7162. algae 3004 ‘S00 1°669 1:336 “801 1:2 3320 301 1°811 1:509 "834 1:3 3669 Fle 1971 1:698 "862 1-4 4°055 AT QTL 1:904 885 1:5 4°482, 223 F352, 2°129 905 1-6 4°953 202 2577 2°376 "922 17 5474 "183 2°828 2646 "935 18 6:050 "165 3°107 2°942 ‘947 1:9 6°686 *150 3°418 3°268 "956 20 7°389 "135 3°762 3627 964 pra 8166 "122 4°144 4:022 ‘970 a 9°025 ‘111 4568 4°457 ‘976 23 9°974 ‘100 5:037 4-937 "980 2-4 11:023 091 5557 5°466 "984 2:5 "987 12°182 082 6°132 6:050 Fr. Table of Logarithms to Base e. 0 ‘1 o ie ‘4 ‘5 O | -095| +182] -262] -3836] -405 693 | 742] -788| -833| °875| -916 1-099 | 1-131 | 1163 | 1-194 | 1-224 | 1-253 1-386 | 1-411 | 1-435 | 1-459 | 1-482 | 1-504 1-629 | 1-649 | 1:668 | 1°686 | 1-705 1-792 | 1:808 | 1825 | 1:841 | 1-856 | 1:872 1-946 | 1-960 | 1-974 | 1:988 | 2-001 | 2-015 | 2:079 | 2-092 | 2-104 | 2-116 | 2-128 | 2140 | 2°197 | 2-208 | 2-219 | 2-230 | 2°241 | 2-251 SOSHOHNOOPWNe — on) S we} Ce Oe aa a ee INDEX. [The numerals refer to the pages.] Acceleration, 68 angular, 69 tangential and normal, 397 Accidental convergence, 10 Algebraic functions, continuity of, 26 Amsler’s planimetre, 250 Anchor-ring, surface of, 272 volume of, 259 . Approximate integration, 275 Arbitrary constants, 166 Are of a curve, formule 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, formule 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, 386 Calculation of 7, 273, 554 Cardioid, 356, 358, 369, 370, 383 Cartesian ovals, 388 Cassini, ovals of, 389 Catenary, 342 are 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, 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 Conjugate point, 333 Contour lines, 98, 605 Convergence of infinite series, 4 essential and accidental, 10 uniform, 543 Convergence of a definite integral, of power-series, 543 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, 395 radius of, 395, 397, 400, 402, 406 Cusp, 336 Cusp-locus, 420 Cusps, circle of, 444 Cycloid, 347 are of, 348 area of, 350 curvature of, 398, evolute of, 423 442, 444 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 Differential coefficients, 64, 95, 144 Differential Equations, 456 exact, 466 homogeneous, 469 integration of, by series, 559 linear, 473, 476 of first order and first sme 462 of first order and higher degree, 484 of second order, 490 simultaneous, 529 Differentials, 132, 138 Differentiation, 69, 71 of a sum, product, etc., 73, 74, 76 of a function of a function, 80, 139 of a definite integral, 219 of implicit functions, 98, 139, 607 quotient, INDEX. 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 Klasticity of volume, 69 Elimination of arbitrary constants, 456 Ellipse, area of, 243, 248 perimeter of, 267, 548 curvature of, 398, 402, 404 evolute of, 499, Ellipsoid, volume of, 260 radii of gyration of, 317, 321, 328 of revolution, surface of, 273 Elliptic disk, radii of gyration of, 316, 327 Elliptic segment, mass-centre of, 326 Elliptic-harmonic motion, 346 Elliptic integrals, 268 Envelopes, 413, 415 contact-property of, 418 Epicyclics, 359 as roulettes, 448 Epicycloid, 350 are 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 Kvolute, 421 are of, 425 Exact differential, condition for, 598 Exact differential equations, 466 Expansion, coefficient of, 69. Expansions by means of differential equations, 562 by Maclaurin’s theorem, 573, 583 Exponential function, 35, 560 graph of, 38 Flexure of beams, 299, 323 Function, definition of, 12 ' graphical representation of, 13 INDEX. Functions, algebraic and transcen- dental, 26 implicit, 97 inverse, 44, 85 Geometrical 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, 552 Guldin, see ‘ Pappus’ Gyration, Radius of, 313 Hart’s linkage, 381 Hemisphere, mass-centre of solid, 304 mass-centre of surface of, 301 Homogeneous differential equations, 469 Homogeneous functions, theorem, 603 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 ‘ Epitrochoid’ Euler’s Implicit functions, 97 differentiation of, 98, 140, 608 Indicator diagram, 211, 247 Inertia, moment of, 313 Infinite series, 4, 9 addition and multiplication of, 11, 35 differentiation and of, 549, 556 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, 165 by parts, 187 integration 621 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 Limacon, 357, 368, 383 Limit, upper and lower, of an as- semblage, 3, 51 Limiting values, 8, 54, 55, 58 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, 552, 575, 585 Maclaurin’s Theorem, 573, 576, 582 Magnetic curves, 389 Mass-centre, 291 of an arc, 293 of an area, 293, 295 of a surface of revolution, 301 of a solid, 303, 306 Maxima and minima, 106, 160, 588 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 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, 333 Order of a differential equation, Orthogonal projection, 324 Orthogonal trajectories, 478 Pappus, theorems of, 307 Parabola, are 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, 595 Partial differentiation, commuta- tive property of, 596, 600 Partial Fractions, 172, 194, 195, 197 Particular integral of a linear dif- ferential equation, 474, 502 Peaucellier’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 differentiation 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 Radius of curvature, see ‘ Curva- ture’ INDEX. Radius of gyration, 313 Rational fractions, graphs of, 28 integration of, 171, 193, 195, 197° Rational integral functions, 26 Reciprocal polars, 384 Reciprocal spiral, 368 Rectification of curves, 264, 266, 268 of evolutes, 425 Reduction, formule of, 189, 190, 229 Remainder in Taylor’s and Mac- laurin’s Theorems, 581, 582 Ring, radius of gyration of a, 317 surface of a, 272, 308 volume of a, 259, 308 Rolle’s theorem, 105 Roots of algebraic equations, sepa- ration of, 105 Roulettes, 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, 575, 605 Sin-! w, expansion of, 556 Singular solutions, 487 Rpheres retina of gyration of, 317, 2 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, INDEX. Spiral, equiangular, 366 of Archimedes, 354, 367 reciprocal, 368 Stationary point, 396 Stationary tangent, 157, 396 eee values of functions, 110, 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, 582 expansions by, 574, 583 extension of, 599 Teeth of wheels, 450 Tetrahedron, volume of a, 257 Theory of equations, 104, 161 Time-integral, 210 623 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 Agnesi, 337 Work, 210 CAMBRIDGE: PRINTED BY J. & C. F, CLAY, AT THE UNIVERSITY PRESS, »* ky Spans ny 3 0112 017232841 a rn :