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'. ve _ bg 2 ee Md ‘ e MATHEMATICS LIBRARY Return this book on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. University of Illinois Library L161— 0-1096 Digitized by the Internet Archive in 2022 with funding from . University of Illinois Urbana-Champaign https://archive.org/details/firstprincipleso0O0bezo FIRST PRINCIPLES OF THE DIFFERENTIAL AND INTEGRAL CAE OU EL Us, | OR THE DOCTRINE OF FLUXIONS, INTENDED AS AN INTRODUCTION TO THE PHYSICO-MATHEMATICAL SCIENCES ; TAKEN CHIEFLY FROM THE MATHEMATICS OF BEZOUT, AND TRANSLATED FROM THE FRENCH FOR THE USE OF THE STUDENTS OF THE UNIVERSITY AT ‘CAMBRIDGE, NEW ENGLAND. SECOND EDITION, BOSTON: PUBLISHED BY HILLIARD, GRAY, & CO. 1836. , o ' vee “ x ‘ ¥ § 7 { 4 ry rea ee 2 ee 1 / . , , Y [ es Li - 4. < \t f GALE T \ ¥/ ; ‘a i. eo" - 53 t p et te ie it \ 1 4 sa, VR atharete MAY HS) eh hy ; td : fy" ae " : : 4 aA a 7 ‘ i et x -* » v i ACS » Sana te ee halt 7 . > ‘i by ' ; b, i & wa 4 5 i x ; i 4 t ’ * 4, ’ ‘ a rf ooy my aoe # *- ng | ) x ‘. 1y } ' ; ras . " f vi a ‘a & ’ i ? _ \ in’ P i ‘ a 4 M4 y oa ‘ ‘ bon 4 9 ' , a | @ i , ; f g ‘ oa | : " BN} i : r \ , : 7 et ‘ y , * wy ’ > j Bere)" \ ae eh f ’ . ; ; Bi . Entered according to act of Congress in the year ono thousand: eight hundred and thirty-six, cay by Hiturarp, Gray, & Co, | s « ' in the Cle office of. the. District. Court of the District of Massachusetts. > & si Tia i jy es ned | ba ee CAMBRIDGE PRESS: ea METCALF, TORRY, AND BALLOU. it. sys ‘ ? €\ oF by ‘ ae ts ie wi é vey ae e. = ADVERTISEMENT TO THE FIRST EDITION. \ THe following treatise, except the introduction and notes, is a translation of the Principes de Calcul qui servent d’ Introduction aux Sciences Physico-Mathématiques of Bézout. It was selected on account of the plain and perspicuous manner for which the author is so well known, as also on account of its brevity and adaptation in other respects to the wants of those who have but little time to devote to such studies. ‘The easier and more impor- tant parts are distinguished from those which are more difficult or of less frequent use, by béing printed in a larger character. In the Introduction, taken from Carnot’s Reflexions sur la Meta- physique du Calcul Infinitesimal, a few examples are given to show the truth of-the infinitesimal method, independently of its technical form. Moreover in the 4th of the notes, subjoined at the end, some account is given from the same work, of the methods previously in use, analogous to the Infinitesimal Analysis. The other notes are intended to supply the deficiencies of La- croix’s Algebra (Cambridge Translation), considered as a prepara- tory work. Since this treatise was announced, the compiler of the Cambridge. Mathematics has been obliged, on account of absence from the country and infirmity of sight, to resign his work into other hands. This circumstance is mentioned to account for the delay attending the publication, as well as the occasional want of conformity to other parts of the course in the mode of rendering certain words and phrases which a revision of the translation, had it been prac- ticable, would have easily remedied. Cambridge, July, 1824. ef oe 7 Ea Py 3 i a LApon! hes oy CFORN, PEON LS INTRODUCTION. Preliminary Principles ELEMENTS OF THE DIFFERENTIAL CALCULUS. Of Second, Third, &c. Differentials Of the Differentials of Sines, Cosines, &c. Of Logarithmic Differentials Of the Differentials of Exponential Omics Application of the preceding Rules Application to the Subtangents, T’angents, Sein dbntal %e of Curved Lines Of Multiple points ’ Of the visible and invisible Points of ride Observations on Maxima and Minima Of Cusps of different Species, and of the aifitennt Sorts of Gone tact of the Branches of the same Curve : On the Radii of Curvature and the Development or Prolite ELEMENTS OF THE INTEGRAL CALCULUS. Explanations ; * Of Differentials with a ares evades ss hates an pif ical Integral; and first, of simple differentials : Of Complex Differentials whose Integration depends on the fundamental Rule Of Binomial Differentials which may - Piedtatea areraically Application of the preceding Rules to the Quadrature of Curves Application to the rectification of Curved Lines Application to Curved Surfaces Application to the Measure of Solidity On the Integration of Quantities containing Sines id Cavan On the Mode of Integrating by Approximation and some Uses of that Method ; Uses of the preceding Approximations in the Integration of Different Quantities 85 91 93 95 104. 106 119 vi Contents. i. \ By the Table of Increasing Latitudes or Meridional Parts By Reduced Maps or Mercator’s Chart : On the Manner of reducing when it is possible, the Tntegeation of a proposed Differential, to that of a known Differential, and distinguishing in what Cases this may be done On Rational Fractions 4 . On certain ‘Transformations by ach ihe ine rain may ms facilitated | On the Integration of Evponehta (Quidutitias On the Integration of Quantities with two or more Martnhles On differential Equations net Sh On Differential Equations of the second, third, and nite braided NOTES. 1. Nature and Construction of a Curve passing through certain given Points ‘ ’ 2. General Vemonstration of the ‘Binaibial eels 3. On the Method of Indeterminate Coefficients : A. On the Methods which preceded, and in some measure sup- plied the place of the Infinitesimal Analysis. y Ist. On the Method of Exhaustions 2d. On the Method of Indivisibles ; 3d. On the method of Indeterminate Quantities Ath Of Prime and Ultimate Ratios INTRODUCTION. Tue Infinitesimal Analysis, as presented in the following Treatise, proposes to ascertain the relation of definite, assignable quantities, by comparing them with quantities which are here called mpfinitely small. But by infinitely small quantities is meant quantities which may be made as small as we please, without alter- ing the value of those with which they are compared, and whose ratio ws sought. ‘The first idea of this calculus was probably sug- gested by the difficulties which are often met with in endeavouring to express by equations the different conditions of a problem, and in resolving these equations when formed. When the exact solution of a problem is too difficult, it is natural to endeavour to approximate as nearly as possible to an accurate solution, by neg- lecting those quantities which embarrass the combinations, if it is seen that they are so small, that the neglect of them will not materially affect the result. ‘Thus, for example, it being. found very difficult to discover directly the properties of curves, mathe- maticians would have recourse to the expedient of considering them as polygons of a great number of sides. For, if a regular polygon be inscribed in a circle, it is manifest, that these two figures, although they can never coincide and become the same, approach each other the more nearly in proportion as the number of the sides of the polygon increases. Whence it follows, that, by supposing the number of sides very great indeed, we may, without any very sensible error, attribute to the circle the proper- ties which are found to belong to the inscribed polygon. And if, in the course of a calculation, we -should find a circumstance in which the process would be much simplified by neglecting one of these exceedingly small sides, when compared with a radius, for example, we might evidently do it without inconvenience, since i Oa 2 . Introduction. the error which would result would be so extremely small, that it need not be noticed. Let it be proposed, for example, to draw a tangent to the point M of the curve AMB (fig. 1) considered as part of the circum- ference of a circle. ' Let Q be the centre, and AP @ the axis; call the absciss AP, x, and the corresponding ordinate PM, y, and let T'P be the subtangent sought. | To find this, we consider the circle as a polygon of a very great number of sides, and Mm as one of these sides; we produce Mm until it meets the axis, and it is evidently the tangent in question, since it does not penetrate the polygon. We let fall upon AQ the perpendicular m p, and call the radius of the circle, a. The similar triangles Mrm, TPM, give M 7: 7m ok PAPA eet rm y Now, since the equation of the curve for the point M is y? = 2ax—x?, (Trig. 101,) it will be, for the point m, (y+rm)? =2a(¢+ Mr)—(c#+Mr)?, developing, we have y? 4 Qy.rmtrm= 2ax+2a _Mr—a2 —22 Mr Mr, from which, if we subtract the equation for the point M, we have A OMe STR map Tewari ost pan age Whence, by reducing, | | Mr 2y+rm rm 2a—2c%—Mr Substituting for ae its value found above, and multiplying by y, we have FTP eae (Ryrm) _ Q2a—2r21—Mr If now r m and Mr were known, we should have the value of TP sought ; they are, however, very small, since they are each less than Mm, which is itself, by supposition, very small. ‘They are, moreover, perfectly arbitrary, since there is nothing in the supposition to limit their magnitude, and they may be rendered indefinitely small without affecting the lines TP and PM, with “ Introduction. Bs which they are compared. We may: therefore neglect, without any material error, these quantities so small, compared with the quantities 2 y and 2 a— 2, to which they are joined; and the raat & bn equation is thus reduced to T'P = —% a—tz This value, thus found by neglecting the very small quantities mr and Mr, is not only nearly accurate, as might be supposed, but absolutely exact, as is thus shown. ‘The similar triangles QPM, MPT, give EM a1 Teg Mages aad log whence pTa=t = _¥ QP a— x \ The above result is thus obtained by a balance of errors; the error made in supposing the circumference to be a polygon being compensated by neglecting the small quantities mr and Mr to obtain the final value: and the omission of these’ quantities is not only allowable, but is absolutely necessary to fulfil the conditions of the problem.+ Mey As a second example, we suppose that it is required to find the surface of a given circle. We here also may consider the curve as a regular polygon of a’ great number of sides. The area of a regular polygon is equal to the product of its perimeter, by half of the perpendicular let fall from the centre upon one of the sides. Therefore the circle, considered as a polygon of a great number of sides, is equal to the product of its circumference by half the radius; a proposition which is no less exact than the result found above. In the examples just given, it is seen that great advantage is obtained by employing quantities which are very small com- pared with the principal quantities in question. The same _prin- ciple once admitted may be very generally applied; all other curves may, as well as the circle, be considered as polygons of a , as above. + If it be asked, how we may be sure in similar cases, that the compensation of errors has taken place, it may be observed, that the error, if any exist, depends upon the arbitrary quantities 7 m and Mr, and may be made as small as we please by diminishing these quan- tities ; but as these disappear in the final result, the error disappears with them, and leaves the result perfectly accurate. . = o = 7 a a ae i ee bt "os 4 Introduction. great number of sides. All surfaces may be considered as divided into a multitude of zones, all bodies into corpuscles; in short, all quantities to be decomposed into small parts of the same kind as themselves. Hence will arise many new relations and combina- tions, and it is easy to judge, from the examples given above, of the resources furnished to the calculus by the introduction of these elementary quantities. The advantage obtained is even much greater than we should at first expect, since, in many cases, as we have seen above, the method employed is not merely an approximation, but leads to perfectly accurate results. It becomes therefore an interesting object, to ascertain when this is the case, to extend the application of the principle, and to reduce the methods employed to a strict and regular system. Such is the object of the infinitesimal analysis. We shall now give some problems, tending to throw light on the mode of reasoning employed in this calculus. 1. To draw a tangent to the common cycloid. Let AEB (fig. 54) be a common cycloid, of which the gene- rating circle is Ep q F.. The principal property of this cycloid is, that for any point m, the portion m p of the ordinate, comprised between the curve and the circumference of the generating circle, is equal to the arc E p of. that circumference. Draw to the point p of this circumference a tangent p T, and let it be required to find the point Z’ where this tangent is inter- sected by m T’, the tangent of the cycloid. In order to this, we draw a new ordinate n q infinitely near to the first m p, and through m draw m r parallel to the little arc p q, which, as well as m , we consider as a straight line. It is evident that the two triangles m n r, ‘I'm p, will be similar, and we shall consequently have mr:nr:: Tf p:myp. But since, by the properties of the cycloid, we have Eg =n q and Ep =m p, we shall have, by subtracting the second of these equations from the other, Eg—Ep=nq—mp, orpq=nr, ormr=mnr. Wherefore, by reason of the proportion found above, we have T'p = m p, or Tp = E p, that is, the subtangent \ T pis always equal to the corresponding arc Ep. This equation Ss = Ferry Introduction. 5 is disengaged, by the disappearance of mrand nr, from every consideration of infinite or arbitrary ; whence the proposition is rigorously and necessarily exact. 2. To show that in motion uniformly accelerated, the spaces de- scribed are as the squares of the times, reckoning from the beginning of the motion. In this motion, the accelerating force acts constantly in the same manner, wherefore, if we suppose g to be the velocity communi- “cated in each unit of time, the successive velocities will evidently ~ form the series 2°, 2 2,3 g, 4 2, &c.; so that after a number of units of time marked by 7, the velocity acquired will be as many times g as there are units in ¢, that 1s, calling the velocity u, u will equal g t. Since the velocities g, 2 g, &c. are each nothing but the space which the moving body describes in the corresponding interval of time, the total space described during the time ¢ will be the sum of the terms of this arithmetical progression. But the sum of the terms of such a progression is found by multiplying the sum of the first and last terms by half the number of terms. Whence, this sum will be, (substituting. « for its value g ¢, which is the last term,) (g +4) xX i: Whence, if we represent the space by s, we have \ Sor = (g +- 1) x a : Let us now conceive that the accelerating force acts without interruption, or, which is the same, that the time is divided into an infinite number of infinitely small parts called instants, and that, at the beginning or end of each instant, the accelerating force gives a new impulse to the moving body. We conceive, more- over, that it acts by infinitely small degrees. ‘Then g, being infi- nitely small compared with w, which is the velocity acquired in the infinite number of instants indicated by ¢, we must, in the equations =(g¢+u Zo omit 2, and we shall have s = af q 5 Q § D If we call the velocity acquired at the end of a second, p, the velocity acquired after a number, ¢, of seconds, will be ¢ p. Whence u== pt. The equation s = = found above, will thus 6 Introduction. 2 become oe If, therefore, we represent by S' another space described in the same manner during the time J’, we shall in like pT? nae whence we may conclude | ~ manner have S = pt? Cea ehae hes te 2 2 : which was to be proved ; and which, being freed from all conside- ration of infinite, is necessarily and rigorously exact. 3. To determine in what manner to dwide a quantity, a, into two parts, im such a manner that the product of these parts shall be the greatest possible. Let x be one of the parts, the other will be a—a, and the product will be ax—w?. Let this product be supposed the greatest possible product of the two parts of a. Suppose x to take a new value infinitely little differing from its present value. Let this be «+ Ah. ‘This value, substituted in the above product, givesaxv +ah—w?—2x2h—h?. If we subtract the former value ax —a? from this, there will remain ah—2h«v—h2. Now since a « — x? was by supposition the greatest product possi- ble, this increase must be nothing ; therefore ah—2hx—h? =—0; orah=2he +2. But h? is infinitely small compared with 2 h a, since it is an infi- nitely small part of an infinitely small quantity,t and may there- fore be neglected. We therefore have oh =D hx, or d= 22, ora—= 5, on the supposition that the product is the greatest possible. Whence we conclude, that each of the two parts is one half of a. These examples are introduced to show how the principles of the infinitesimal analysis may be employed in ordinary reasoning and in common algebra. In the following treatise the same prin- ciples are reduced to a system in the Differential and Integral _ Calculus. +If his only —+_, h2 : 1,000,000 ? “" ~~ 7,000,000,000,000° PRINCIPLES OF THE CALCULUS, SERVING AS AN INTRODUCTION TO THE PHYSICO-MATHEMATICAL SCIENCES. Preliminary Principles. Axersra and the application of Algebra to Geometry contain the rules necessary to calculate quantities of any definite magni- tude whatever. But quantities are sometimes considered as varying in magnitude, or as having arrived at a given state of magnitude by different, successive variations. ‘The consideration of these variations gives rise to another branch of analysis, which is of the greatest use in the physico-mathematical sciences, and especially in Mechanics, in which we often have no other means of deter- mining the ratio of quantities, which enter into questions relative to this science, than that of considering the ratios of their varia- tions, that is to say, of the increments and decrements which they each instant receive. As an introduction, therefore, to Mechanics and the other branches of Natural Philosophy, it is well to obtain some knowl- edge of this part of the calculus, the object of which is, to decom- pose quantities into the elements of which they are composed, and to ascend or go back again from the elements to the quantities themselves. This is, strictly speaking, rather an application of the methods, and even a simplification of the rules of the former branches of analysis, than a new branch. Mes «| 2. We propose to ourselves two objects.» The first is, to show how to descend from quantities to their elements ; and the method ' of accomplishing this, is called the Differential Calculus. ‘The 8 Differential Calculus. ‘second is, to point out the way of ascending from the elements of quantities to the quantities themselves ; and this method is called the Integral Calculus. So much of these two methods, as is of essential importance, may be easily understood, as it is but a consequence of former parts of analysis. ‘Those branches of these methods, which require more delicate researches, or which are of a less frequent applica- tion, will be distinguished by being printed in a smaller type. 3. As we are about to consider quantities with relation to their elements, that is to say, their infinitely small increments, it 1s _ necessary, before proceeding farther, to explain what is meant by quantities infinitely small, infinitely great, &c., and to point out the subordination which must be established between these quan- tities in calculation. ~ 4, We say that a quantity is infinitely great or infinitely small with regard to another, when it is not possible to assign any quantity sufficiently large or sufficiently’ small to express the ratio of the two, that is, the number of times that one contains the other. Since a quantity, as long as it is such, must always be suscep- tible of increase and diminution, there can be no quantity so small or so great, with regard to another quantity, but we may conceive of a third infinitely smaller or greater. For example, if w is infinitely great with regard to a, although it be then impos- sible to assign their ratio, this does not prevent our conceiving of a third quantity, which shall be to w, as v is to a, that is, which shall be the fourth term of a proportion, of which the,three first ; : . 22 area: a@::a:3 this fourth term, which is —, must therefore be a infinitely greater than «, since it contains x as many times as @ is supposed to contain a. In the same manner, nothing prevents our conceiving of the fourth term of this proportion, ®:a@::a:2; and this fourth term, which is a will be infinitely smaller than a, since it is contained by @ as many times as a is supposed to be contained by x ‘There are no bounds to the imagination in this respect; and we may still conceive of a new quantity, which e . e 2 2 e . shall be infinitely smaller with regard to “than “is with re- x x Preliminary Principles. | 9 gard toa. | We call these quantities infinitely great or infinitely small quantities of different orders. In general, the product of two infinitely great quantities or of two infinitely small quantities of the first order, is infinitely greater or infinitely smaller than either of the two factors; for,vy:y::@:1; but, if w is infinite, it contains unity an infinite number of times, ® y, therefore, contains y an infinite number of times. A similar course of reasoning shows that a product or a power, of any num- ber of dimensions whatever, and all of whose factors are infinite, is of an order of infinite marked by the number of its factors ; thus, when « is infinite, x* is an infinite of the fourth order, that is, infinitely greater than «%, which is infinitely greater than x?, which is itself infinitely greater than vw. For epee e8 et ae se hee ST. On the contrary, if « were infinitely small, «* would be then an infinitely small quantity of the fourth order, that is, infinitely smaller than x°, while «* would be infinitely smaller than 22, which would be infinitely smaller than «. Again, a fraction, whose numerator is a finite quantity, and. whose denominator is any power of an infinite quantity, is of an order of infinitely small quantities, marked by the exponent of that power. ‘Thus, ne for example, is an infinitely small quantity of the second order, if x is infinite ; and —— an infinitely small quan- tity of the third order. For a ae ; eA Tied oy y? x t But if a product have not all its factors infinite, then its order of infinity is to be determined by the number of those factors only which are infinite; thus a wv y, for example, is of the same order asxy; sinceaxy:ay::a: 1, and this last ratio, a: 1 or ao is a determinate ratio, if a is a finite quantity. * This difference is worthy of observation in the comparison of infinitely great or infinitely small quantities with each other or with other quantities, with regard to which they are infinitely great or small. If x be infinite with regard to a, nothing can measure their ratio ; but, on the same supposition, the ratio of x, to 7 mul- ) | 10 Differential Calculus. tiplied or divided by any finite number whatever, is a finite ratio. For example, x being infinite or infinitely small, is not comparable to a, a being supposed a finite number; but it may be compared with az, sincex:axv::1l:a. | 5. To express, by the calculus, that a quantity @ is infinite with regard to another quantity a; or, which is the same thing, to ex- press that,a is infinitely small with regard to 7, we must, in the aleebraical expression where these quantities are found together, reject all the powers of x lower than the highest, and consequently | all those terms without z. If, for example, in pti i x is sup- 52+ b posed infinite with regard to a and b, we must suppress a and 6, sta 52+ 6 pot Bo numerator and denominator by 2; and, when we suppose z to be and we shall have oe or : for the value of , when 2 is dzta a an infinite. is the same thing as dividing the see's PRN Ay b infinite with regard to a and 6, the fractions — and —, which repre- x x sent the ratios of a and 6 to x, must necessarily be suppressed, since, by this supposition, these ratios are less than any quantity whatever; wherefore, in this case, the proposed quantity is reduced 02 5 The quantity z? + ax -+ 6 would, in like manner, be reduced to z?, on the supposition that «# were infinite. For it is only on the supposition that 6 adds nothing to the value of a « + 0 that & can be said to be infinite: and in the same manner, it is only by supposing that a x adds nothing to the value of #2 + a z that ¢? can be said t6 be infinite. Wherefore, both a x and b must be considered as of no value by the side of x?, and are therefore to be rejected, and the quantity is reduced to #2. If, on the contrary, 2 were infinitely small, it would be neces- sary to retain those terms only in which the exponent of & is smallest. ‘Thus #? + az is reduced to az, when x is infinitely a+ b : a ae : b small; and eaetd upon the same supposition, is reduced to = Preliminary Principles. 11 It need not be apprehended that these omissions will affect the consequences to be drawn from the calculations in which they may be made. On the contrary, it is only by these omissions that we can express what we mean to express, viz. that w is infi- nitely great or infinitely small. It is only by these omissions that we can arrive at a conclusion conformable to the supposition which we have made. For if, when we supposed z infinite, we should not reject the terms just pointed out; if, for example, in i pate dtta t Sats BR as We should not reject . © and ° —, then the calcu- es : a b ; lus, not expressing that — and — are ratios less than any assigna- . x x ble quantity, would not answer what it is required to know, viz. what is the ae of that quantity when 2 is infinite; in short by allowing -— © and = to have any effect on the value sought, we con- tradict Ae os. which we have made, that z is infinite. We shall not want occasions for verifying the exactness of this principle of neglecting infinite quantities of the inferior orders. For the present, the following example will confirm the reasoning just made use of. Let there be the series 4, 2, 2, #, 2, #, Xe. ; the terms of this series evidently approach nearer and nearer to unity, yet without ever being ie to pass this limit. Now each term may be represented by ——--, by substituting for x the an il; number expressing the place of that term. Since then the terms continually approach unity, and that the more nearly, as they are farther from the beginning of the series, they can reach that limit only at an infinite distance from the beginning of the series; in order, therefore, to express the last term of this series, we must suppose in x i that x is infinite; but, conformably to the prin- . . . v . ciple, this quantity must then be reduced to mt that is to say, to . 1; the omission, therefore, of the term +1 in the expression i 7? 8° far from making the conclusion false, is, on the contrary, x } that which makes it what it ought to be. In short, by making this 4 12 Differential Calculus. omission, we act conformably to the supposition which has been made. “Such is the subordination which must be established in the cal- culus, between infinitely great and infinitely small quantities of different orders. But in the application of this principle of the omission of quantities, certain particular cases may occur which it will be well to notice. Suppose we have the two quantities z?-+ax+6, and s?--+ar+c; when wx is infinite, each of these is evidently reduced to #?, so that their difference, in this case, seems to be nothing. But if we take this difference according to the common rules, we find it b —c, or c—b, whether x be infinite or not. This seeming dif- ficulty, however, is easily solved. For the difference of these quantities is really 6 — c or c—6; but when we seek this differ- ence, after having supposed z infinite in each, it is the same as asking what this difference is, compared to the quantities them- selves; and, since each of them is infinite, we ought to find, as we do in fact, that the difference is nothing in comparison with them. When, therefore, it is asked what the result of certain operations on several quantities becornes, on the supposition that is infinite, it is to the result that we must apply the principle stated above, and not to each of the. quantities taken separately. ‘Thus we shall find that the sum of —«2? +a 2-4-6, and 72+ 676, is reduced, when z is infinite, toaz-+ 6-2; for, by the general rule, it is az-+ 62+ 6+ -¢, which, when z is infinite, is reduced toax+ bz. In like manner, if we had ¢—4/xz2 —62; this quantity, when @ is infinite, seems to be nothing. But as 4/z2—o2 is only an indication of the root of #2 — 6?, we must, in order to find the difference between this quantity and x, reduce — 4/x2 — b2 to a series (Ale. 144); the quantity c—»/z2—bd2 will then b2 b4 b? b4 ; become 7 — # Wea So: + &c., or OE + ae + &c., which, ob ey te é ; b? when z is infinite with regard to 6, is reduced to ay" x Simple Quantities. 13 Elements of the Differential Calculus. 6. When we cénsider a variable quantity as increasing by infinitely small degrees, if we wish to know the value of those increments, the mode which most naturally presents itself is, to determine the value of this quantity for any one instant, and the value of the same quantity for the instant immediately following. The difference of these two values is the increment or decrement by which this quantity has been increased or diminished. ‘This difference is also called the differential of the quantity. 7. To mark the differential of a simple variable quantity, as z or y, we write d ord y; that is, we place before the variable the initial d of the word difference. But when we wish to indi- cate the differential of a compound quantity, as 72, 522 +322, or 4/x 2—a2, &c., we enclose this quantity in a parenthesis, before which we write the letter d ;. thus we write d (27), d (54° +3 22), d (4/z2—a2), &e. The differential of a compound quantity is also sometimes expressed by a point between d and the quantity, asd.#?,d.aryz, &c. We shall hereafter represent the variable quantities by the last letters of the alphabet, ¢, u,v, y, z; and the constant quantities, or those which always preserve the same value, by the first letters, a, 6, c, &c., and if they are used otherwise, notice of it will be given. As tothe letter d, it will be used only to designate the differential of the quantity before which it is placed. 8. Agreeably to the idea which has just been given of the dif- ferential of a quantity, we see, that to get the differential of a quantity which contains only variables of the first degree, and neither multiplied nor divided by each other, we have only to write the characteristic, d, before each variable, leaving the sign of each unchanged; for example, the differential of « + y—z will be de +d y— dz. For, in order to obtain this differential, we must consider z as becoming + dz; y as becoming y+ dy ; and z as becoming z-+-dz; then the quantity proposed, which is z+ y— z, would become e+da+y+dy—z—dz; and, taking the difference of these two states, we shall have atdertytdy—z—dz—2£2—y+2; that is, d(a+y—z)=da2+dy—dz. ee eee et See ‘ 14 Differential Calculus. The case would be the same, if the variables, which enter into the proposed quantity, had constant coefficients; thus the differen- tialof5ce+3y,s5dz+3dy; thatofaz +b y, is adztbdy; for, when w and y become «+d and y+dy, the quantity ax-+b y becomes a («+ dx) +6 (y+dy), that is axtadi+by+tbdy; the difference of the two states, or the differential, is ada +bdy; that is, generally, each variable must be preceded by the character- istic d. 4 If in the proposed quantity there be one term entirely constant, the differential will be the same as if there were no such term. That is, the differential of that term will be nothing. This is evident, since the differential being nothing else but the increment, a constant quantity cannot have a differential without ceasing to be constant ; thus the differential of a +6 is simply ad x. 9. When the variable quantities are simple but multiplied to- gether we must observe the following rule. ; Find the differential of each variable quantity successively, as if all the rest were a constant coefficient. For example, to find the differential of x y, we first consider «x as constant and obtain x d y}, we then consider y as constant and have y d a, so that,d (vy) =xdy+ydea. The reason of this rule will be seen by going back to the prin- cuple upon which it is founded. ‘To find the differential of x y, we must consider w as becoming w + da, that is, as increasing by the infinitely small quantity d x; and yas becoming y + d y, that is, increasing by the infinitely small quantity dy; then aw y be- comes(x-+ da) X (y-+dy), that is,e7ytaedytydrtdydz; then the difference of the two states, or the differential, is vy tadytyde«e+dydi—cy, or odytydx+dydz; but in order that the calculus may indicate that d y andd @ are infinitely small quantities, as they are supposed to be, we must (5) omit dy dz, which (A) is an infinitely small quantity of the second order, and therefore infinitely small compared with w dy and y dz, which are infinitely small of the first; therefore the differential of vy,ord.czyisxdy-+yd zx, agreeably to the rule. + To avoid obscurity, in the manner of writing, it is best to write last the variable affected with the characteristic d. | Compound Quantities. 15 We shall find, by the same rule, that the differential of x y z isrydz+azdy+yzdz, by differentiating as if «y, x z, and y 2 were successively constant quantities. ‘This may be demon- strated, as above, by regarding «x, y, and z as becoming, respec- tively,wt+da,y+d y,andz+ dz; in which case w y z is changed into (w-+dz) (y+ dy) (e+dz)=axyz+aydz+ wzedytyzdetydedzt+zdydu+adzdy+dadydz; then the difference of the two states will become, by reducing, and rejecting the infinitely small quantities of the second and third orders, d.vnyz=xcydz+aurzdytyzdua. 10. If the quantity proposed be any power of a variable quan- tity, observe the following rule. !’ Multiply by. the exponent, diminish this exponent by unity, and multiply the result by the differential of the variable. Thus, to find the differential of 2?, we first multiply by the exponent 2, diminish this exponent 2 by 1, then multiplymg by dx the differential of the variable «2, we have 22 dz. We shall find, in the same manner, that the differential x is 322 dx; that of 24,403 da; that of a 1,—a-2 da; that of x-3,—3a-* dx; that of x, 4 w 2d ae that of #2, iS aus MMi and, generally, that of 2”, is m a”—1 da; whether the exponent m be positive or negative, a whole number or a fraction. To find the reason of this rule let us go back to the first princi- ples. Let us consider x as becoming x-++ d x (dx being infinitely small) ; then 2” becomes (x 4- dx)", which, being reduced to a series (Ale. 144), becomes a” +-ma™—1dax+m m—I1 A Obi Sah or, because the term m. aes KL a” —2 d x is infinitely small of the second order, and the following terms would be of still lower orders, the series is reduced tow"-+ma”"-!dz«a; then the difference of the two states is a+ mx"-1dx—x"; and, therefore, d.a”= rm Teer 2 dx? 1. Xe. mx™-1dx. If there were a constant coefficient or multiplier, the case would not be altered, the constant coefficient would remain in the differential the same that it is in the quantity ; d.ac«™, therefore, = maxr™—1 dx. 16 | Differential Calculus. We have thus given whatever it is necessary to know, in order to be able to differentiate all sorts of algebraical quantities. What follows is only an application of these rules. : . . : . x 11. Suppose it were required to differentiate the fraction —, we bf should write it wy—1 (Alg. 133); and then, applying the rule given (9), we have d.cy-1=2d.y-1+y-1dz, and, consequently, (10) d.vy-1= y-1dx—ay-2 dy =, by ydxu—ady reducing to a common denominator, 5 Therefore, to find the differential of a fraction, werrmultiply the differential of the numerator by the denominator, subtract from the product the differential of the denominator, multiplied by the numerator, and divide the whole by the square of the denominator. This is the rule usually given for the differentiation of fractions ; but we easily perceive that we may dispense with charging the memory with this new rule ;*as it is sufficient to raise the denomi- nator into the numerator (A/g. 133), and then differentiate by the general rule. 12. If we wish to differentiate a x° y?, we first consider v* and y2 as two simple variables, and (9) we have d.axv®y2=—a0r°d.y? +ay?d.x°; then (10) we have d.axvt y?=2areydy+d3ay* x2 dx. In general, d .aa"ay® ae" dy ay" bd. o=—nar™y"—-* dy-+-may" 2" Vd 2. 13. If the quantity, which we wish to differentiate, is complex, but without containing any powers of complex quantities, we differentiate separately each of the terms of which it is composed. Thus, d(ax? + b2?-+cxry)=dsax?dx+2bedxu+cudy+cydz. Ip like manner, d (au? +bo04+ ad (an®-- ba--cax-?'y) =2acdx+bdx—2cxyde+cu-? dy. In like manner, d(x? y+ay?+ 63)=322 ydr+nx2°dy+2aydy, observing that the constant quantity 6° has no differential. i te tall Complex Quantities. 7 17 14. If the exponent of the quantity be a whole number, as in (a+ ba-+cx?)®, we regard the whole quantity affected by this exponent as a single variable, and differentiate by the rule for powers (10). Thus, d(a+ba+ca?)'=5(a+6x-+cx?)* Xd(a+ba+c27) = 5(a+ bx+cx?)* X (bdx X 2cxdz). In like manner, | d(a+bx2)*=s(atba2)* x d(atba2) =$(a+ba2)3 XVbrda=mPbadax (a+ b x2) 3, d(x? +2axc-+a?)? = 4 (a? + 2ax+a?) (wn+a) da. 15. When a complex quantity is composed of different factors, we regard each factor as a simple variable, and follow the rule given (9) for a product of several simple variables. ‘Thus x? (a+ 6 x?) 5 which we may consider as composed of the two factors x° and (a+ ba2)*, will give d(x? (a + 622) 8)— (a+ bw?) 2 d(x?) +03 d(a+bx2)8, which, by the preceding rules, becomes 302dx(a+ba2)?+ 19 b xt dx (a+ ba2)s, And a(t )= a5 (0+ a)* (»+d)-? = (wa)? d (w+ by (Fb) H(@+ 6)? d(a pa)? ; that is, —— 2(e#+a)?(x+6)-?dx+3 («c+ 56)-? (x +a)? da, which, by restoring the denominators, becomes Q2(r+a)>dx , 3a+a)y?dx Beis Ahh ity RCE FSI (reducing to a common denominator) 2(¢+ a)? (ex+a)dz , 3(4+a)? (x+6)dzx PEs!) scmneriyr ooo GELS _ (Br+Bb—21—2a) (ef a)2dr _(x8b—2a)(r4a)?dz (x 4-6)? ae (x +6)3 . Also, dd —_..) oo aledtyydy) Cet oe 18 Differential Calculus. 16. If the proposed quantity is radical, we substitute fractional exponents in place of the radical signs (Alg. 132), and differen- tiate according to the rules already given. ‘Thus, 1 mi d(/z) = d (a?) = fa? de=— = d Q/aa )ad(a®) = 40°? da; d(/a2—22)= d (a? — w?) 2 = 1(a2 —a2) 2d (a? —2?) = = d'2(a? —x?)? = —rdt | gies df um/(a-bery $d Lam(at bor § —a«" d(a+ ba" )e+(at ba" )eda scr “EP amide X (a+b x" y= +ma"—!dax(a+ 6x") P q In hike manner, vas d.(otyt =z otypytd(e ty) =e ete i (xy + y¥7)? ‘(=)= aan (rz) aF5 (ered patted Of Second, Third, &c. Differentials. 17. In addition to the differentials which we have just been con- sidering, and which are called first differentials, we consider also second, third, a&c. differentials. These are indicated by writing twice the characteristic, d, before the variable, for the second differ- entials, three times for the third, &c. For example, d d x indicates the second differential of +; dddzx, the third differential. When we speak of second differentials, we consider the variable as increasing by increments which are’ unequal, but whose differential is infinitely small with regard to these increments themselves. Thus d dx is infinitely small compared with da. In the third differentials also, dd dxor d® x, (for they are indicated in both ways), is infinitely small compared with ddz, and so on. ‘To indicate the square of d z, ae Se eaiaee » + From this expression we may deduce the rule, to differentiate a radical of the second degree, we divide the differential of the quantity under the radical sign, by double the radical itself. Successive Differentiations. 19 7 we should naturally write (dz)?; but, for greater simplicity, we write dx?, which cannot be mistaken for the differential of x2, as that is Ferrin tcd by d (z?) or d. 2?. We observe that although dd « and dz? are both infinitesimals of the second order, they are nevertheless not equal; for dd x is the second differential of x, or the difference of two successive differen- tials of x; and d x? is the square of dz. In order to determine the second differentials, it is most natural to consider the variable quantity in three successive states, infinitely near each other; to take the difference between the second state and the first, that between the third and second, and then take the differ- ence of these two differences. [or example, the first state of xz is 2; at the second instant it has increased by the quantity d 2, and become a+-d2x; the following instant «-+-d x increases by dx +d (d 2), d (d x) marking the quantity by which the increment of the second instant exceeds that of the first, or the differential of dz. Thus the three successive states of the quantity x are aa+dz1+2d2r+d (dz). The difference between the second and first is d a; that between the third and second is dx-+-d(dzx); finally, the Mececeit: between these two differentials, or the second ‘differential of x, is d (dx); we have therefore ddx—d(dz).\\ Therefore, the second differentials are obtained by differentiating the first differentials according to the rules already given. | For example, to get the second differential of x y, we take the first differential, which is 7dy—+-ydx; we then differentiate this quan- tity as if x and dz, y anddy, were so many different variables, and we find tddytdydzr+yddi+tdydz; or dd.cy=2tddy+2dydzr+yddx. In like manner the second differential of «? is found by first differ- entiating x2, which gives 2d; then differentiating 22 dz as if x and d x were both finite Paranles which gives 2rddzx+-2d x?. We shall find also, that dd.axw==d.maa""de=m.m—law dx? +max"—ddz.t + A difficulty may present itself in this mode of taking the second differentials which it is best toexplain. When we determine the first differentials, we reject the infinitely small quantities of the second order; but the second differentials being infinitely small of the second order, is there not reason to fear that what we have rejected in the valuation of the first, will render the second defective ? .We answer, no: for this infinitely small quantity of the second order, which has been rejected, can have for its differential only an infinitely small quantity of the 20 Differential Calcults. We should proceed in the same way to differentiate a quantity in which there were already first differentials, whether it were the result of an exact differentiation or not. ‘Thus. d.rdy=xrddytdzidy. dU Hd. x tdy=—12dedy 41 ddy, dt WPAN. bale rs, pat ddz drddy GR apy Tes te SY 1— ddxdy—drd y—ddy = Py Tp ya 18. It often happens that in calculations into which several varia- bles enter, the first differential of one of the variables is supposed constant. ‘This supposition is allowable, because we may always take one of the first differentials as a fixed term of comparison for the other first differentials, and it simplifies the calculation, inasmuch as the terms affected by the second differentials of this variable will not afterwards occur in the course of the operation, since if dz is con- stant, dd — 0, which makes all the terms affected by dd « disap- pear. We have only to take care, in this case, not to differentiate dx (or the constant differential) in the terms in which it occurs. , 1% . Thus, the differenflal of aa or d.dxdy- is on the supposition tdd that d x is constant, —dady—ddy or — a ila cae If on the Ree E contrary, we suppose d y constant, it is a 19. By reasoning in the same way, we perceive that the third dif- ferentials are found by differentiating, in the usual way, the second differentials, considering the variables and their first and second differ- entials as so many different variables; and so for higher differentials. /It is only necessary to observe, that when a differential has been con- sidered as constant in the passage from the first to the second differen- tration, 1 must be considered so in all the successive differentiations. Remarks. 20. We have supposed, in the preceding pages, that the varia- bles x, y, &c. all increased at the same time ; that is to say, that x becoming « + dx, y became y-+ dy, and so of the rest. But it may happen that some diminish while others increase. In this case we must, after the differentiation, change in the result the third, which would be rejected by the side of the second differential, since a second differential is infinitely small of the second order. P Differentials of Sines, Cosines, &c. Q]1 sion of the differential of the variable which has diminished. Or indeed, we may let the differential remain as given by the pre- ceding rules, but, in the application which we make of it to the question, observe to take negatively the quantity which represents the differential of the variable which has diminished. For if y has been diminished by a quantity, g, and if, in the differentiation, you have tacitly supposed y to become y+ dy, it must follow that y—gq=y+d y, or thatt—q=—d y, or g=—d y; in such cases, therefore, what has been called d y must be called —dy every where else but in the differentiation. We shall see exam- ples of this hereafter. It will be the same with second differentials compared with first. If the first differential diminishes, you will nevertheless differentiate in the usual way, but in the application to any question, you will, if dy, for instance, is the differential in question, call that—dd y, which you would otherwise have called dd y. Such are the rules for differentiating quantities, when they are presented directly. But it often happens that ‘i is not so much upon quantities themselves, as upon certain expressions of those quantities that we have to operate. || Instead of angles, for example, we often employ their sines, tangents, &c.; often, also, we meet with the logarithms of quantities instead of the quantities them- selves. We proceed to%show how to differentiate these kinds of expressions. . Of the Differentials of Sines, Cosines, &c. 21. If we have to differentiate such a quantity as sin 2.(or the sine of the angle or arc z), we must conceive that the angle z becomes z + d z,and then sin (z +d z) — sin z is the differential of sin z. Now according to what has been laid down (Trig. 11), sin (z + dz) = sin zcos d z + sind z cos z, supposing the radius = 1. But the sine of an infinitely small are is the arc itself, and its cosine does not differ from the radius; we have therefore sind z= dz, @0CLCOS (he ars then sin (z-+dz)=sn z-+d z cos 2; therefore, sin (2 +d z)—sin z =d/(sin z) = d 200s z; that is to say, the differential of the sine of an arc or angle, whose 22 Differential Calculus. radius is unity, is found by multiplying a differential of the angle by the cosine of the same angle. 22. In like manner, the differential of cos z, or cos (z +d z)—cos z = cos z cosd = —sin z sin d Z — Cos Z, since (Trig. 11) | cos (2 +dz) = cos zcosd z—sinzsin d z; therefore, as sin). 2—=0.2, ARO COs dyes we obtain d (cos z) ==cos z —d z sin z —cos z= —d zsin Z; that is to say,\ The differential of the cosine of an angle, whose radius is 1, is found by multiplying the differential of the angle taken with the contrary sign, by the sine of the same angle. Thus, to recapitulate, we have d (sn z)==d zcos z; ad (cos 2) BisSavg 2 sing. | By means of these two nance we may differentiate any quantity composed of sines and cosines, without any other rules than those already given. Thus, to differentiate cos 3 z, we have d (cos 3 7) =—3dzsn3 z. Universally, if m is a constant quantity, d(cosmz)—=—mdzsnmz; d (sin mz) =m d & COs 1 Mm Z. Tn like manner d (sin z cos t) = cos ¢ d (sin z) + sin z d (cos f) —=dzcostcos z—dtsin z sint. And as(sine)* =m sine a (sin 2) wea 2 cose (810 2h oa. | 23. If we had a, which is the expression for the tangent of an angle when radius = 1, since (Trig. 8) COSi2 ulin seen alle 2, we should have d ic # ) =d . sinz(cosz)—'=dz cos z(cosz)—!+-dz sinz?cosz—* Cos Z dz cosz dzsin z2 dz cos z2-++-dz sin z? dz COs Zz cos 22 cos 22 cos 22 ? because (Trig. 10) cos z? 4+-sin z? = 1. Therefore, ' The differential of the tangent of an angle, whose radius is 1, is equal to the differential of the angle, dwided by the square of the cosine of the same angle. Logarithmic Differentials. Q3 Whence we may also conclude, that the differential of an angle as equal to the differential of the tangent of that angle, multiplied by the square of its cosine ; for, since a(= s)=4 (tang z) = ade COS Z cos z2 : we have ad z==cos 2? d tang z. COS Z 24. If it were required to differentiate which is the ex- sin z pression for the cotangent of the angle z, we should have COS Zz d.——_=d . cos z sin z—! = — d z sin z sin z—!—dz cos z2sin 2 sin Z , dzsnz dzcosz? — dzsin z2 —dzcosz? — Az sin z sin z2 sin z2 ~ gin 22 Therefore, The differential of the cotangent of an angle, is equal to the differential of the angle, taken negatively, divided by the square of the sine of the same angle. ‘The use of these differen- tiations will be exemplified hereafter. Of Logarithmic Differentials. 25. According to the description already given (Ale. 238), logarithms are a series of numbers in any arithmetical progression, answering, term by term, to a series of numbers in any geometrical progression. This being laid down, let y and y’ be two consecutive terms of a geometrical progression, of which r is the ratio, and @ and a’ the two first terms. Let, also, x and x’ be two consecutive terms of an arithmetical progression, of which 6 and 0’ are the two first terms. Let us suppose, moreover, that x and a are in the same place in the arithmetical progression that y and y’ are in the geo- metrical progression ; in which case, x and 2’ are the logarithms of y and y’. By the nature of geometrical progression (Ale. 231), we have y’ =ry, and a’ =r a; substituting in the first of these equations the value of r deduced from the second, we have ay =— pea Let us now suppose that the difference between y’ and y is Z, or 24 Differential Calculus. that y’ = y+ 2; we shall have ——— yee ri+ Ae <, and conse- a! ne az —l|=> , or — =a’ —a. quently, Again, the nature of arithmetical progression gives (Alg. 228) a’ —x=—b/ —b. In order to find then the ratio of these two progressions, let us suppose that the difference a’ —a of the two first terms of the geometrical progression, is to the difference 6’— 6 of the two first terms of the arithmetical progression as unity is to any number m; that is to say, that a’ —a: b’—6::1:m; we shall have m (a’—a)== 6b’ —b; substituting then, in this last equation, in- stead of a’ —a and 6/—8, the values which have just been found, ‘we shall have ae —®, an equation which expresses gene- rally the ratio of any geometrical progression to any correspond- ing arithmetical progression. Let us imagine that, in each of these progressions, the consecu- tive terms are infimtely near each other; then z, which marks the difference of y’ and y, will be dy; and «’ —«x, which marks the difference of x’ and a, will be dw; whence, the equation will be changed into mG s e | y With regard to m, which indicates the ratio of the difference _ of the first two terms of the arithmetical progression, to the differ- ence of the first two terms of the geometrical progression, it will nevertheless be a finite number, although these two differences be infinitely small, because we easily conceive that one of two infinitely © small quantities may contain the other as many times as one of two finite quantities can contain the other. mad y _ 4 “The equation ~ == d « shows, therefore, that d a, the dif- ferential of the logarithm of a number represented by y, is equal to d y, the differential of that number, divided by the same number y, and multiplied by the first term a of the fundamental geometrical progression, and by the number m, which represents the ratio of the difference of the first two terms of the arith- metical progression to the difference of the first two terms of the geo- metrical progression. As this number, m, determines, in some meas- ure, the relation of the two progressions, it is called the modulus. ‘Logarithmic Differentials. Q5 We see then, that according to the value which m and the first term a of the geometrical progression are supposed to have, the same number y may have*different logarithms. But of all these different systems of logarithms, the most convenient in algebraical calculations, is that in which the first term of the geometrical progression is 1, and in which the modulus is 1. In that case, : ad the equation ~~" 7 - ao x, which comprehends all the different systems of logarithms, becomes R 26. In the system of logarithms, therefore, used in algebraical calculations, the differential d x of the logarithm x of any number y, ts equal to d y, the differential of that number, divided by the number itself. ‘This is the principle by which we may easily find the differential of the logarithm of any alyebraical ESI But before making use of it, we must observe, Ist. That the logarithms here spoken of are not those of the tables; but it is easy to deduce the one from, the other, as will be seen hereafter. Qd. That since the first term.6 of the arithmetical progression is not found in the equation aoe = d«, this equation, as well as dy Megas : the particular equation at w just deduced from it, are always true, whatever may be the first term 8, that is to say, the loga- rithm of the first term a of the’ geometrical progression. We are therefore at liberty to suppose, for the sake of greater simplicity, » that the first term of the arithmetical progression is nothing; and, as the geometrical progression which has been fixed upon has unity for its first. term, we shall take zero or O for the logarithm of 1; but it should be observed that this is entirely arbitrary. By thus taking unity for the first term of the geometrical pro- gression, and zero for the first term of the arithmetical progression, or for the logarithm of unity, the rules already given (Alg. 241) for the application of logarithms, will equally well apply here. If ‘we generalize these rules, designating logarithm by J, we shall see that, instead of J (a 6) we may take J a+ 6; instead 4 : 26 * Differential Calculus. of / o Za—Ib. In the same manner, /a"—mla;_ finally, L/a lan =—la. This being laid down, if we apply the principle which has just been established concerning the differential of the logarithm of a - number, we shall find that a ee . Pie Veal HE mis ee ince) mal ax ait )=d (as one)» at 2% ata observing that the differential of the constant, / a—0. We have also, d. I= a (he) =— dor d.2tn ==, d.l@y)=d(le tly) = Sie dl ta do-inatt_ te dl (F)\=4 (1@+2)—-1@—2))= oo eee dl (a? + mE) = ape an 55 dl fare Rees = Vea oe or, more directly, dl fara —d.pl(a toys, dl (u(a+ bat Pd (Le™+-1 (a+b 2" )?) eet n)\ mae, npbalde (mlx +pl(a+ba )) : cranny sae These examples _are sufficient to show how other logarithmic quantities may be differentiated. ®. Exponential Quantities Q7 Of the Differentials of Exponential Quantities. 27. We sometimes meet with quantities of this form, c* , 2; that is to say, quantities whose exponent is variable. ‘They are called Exponential Quantities. In order to find how to differentiate these quantities, let us amnines WY == 2; then, taking the logarithms of each member, we have Eny =a lz dz and consequently — a: Ee) ori then dz—=zdl (x), and, substituting for z and d z, their values, d (zy )== 29 d1 (x¥.); ‘that is to say, the differential of an exponential quantity is found by multiplying that exponential quantity by the differential of its loga- rithm. 'Thus, TARE A ers CYR eas ylemm (dyla +27"), x ite like manner, d(@+y')=d.e+d.yroedle+tyd.ly =a@d.tle+yd.zly=@dizrla+y (daly +72), So . iy » d(a?+2?)? = (a?+ 2?)2 d1(a?+x2)* = (a? +27)? d Jal (a?-+ 2?) = (a? + 22): (axi Cob et 04 ay 4 a) a? 2 and so of others. Frequent use is made, in calculations, of the exponential quantity e* ,e being the number whose logarithmas —1. ‘The differential of this quantity is, according to what has just been laid down, i =e 0 (ne) ee .0, 1a; and since J ¢ is laa eg = 1, we have simply d.e&=edt. \ That is to say, this particular exponential has, for its differential, the exponential itself multiplied by the differential of its exponent. This exponential will be found hereafter. 28 ‘Differential Calculus. Application of the preceding Rules. 28. In order to show, by some examples, the use of the rules which have just been given, and the advantage which they have over common algebra, we shall now apply them to subjects with which we are already acquainted, viz. questions in Geometry and Algebra. Application to the Subtangents, Tangents, Subnormals, &c. of Curved Lines. 29. To draw a tangent to any curved line AM (fig. 1), we con- ceive this curve to be a polygon of an infinite number of infinitely small sides. ‘The prolongation MT’ of Mm, one of these sides, is the tangent, which is determined for each point MM, by calculating the value of the subtangent PT’, or that part of the line of abscissas, which is comprehended between the ordinate PM and the point TZ, the intersection of this tangent. .The subtangent is determined in the following manner. : Through the extremities M and m of the infinitely ‘Small side M m, we suppose the two. ordinates MP and m p, to be drawn, and through the point M, the line Mr parallel to AP, the axis of the abscissas. 'The infinitely small triangle M rm is similar to the finite triangle Z'PM, and gives this proportion, 7 Ons ME eae EVA ck ls Now if we call AP, c; PM, 7; it is evident that Pp, or its equal r M, will be d x, andr m will be dy; we shall therefore have dy: da: es 1a, ee This is. the general formula for determining the subtangent of any curve whatever, whether the y’s and z’s are perpendicular to each other or not, provided that the y’s are parallel among them- selves. We shall now give an example of the application of this formula to any curve of which we have the equation. Let us suppose that the nature of any curve AM were ex- pressed by an equation containing x, y, and constant quantities. If we differentiate this equation, there can never be more than two kinds of terms, those multiplied by dw and those by dy. It will then be easy, by the common rules of Algebra, to deduce, from Tangents and Subtangents. 29. x ga : dx this differential equation, a value for 7? which shall contain ay only terms of x, y, and constants; by substituting this value in the yd% x 7 | . formula ory X 5—-, we shall have a value for the subtan- d dy dy gent in a, y, and constants; finally, putting instead of y its value in terms of x, deduced from the equation of the ctirvé, we shall have the value of the subtangent expressed in terms of v and con- stant quantities only. So that to determine the position of this line for any point whatever M, we have only to substitute, in this last result, in place of x, the value of the abscissa AP, which answers to that point. Suppose, for example, that the i curve is an ellipse, of which the equation is-(Ap. 112) y2 = — a ( «—w?). Differen- tiating this ee we have ee esc e dx—2b2xdzx; from this we deduce the value of 5 z, by dividing first by d y and then by the multiplier of d a, in vihe second member, and find dz 2a2y d : Fie eas : substituting this value in ~ es 9 we shall have dy yao 2 a? y? dy abt—2b2a° 62 finally, substituting for y*, its value ~> (a «— w?) given by the equation of the curve, and reducing, we have >) 2(ax—x2x2) ax—zx? a value. which is. precisely the same as that which was found by Algebra (Ap. 119), but which is obtained here in a more expedi- tious manner. We may~observe here how this result justifies what was said (5) concerning the quantities which are to be rejected in the culculation ; for, by employing here the differential calculus, the rules of which, in this example, suppose the omission of the infinitely small quantities of the second order, by the side of those , Of the first, we arrive at the same result as in the Application 30 | Differential Calculus. of Algebra to Geometry, where this subtangent was détermined in the most exact and mgorous manner. We see that by thus rejecting the quantities which were pointed out to be neglected, we only impress upon the calculus ,the character which it ought to have, in order to express the ohatinons of the question. We pursue a similar course in determining the tangents, subnor- mals, normals, &c. at _ Let us suppose, for the sake of greater simplicity, that the ab- scissas and ordinates, (the x’s and y’s), are perpendicular to each other. In order to determine the tangent, we compare anew the triangle M mr with the triangle T’PM, and we have rm:Mm::PM: TM; but by a property of the right oe triangle, Mr m, we have Mg BER MLE pee eae Page +dy2 5 therefore, dy: /dz2+dy2::y: TM; TM = Ya/daztdy2 _ Yrx/dartdy2 dy a/ dy? Si dx+dy Taz ig y, |e ay Thus, after differentiating the equation af the curve, we shall therefore, d x deduce thence the value of — iy , the square of which we substitute in this expression for the tangent; after which, putting in place of. y its value, in terms of « and constants, drawn from the same equation, we shall have the tangent expressed in terms of x and constant quantities. An application of this may be made to the equation of the ellipse, and the same value will be found as was formerly obtained (Ap. 121). Ifthe subnormal is required, we suppose the line MQ perpen- dicular to the tangent 7'M, and observing that the.triangles Mr m, MP Q, which have their sides respectively perpendicular each to each, are similar; we have | Mr:rm:: PM: PQ; that is to say, re a eae _ydy aeitneg ete Te aoe Then, after having nee: the equation of the curve, we de- d duce thence the value of © ag which we substitute in a 2, and, ’ a i Normals, Subnormals, &c. . 31 completing the operation as before, we have the value of the sub- normal in terms of and constant quantities. 2 , b In the ellipse, for example, the equation y2 = qa (ae — 0), being differentiated, gives 52° b2 —~ (a—22) eydy=-. (adx—2Qadza); then?! — saa rie - consequently He subnormal, - l b2 — 2x 2 dy ode Me dx a Be aes : as was , formerly found (Ap. 118). If the normal MQ were required, it might ie found by com- paring anew the triangle Mr m with the triangle MP Q. Let us take, as a second example of the formula for subtangents and of that for subnormals, the equation of the parabola, which is y2 =p ux (Ap. 172). . By differentiating, we have ‘ Dee d-y uh Bi , 2ydy=—pdz; then 7 =p and Ge ay the subtangent, therefore, Mike (ane sane acta hd i eames p and the subnormal YAY PY LD ae a ee which agree perfectly with what have been already found (Ap. 179, 180). We shall take, as a third example, the equation y™+"™—=a™ a" , which is a general expression for parabolas of every kind. The name of parahola is:given to every curve, whose equation, such as y™+"— aa", has only two terms, but in which, the exponents of x and y, in the two members, have the same sign. By differentiating this equation, we have (m-+- 1) yr Bold yn a™ old x; dx os ese +n) ymrnr—l dy ry oie ek from which and the subtangent . Ms ydx (m+n) ymtn. dy nangr—l ? and substituting for y+” its value a” «” , we find 32 | Differential Calculus. er ee ee ee DS RRA se ne n Whence we may see that the subtangent, in these curves, is equal to as many times the abscissa x, as there are units in the exponent of y divided by the exponent of x. ‘This holds true, as we have already seen, in‘the common parabola, where the subtangent is 2x, and where the exponent of y divided by the exponent of a, Is in fact 2 or 2. . Let us now take, as an example, a curve the nature of which is expressed by an equation in terms of:the differentials of the codrdinates. - Let us suppose, for instance, the curve BM (fig. 4) to be such, that the abscissas 4P, Ap, &c. being taken in arithmetical progression, the corresponding ordinates PM, p m, &c. are in geometrical progression. This is called the loga- rithmic curve, because, while the ordinates represent successively all imaginable numbers, the abscissas are their logarithms, and ‘it . m eee will have for its equation — = d x, since we have found (25) that this equation expresses the relation of numbers to their loga- , a if dx am rithms.. We shall have, therefore, - == ——, and consequently BYES RY | Ph aa. the subtangent a will become —~ =a m3;*that is to say, y, 7, ' for each point of the’same logarithmic, the subtangent PT" is always the same, and is equal to as many times the first ordinate AB, or a, as there are units in the modulus m. 30. When the equation of the curve is ‘such that, x increasing, y diminishes, as in figure 2, then the line r m must be expressed by —d y (20); and the proportion r M:rm:: PM: PT, which serves to find the subtangent, becomes ih YE ei ka Pee 4 aENt Thus there will be no difference in the calculation; the only change is that the tangent, instead of falling on one side of the point A, the origin of the abscissas, with regard to the ordinate PM, will fall on the opposite side. This is the reason why we ty 3 may always take ay as the formula of the subtangents; if the dz. (eh dae ordinates decrease, the value of Pris have a negative sign, Tangent to the Hyperbola. 33 which indicates that this value must be referred to the side opposite to the origin of the abscissas. If, for example, we take the equation of the circle, the origin of the abscissas being at the centre, that is to say, the equation (Ap. 103) being y? = } a? —w?, it is evident that, while Cp or x (fig. 3) increases, y or P/M diminishes ; so that the subtangent PT falls on the side of PM oposite to C, the origin of the ab- scissas. ‘This is shown also by the calculus ; for, if we differentiate, \ du —y we have 2 yd y=—2zwd x, and consequently wae aaa then ee =e = pa sen Hass aa a value, of which the sign — in- dicates, that it should be referred to the side opposite to that which is supposed in taking ay for the formula of the subtan- gent. Let us now take, as an example, the equation x y = a?, which belongs to the hyperbola considered between its asymptotes (Ap. 163), we have yd x +.4d yO, and consequently, - ed ; therefore, oa = a ce aan LL é which shows that, to draw a tangent to the hyperbola considered between its asymptotes, we must take upon the asymptote nearest M, the point in question (fig. 7), and on the side of PM, which is opposite to A, the origin of the abscissas, the line Nd A 0 Bom is We see with what facility all these results are obtained by the differential calculus. In the same manner with parabolas of all kinds, we call by the name of Hyperbolas referred to their asymptotes, all those curves whose equation, such as y”==a™t™ax-" , contains only two terms, but in which the exponents of y and x, in the two members, have contrary signs. We leave these curves as an example for the exercise of the reader. ‘The subtangent will be found to be =~ @; that is to say, that it falls on the side opposite to the origin of the abscissas, and that it is equal to as many times the abscissa as there are units in the exponent of y, divided by the exponent of x. 5 34 Differential Calculus. In general, we determine at once, by this calculus, all the sub- tangents, tangents, &c. of all the curves of the same family. Those are said to be of the same family, whose equation is formed in the same manner, and differs only in the magnitude of the exponent. Thus we call by the common name of circles, all the curves in which any power of the ordinate is equal to the product of any two powers of the two distances of that ordinate from the extremity of the line a, on which the abscissas are taken. Their equation is y™+™== a" (a—w)", which comprehends the circle properly so called, when m =n = 1. ‘The equation ane ae ; am (a —wx)" represents ellipses of all kinds; and y™*" = ; a” (a+ cv)", as well as y™t" = i («— a)" belongs to the family of hy'perbolas. The figure of these curves is determined by means of their equation, as we have seen in conic sections (Ap. 101) ; observing that y has only one real value when its exponent is odd, and two when it Is even (Ale. 157); which determines that to the same abscissa, there corresponds only one branch of the curve in the first case, and in the second, two, which fall on different sides of the same axis. 31. When we know how to determine the tangents, normals, &c. we may easily solve the two following problems. Ist. From a given point without a curve, to draw a tangent to this curve. 2d. From a point given any where in the plane of a curve line, to draw a perpendicular to that curve. For the first of these, let us suppose that DM (fig. 8) is the tangent required, which passes through the point D. The gene- i ral value of PT’ or ay will be easily found by means of the equation of the curve. If BD be drawn parallel to the ordinate PM, the line BD and its distance BA from A, the origin of the abscissas, are considered as. known, since the point D is given in position. Calling, then, DB,h; AB,e@; AP, x; and PM, y; we shall have ziti Saha _ ydt if 5 P's peeerend Lb = PT oeBe = “dg hae ee Perpendiculars to a Curve. 35 Now the similar triangles TBD, TPM, vive ee Lees) PIP AM, That is to say, yd : ydu ht. 7 —xr—eih ath JPebel ee pai Therefore, Maes s= ise dy dy . : ; dx Putting, therefore, in this equation, the value of dy’ drawn from ‘the equation of the curve which has been differentiated, and then substituting for y its value in terms of w and constants, also given by the equation of the curve, and we shall have the abscissa « of the point of contact. And if there should be more tangents than one, drawn through the point D, the last equation, in terms of x, will give all the abscissas of the points in which these tangents must end. | As to the case of the perpendicular, let us suppose DQ (fig. 5) to be the perpendicular required; the subnormal PQ will be d 2 a —" (29) ; now the similar triangles DBQ and MPQ give PD Oe bv: POs that is to say, employing the same denominations as above, fy ydy dy Piyt SS, ori:l: x, therefore, ame ydy Same Pee -an equation of which the same use may be made as in the pre- ceding case. The two solutions just given may be simplified by making the axis of the abscissas pass through the given point D, in a direction parallel to its former position; that is to say, by taking DK (fig. 8) for the axis of the abscissas instead of J7'P, which only requires, if we make KM= z, and consequently, z = y —h, or y==z-+A, that we substitute, both in the equation of the curve and in that of the problem, z +h instead of y. We may also take the point D for the origin of the abscissas. When the curve has a centre, as the circle, the ellipse, é&c. we may always suppose that the point D is in a diameter, and then the solution becomes much more simple. 36 — Differential Calculus. ae 32. We may remark here, that —— expresses the tangent of dy the angle which the curve makes, at each point, with the, ordi- dy nate ; and Pie the value of the angle made by the element of the curve with the axis of the abscissas: For, in the right angled triangle Mrm (fig. 1), we have, supposing the radius of the tables = 1, Nora Ted: Tange eae: r ste ‘dx Therefore tang rm M = —— a rm — dy If then we wish to know in what place, a curve, or its tangent, makes with the ordinate a given angle, or one whose tangent is known, representing this tangent by m, we shall have dx ~ dy so that, determining, by the differentiation of the equation of the ? fee curve, the value of —— iy and making it equal to m, we shall have an equation, which, alter having substituted therein for y its value in terms of w and constant quantities, deduced from the equation of the curve, will give the values of @ answering to the points where the curve makes such an angle with the ordinate ; and if the curve no where makes with the ordinate an angle equal to the given angle, the value or values of w will be imaginary, or the equation will indicate a manifest absurdity. For example, in the hyperbola, having for its equation y2 —2 (ax-+ 7), we should have d x 2y dy 2a} 4x’ which, being equal to m, gives ay y 2 a ialg Re eae whence we deduce i y= ma+2rma; but the equation of the curve gives Y =A/2 (a at 22) 5 therefore ma+22mae=/2(axr+22), or, squaring both members, m? a? + 4m? ax+ 4m? x? oi Oe If itis asked, now, in what place this hyperbola makes with the wo Rectilineal Asymptotes. 37 ordinate an angle of 45°; as the tangent of 45° is equal to radius, we shall have m= 1, which reduces the equation to a+4dav+4a? =2arv+ 22? or — 2e?+2an+a?—0, _ which being resolved, gives e=—fsatr/ta?—ta®?=—fat/J/—}a?; ~ these values are imaginary, and show that the hyperbola, whose particular equation is y? =2(av+22), no where makes with the ordinate an angle of 45°. Although the methods employed conformably to the above remark, and in the resolution of the two preceding problems, seem equally appli- cable to curves of which we have only the differential'equation ; yet on close examination it will be found, that by this last equation the question cannot be perfectly satisfied by the calculus, as in the case where we have the equation in finite terms. Indeed, when the equa- tion, which resolves the question, still contains « and y, after the “AS dx ; ; substitution of the value of Ty the differential equation of the dy curve will not serve to eliminate y, since, by supposition, it contains dzxanddy. And even if it should contain ‘only 2, we could not ‘be sure that this value of x would satisfy the question. 'T’o be sure of this, it would be necessary to substitute this value of x in the equa- tion of the curve, and deduce thence a real value for y. But we cannot, in any way, deduce from the equation in question, the value of y. The only means of resolving these questions, in such a case, is, supposing the curve already constructed, to construct also the equa- tion we have obtained, expressing. the conditions of the question, which gives a second line, whose intersection or intersections with the first, furnish the solution or solutions required. 33. We may also, by the same principles, determine the rectilineal asymptotes of curved lines. A curve has a rectilineal asymptote, when, having some branch which is infinitely extended, the tangent at the extremity of this branch meets the axis of the abscissas or that of the ordinates at a finite distance from their origin. Thus (jig. 6) if, from the subtan- gent pra" we take the abscissa AP =x, we shall have Ade — 1x for the value of AT’, or the distance of the origin A from y the intersection of the tangent. After having calculated, therefore, ydx the value of dy — x, we have only to represent it by a finite quan- tity s; by means of this equation combined with that of the curve, we may eliminate y or x, and we shall have s in terms of 2 or y. Then supposing y or x infinite, if there result one or more finite 38 Differential Calculus. values for s, they will be distances AC from the origin to the intersec- tions of the axis and the asymptotes. But as a single distance does not fix the positions, we imagine, through the origin A, a straight line AK to be drawn parallel to the ordinates, and observe that the similar triangles 7'PM, TAK, Bite TP:PM::TA: AK; that is to say, eee wits: EER ree ek We therefore calculate, in the same way, the value of y — oy and having supposed it equal to a finite quantity ¢, by means of this equation and that of the curve, we eliminate # or y; and, supposing the remaining zor y infinite, the value or values of ¢ which result, will give the distances AK, which, with the distances AC, determine the position of the asymptotes. For example, if the equation of the curve were y? =x? (a-+-2), we should have dy? dy=2rdzx(a+27)+7? dx—=2ardt+3 x? aa: ydu oY? then Mee ee if ere Be dy 2ax+3 2? ; or substituting for y® its value, yd sax? +345 GOs DAR Ft 8 i dy eS Bas +32? "= artB1 2ap3z then supposing « infinite, that is to say, neglecting 2 a by the side of 3 z, we shall have s =o We find, in like manner, ady 2a72 +3273 Sy —2Rar? —3x3 PE eT Tua a Biya a MNT, bay 2 ase AHO which, if we substitute for y its value, is reduced to Qi" 3 eR, 30 /utaee supposing then x to be infinite, we have ¢='1 a. If, the value of s being finite, we find that of anit it would prove that the asymptote is parallel to the ordinates. If, on the con- trary, s being infinite, ¢ were finite, or zero, or infinitely small, the asymptote would be parallel to the abscissas., 34, In all which precedes, we have supposed that the ordinates were parallel, and moreover that they all issued from the line on which the abscissas are reckoned. But it often happens that we make the ordinates issue from a fixed point. Sometimes we take as abscissas the ares of a curved line, and as ordinates, straight or curved lines. But, in general, to whatever lines the points of the principal curve be referred, we always have or may have an equation which expresses the ratio of the abscissas to the ordinates. When we wish to make use of it to determine the tangents or other lines, we must take care that the lines we employ to determine these tangents contain Drawing Tangents. 39 no other differentials than those of the variables which enter into the . equation of the curve. We shall now illustrate this by some examples. 85. Let us suppose, first, that AM (fig. 9), being a known curve of which we know how to draw the tangents, BS were a curve having for its abscissas the arcs AM of the first, and for its ordinates the lines MS parallel to a given line; the ratio of AM to MS being expressed by any equation, it is required to draw a tangent to the point S of the curve BS. We imagine, as before, the infinitely small are Ss, of which the prolongation SQ, or the tangent, meets at Q the tangent MT to the corresponding point M of the curve AM; and, having drawn the line S& parallel to MT or Mm, the triangle Sks will be similar to the triangle QJIZS, so that we shall have - | sk: Sk:: MS: MQ; now if we call the arc AM,2; and the ordinate MS,y; we shall have Mm=Sk=dz,andsk=sm—SM=dy; then dyidv:zy:MQ=le*, ly taking then upon the tangent MT the abs M@ equal to the value ry dx Wace determined by the differentiation of the equation of the curve, we shall have the point Q, through which and the point S, drawing QS, this line will be the tangent required. Let us suppose, for example, that the curve BS is described so that the ordinate MS is always equal to a determinate part of the arc AM; that is to say, that MS is always to AM in the given ratio of ato 6, we should then have x: %::a@:6, and the equation of the curve isby—=az, Differentiating, bd y= adzx, and consequently O tae d x b Egg) 3 Dale hoe MQ= ge but by the equation of the EPMA a dy a b / curve, ae therefore MQ—-x. Thus in all the curves whose parallel ordinates have always the same ratio to their corresponding abscissas, whether straight lines or curves, the subtangent QM will always be equal to the corresponding abscissa AM. 36. When the curve AM, upon which we take. the abscissas, is a circle ; and the ordinate MS is always to the arc AM in a constant ratio, the curve BS is called a cycloid. If the ordinate MS is always equal to AM, it is the common cycloid, or that traced by a point in the circumference of a circle revolving on a plane. If MS is greater than AM, but still having a constant ratio to it, it is called a prolate cycloid ; if, on the contrary, MLS is less than AWM, it is called a curtate cyclovd. 37. If the equation of the curve to which it is required to draw a tangent, instead of expressing the ratio of AM to MS, expressed that of AM to PS; that is, if the arcs AM were the abscissas 7, and the ordinates PMS, or y, were reckoned from a determinate straight line AP; then Su being drawn parallel to AP, the subtan- _ gent PI would be determined on the line AP, in the following man- ner. As the curve AM is supposed to be known, the subtangent and 40 Differential Calculus. tangent to each point of it, are also considered as known; so that, making PT'=s, and T'M=t, we shall have, by drawing Mr paral- , lel to fv Wg and comparing the similar triangles TPM “and Mrm, UP: TM v2 Mire dm: that ispsid.: : Jiri dee therefore Mr = nee == Su, supposing Sw to be parallel to AP. Again; the similar triangles Sus and LPS give su Sor 0 UPS eee and as PS is y, us is dy; therefore sdx sydt Th ee aed omens —. dy t J tdy Then, differentiating the equation of the curve, we find the value of beta will give the value of PJ, tdy se which being substituted in freed from differentials. 38. Sometimes the equation of the curve is not given by the ratio of the abscissas to the ordinates, but by that which each ordinate of the curve is supposed to have to the corresponding ordinates of some other known curve. In that case the tangents are drawn by the following method. Let us suppose, for example, that the curve BMV (fig. 10) depends on the two known curves AZ and CW, by means of an equation between the corresponding ordinates PL, PM, PN, which we shall call respectively x, y, z. Since the curves AI and CW are supposed to be known, their. subtangents PS and ° PR are also considered as known. Call PS, s and PR, s’; and imagine the ordinate pn m/to be drawn infinitely near to PL, and Lu, Mr,no parallel to CPA. The similar triangles LPS, Lul give IPS IPT Gee TNS soe Ta ee 5. therefore Lu = iat Mr. . Now the similar triangles TPM, Mrmgiverm: Mr:: PM: PT; that is, dy: dite :y': PT; therefore PT — ae bi then, if the equation of the curve contained only x and y, we should, by differentiating this equation, have the value of ae which being “A substituted in $Y. 2% ady entials. But as this equation contains 2, y, and z, its differential will contain dw, d y, and dz; we must therefore find the value of dz expressed in terms of d% andd y. Now the similar triangles Non and NPR give No: onor Lu:: NP: PR; that is, ob- serving that while MM increases, PN diminishes, so that its differ- ence No or d z is negative, =a % Nee ier therefore pee S20 r 5) 1G so that, putting in the differential equation of the curve, the quantity “-s2dt —, instead of dz, we shall easily find the value of ee to be s' x dy , would give the value of PT freed from differ- _ Drawing Tangents. 4) sydz - —of the subtangent. As an exam- aG y ple, let us suppose that AZ and CW being any two known curves, the ordinate PM is always a mean proportional between PL and PN, we havex:y::y:2z3 or ez=y? for the equation of the curve BM. Differentiating, it becomes tdztzdz=2ydy; Seas, —, we have substituted in the formula substituting for d z its general value — szdzx -tzdxr=2ydy, ee Pe Vg y da: whence we deduce — = en aythen, Les if Bek becomes dy. z(s'—s) ady i 2s s' y? Mage iM. ——_—_—=—.,, or, substituting for y? its value 2 z, and reducing zz(s'—s)’ ” 8 : : ‘ It is easy to vary these examples by taking any equation we please in terms of x, y, and z. We may, if we please, suppose AZ and CN to be straight lines (fig. 11). In this case, taking always PM a mean proportional between PL and PWN, the curve BAL is a conic section, viz. a parabola, when the point C is infinitely distant, or the straight line CW is parallel to AC; an ellipse, when the two angles HAC and HCA are acute; and, particularly, a circle, when they are each 45°; and an hyperbola, when one of the two angles is obtuse. 39. When the ordinates issue from a fixed point, we take as ab- scissas the arcs of a known curve, which most commonly is a circle; that is, in this last case, the equation expresses the ratio of the ordi- nate CM (fig. 12) to the angle ACM, which that line makes with another, such as AC, given in position; or it announces the ratio of the ordinate CM to the arc OS described with a determinate radius. To draw the tangents, when we have the equation between the ordinate CM and the angle ACM, or the arg OS, we imagine that, for each point M, there is raised upon CM a perpendicular CT’ which meets the tangent 7'M in 7’, then taking the infinitely small are M m, and drawing the ordinate m C, we conceive that with the radius CM, there has been described the are Mr, which may be considered as a straight line ‘perpendicular to Cm at r. As the angle Mm r differs infinitely little from the angle 7'WC, the triangles Mr m and TCM are similar, and give rm (dy): Mr::CM(y): CT; or ae el MET RY. ay: Mire oye CL Lye calling the are OS, z, and its radius, a, the similar sectors CS's and CMr give CS:CM::Ss: Mr; that is, sides dg ae ee a . Substituting this value of Mr in that of CT’, we have CT or the 6 - 42 Differential Calculus. 2dz ‘ : subtangent = on Now as the ratio between 2 and y is sup- a Q posed to be known, it will be easy, by differentiating the equation which expresses this ratio, to obtain the value of a which, being substituted in that of C7’, will give a new expression for CT’, freed from differentials. If we suppose, for example, that the ordinate CM (fig. 18) 1s always to the corresponding arc OS, in the ratio of m to n, that is, if yst.imin, we haven y =m az ;-thenen d= mdz, and conse- quently Hs — = therefore 2d 2 n yidmuyt se ay ny, ady a ma m ” n y x now, by the equation, we have aie a; therefore CT = JS. If, MILT a then, from the point C, as a centre, with the radius C.M, we describe the arc MQ, we shall have C7'= MQ. For the similar sectors COS and CQM give CS: OS: : CM: MQ, that is, dest aye’ MQ=— ; therefore CT’ = MQ. The curve of which we have just been speaking is the spiral of Archimedes. 40. Let us now suppose that, OS ( fig. 14) being a known curve, or one whose tangents may be drawn, the curve BW is constructed with this condition, that C'S, z, and CM, y, have with each other a determinate ratio expressed by a given equation. If we conceive the infinitely small arc Mm, the ordinates CM, Om, and the ares Mr, Sq, described from the centre C and with the radii CM and CS, it is evident that the differentiation of the equation will give only the ratio of dy todz or r m to s q, since y and 2 are the only variables which enter into this equation. But, in order to determine the subtangent CZ’, we must have the ratio of rm tor M. Now r M may be thus determined by means of the conditions of the question. Since the curve OS4s known, the subtangent CQ for each point S is given; now the similar triangles QCS, Sqs give SCO sc qsvgs; calling, therefore, CQ, s, we shall have a:s::dx:gS= sss, but the similar sectors CS q, CMr, give CSO Sig « Mere core: ys: seb, eink i it is now therefore easy to find the subtangent CT, by comparing the similar triangles Mr m and T'C.M, which give syrdx rm:rM::OCM: CT, or dy: * Sle : CT = aoe y To apply this, let us suppose the curve BJM to be constructed by taking SM always of the same magnitude, or equal to a given line 4 Limits of Curves. 43 a, whatever the line OS may be. We shall have then, «+a=y. Therefore d x= d y, and consequently iy ==1,; the subtangent CT' becomes then cT=~.,, which is constructed in the following manner. | Through the point M and parallel to the tangent SQ, we draw a line MN; then having joined the two points S and N, we draw from the same point M a line MT’ parallel to SN, and MT, thus determined, will be the tangent required. For the similar triangle sy CSQ, CMN, give CS:CQ:: CM: CN, ort:s::y: CN=— In like manner, the similar triangles CSN and CMT, give 2 CS: ON:: CM: OT; oz: ::y:0T = 2 We perceive, by these examples, how to proceed in the applica- tion of the same methods to other cases. It is to be observed in conclusion, that when OS is a straight line, the curve BM, which is formed by taking SM always of the same magnitude, is what is called the Conchoid of Nicomedes. Application to the limits of curved lines, and in general to the limits of quantities, and to questions on Maxima and Minima. | du 41. We have seen (32), that jp ARES the tangent of the angle which the curve, or its tangent makes, at each point, with dy the ordinate ; and that = represents that of the angle which the curve or its tangent makes with the axis of the abscissas. To know, therefore, at what point the tangent of a curve be- comes parallel to the ordinates, we must find the values of « and ; dx y corresponding to sea 0, or simply to dw=0; and to find where the tangent of the curve is parallel to the abscissas, we d must suppose — =0, or merely d y =O, and the values of « and y which result, are those of the point of contact. It evidently follows frorn this that, in order to find whether a curve, of which we know the_ equation, has, at any point, its tan- gent parallel to the ordinates or to the abscissas, it is necessary : dx to differentiate the equation, and having deduced the value of dy’ if we make the numerator of this ratio equal to zero, we shall 44 Differential Calculus. have an equation which, with the equation of the curve, will give the value of xv and that of y, which determine at what points the tangent is parallel to the ordinates ; so that if there are more than one of these points we shall have several values for x and y. On the contrary, if we make the denominator equal to zero, this equation, conjointly with that of the curve, will determine the values of x and y, which answer to the points where the tan- gent of the curve becomes parallel to the abscissas. It must however be observed, that although dx is always zero, when the tangent is parallel to the ordinates, as well as dy, when the tan- gent is parallel to the abscissas, still it must not be concluded, when the value or values of x resulting from the supposition that d x = 0, or that d y =O, are found, that the tangent is parallel to the ordinates or abscissas, except when we have not at the same time dv = 0, andd y=—0. To illustrate these rules by a familiar example, let us take the curve which has for its equation y? + a?=3ax—2 a? +2 by—)?, which, on the supposition that a and y are perpendicular to each other, belongs to the circle, the origin of the codrdinates being at the point A. The lines AP ( fig. 15) are x, and the lines PM, PM’ are the two values of y which the resolution of the equation gives for each value of w. If we differentiate this equation, we have 2ydyt+2edx=—3adre+2bdy, dx 2y—2b Fi amr Peau Let us first make the numerator equal to zero; to find the points at which the tangent becomes parallel to the ordinates, we shall have 2 y—26=—0, or yb. Substituting this value in the equation of the curve, it becomes, 6? + «7 = 3a2—20? 1-26? —6?, or x?-—3 an——2a?, which, being resolved, gives c—=2a-+/1a?; that is, ~w—2a, and x==a; which shows that the curve or its tangent becomes parallel to the ordinates at the two points R and R’, which have for the ordinate the line 6, and of which one, R, has for its ab- scissa the line 4Q = a, and the other, R’, has for its abscissa the line AQ’ = Qa. whence we deduce Limits of Curves. 45 : dx Let us now make the denominator of aa equal to zero, to find at what point the curve or its tangent becomes parallel to the ab- scissas. We shall have 3 a— 220, orw=2a. Substituting this value in the equation of the curve, we have y? + 20? = 3 0? —2 a? +2 by — 6?, or y? —2by+ b2=1a?; and, extracting the square root, y— 6== +a; therefore y=b6+44a; that is, y=6b++a4, and y= b—}i a, which shows that the tangent becomes parallel to the abscissas, at the two points Z'and 7’, which have as their common abscissa the line AS'= 2 a, and of which, 7” has for its ordinate ST’ = b+ 23a, and T', the line ST=6—+ a. The points Q and Q’ are called the limits of the abscissas, be- cause, between Q and Q” for each abscissa AP, are corresponding real values PM and PM’ for y; while between Q and A, and beyond Q with reference to A, there is no point of the curve, so that if x be supposed smaller than AQ ora, or greater than AQ or 2a, no real value can be found for y. Indeed, if in the equation we substitute for x a quantity a— gq smaller than a, or a quantity 2 a+ q greater than 2 a, on resolving the equation, the two values of y will be found to be imaginary. In like manner, if through the point A we conceive AL/ to be drawn parallel to the ordinates, that is, to the axis of the ordinates ; and if through the points J’ and 7’, the lines TL, T’L/ be drawn parallel to the abscissas; the lines AL = ST = b— $a, and AL’ = ST =b+ da, are the limits of the ordinates ; for it is evident that there can be no ordinate greater than AL/, nor smaller than AL, the tangent being supposed parallel to the abscissas. In fact, if in the equa- tion of the curve, a quantity, b—3a—q smaller than 6—42 a, be substituted for y, it will be found, on resolving the equation, that the values of x are imaginary. ‘The same thing will happen, if instead of y, the quantity 6+-1a-+q, greater than 6+ 44 be substituted. 42. The ordinate SJ” is the greatest of all those which termi- nate in the concave part RT’ FR’ of the circumference. The or- dinate S’7' is the least of those which terminate in the convex part; and the ordinates QR, QR’ are, at once, the least for the concave part, and the greatest for the convex. 46 Differential Calculus. 43. Thus. the same method serves at the same time, Ist, to assign the limits of the abscissas and ordinates ; 2d, to determine in what cases the tangent becomes parallel to the abscissas or to the ordinates; 3d, to find the greatest and the least abscissas or ordinates. 44. Now in whatever manner a quantity is expressed algebra- ically, the algebraical expression which represents it, may always be considered as the expression for the ordinate of a curved line. et AAD sae) Sh For example, if id Aaa is the expression of a quantity y : 7 42 xX (a—z which we call y, in which case we have y= Eheim we may a consider this equation as that of a curved line whose abscissa is 2, 2 —_— wv and ordinate, y. If then the quantity ima SNe may, in a cer- tain case, become greater or smaller than in any other case (which is called being susceptible of a maximum or minimum), it is evident that we must pursue exactly the same method as above, that is, differentiate the equation, and having deduced from it the value of a7 } a make the numerator or denominator of that value equal to Zero. 45, It is to this that the method which is called that of maxima and minima is reduced. ‘This method is one of the most useful in analysis, and has for its object to find, among several quantities which increase or decrease according to a certain law, that which is greatest or least; or, in general, that, among all similar quanti- ties, which possesses certain properties in the highest degree. We shall now give some examples from Geometry and Algebra. Mechanics will hereafter furnish some at once more curious and more useful. 46. Let it be proposed to divide a given number t+ a into two parts, such that their product shall be a maximum. Call w one of the parts ; the other will be a—2, and the product will be repre- sented bya w— x?. Let this product = y; we have y = ax — 2? ; by differentiating, therefore, d y=adx—2axdvx, and conse- 1 5 Ia gas if we make the numerator equal to zero, Fr dz quently 7 = y t See the Introduction. ae Maxima and Minima. 47 we have 10, which is absurd; consequently if there be a maximum, it will be shown by making the denominator equal to zero; let therefore a— 2 = 0, whence = 1! a, which shows, ‘that among all the different ways in which a number may be di- vided into two parts, the product of the two parts will be greatest, when each is one half of the number.+ 47. When, as in this example, we have the algebraical expres- sion of a quantity, we may dispense with making it equal to a new variable y; we have merely to differentiate, and make the nume- rator or denominator, if the differential is a fraction, equal to zero. Thus, in the same example, we merely differentiate a « — v2, and, making the differential a d x—2ad« equal to zero, we have ad x—2 xd «x=O0, whence We deduce, as before x = +} a. 48. 'To take a more general question, let it be proposed to di- vide a known number a into two parts, such that the product of a determinate power of one of the parts by the same or another power of the other part, shall be the greatest possible. Let x represent the first part, and m the power to which it is to be raised ; the second part will be a—., and if n designate its power, the product in question will be a@(a—wz)". If we differentiate this — product and make the differential equal to zero, we shall have ma™—ld x (a—ax)"—na"d x (a—x)"—!=0. Dividing the whole by a"—'d a (a—«)"—1, we have m (a —x,—nx=O0, or ma—mxc—nx=0, m a which gives == ———.._ lf we suppose, for example, that it is m—+-n required to divide a into two such parts that the square of one multiplied by the cube of the other part, shall be the greatest product possible ; then m= 2,n=3. We have therefore 2a that is, that one of the parts must be 2 of the number or quantity proposed, and the other be consequently three fifths. What has been said above in relation to figure 15, shows that a quantity may become greatest among all quantities of the same xv obo a; + It is here to be observed that those considerations, employed in the solution of this question in the introduction, do not here appear, but are contained in the calculus, being principles which are at the foundation of this mode of considering questions. 48 Differential Calculus. kind, in two different ways ; either when increasing first, like PM’, it diminishes afterwards, or when, like P/M’, it continues to in- crease, until it stops suddenly on becoming Q’f’; but in this last case, it is at once the greatest of all the ordinates which termi- nate in the convex part, and the least of those which terminate in the concave part. In like manner a quantity may become least of all those of the same kind, in two different ways; either, like PM, by diminishing at first to increase afterwards, or, like P/M”, by diminishing until it suddenly stops, and then it is at once a minimum and a maximum ; it is a minimum with reference to the branch W#T’M’”, and a maximum with reference to the branch MTM”. 49. So that, to ascertain whether a quantity is a maximum or a minimum, or both, we must, supposing @ to mark the value of 2, corresponding to the maximum or minimum, substitute suceessively instead of x, in the quantity proposed, a + 9, a,anda—gq. If the two extreme results are real and less than the middle one, the quantity is a maximum ; if, on the contrary, the intermediate result Is the least, the quantity is a minimum; finally, if of the two extreme results, one is imaginary and the other real, the quantity is at once a maximum and a minimum. 50. When, in the determination of a maximum or minimum, the value found for the variable renders that of the maximum or mini- mum negative, we must conclude that the maximum or minimum which it indicates, does not belong to the question under considera- tion, but that it answers to a question in which some of the con- ditions are of a contrary character. If, for example, it were required to divide the line AB (fig. 16) at the point C, in such a manner that the square of the distance AC from the point A, being divided by the distance BC from the point B, gives the least possible quotient; then, calling the given line AB, a, and the dis- tance AC, x; the remainder CB is a—a, and consequently the ’ Dp ‘ quotient is = we have 2edx(a—«x)—!+ x? dx (a—x)—* =0, Ke 2rd2% I kiko a—z ' (a—z)? ” or 2axdx—x? dx =O, or (Qa—v)cx=—0; which gives either « = 0, or 2 a— x =0;; the first value indicates 3 differentiating this quantity or «? (a—)—', Maxima and Minima. 7 49 a minimum, which is evident without calculation. As to the sec- 2 ond, which gives x= 2a, if we substitute it in we find 4a? AN —~, or—4a. The minimum, therefore, does not belong to the acl present question. But if we examine the value of x2 a just obtained, we .see that the point C cannot be between A and Bs but that the question will have a solution, if it be required to find itin AB produced beyond B with regard to A. In that case, if we call AC’, x, the distance BC’ will not be a—a, but x—a, and the quantity under consideration will be ara which, being differentiated and made equal to zero, gives Pn: heat Ae Tate win (Me ce SRG or, after making the reductions, x2dx—2ardx—0O0, which gives «7 = 2a, as before; but this quantity, substituted in Ss , changes it into 4 a. There is therefore a minimum for this case. If the denominator « — a of the -differential be made equal to zero, we have x—a, which indicates a maximum ; and indeed, when « =a, the quantity becomes infinite. But it has nevertheless the true character of a maximum, for whether we suppose « greater or less than a, it gives a less quantity than to suppose v= a. 51. When the expression of a quantity of which we, wish to know the maximum or minimum, contains any constant multiplier or divisor, we may suppress this multiplier or divisor before dif- ferentiating; for if we suppose that oa represents a quantity which is to be a maximum, or a minimum, a and 6 being constant, ad —" must be = 0; but since a and 6 are not zero, d y must ne- cessarily be =0; the conclusion is therefore the same as if the maximum of y only were required, that is, the same as if the con- stant divisors and factors were suppressed.. This remark serves, in many cases, to simplify the calculation. 52. Let it now be proposed to find, among all the lines which may be drawn through the same point D given im the angle 7 50 Differential Calculus. ABC ( fig. 17), that which forms with the sides of the angle the least possible triangle. Through the point D let DG be drawn parallel to the side AB, and supposing EF’ any straight line drawn through the point D, let fall upon BC the perpendicular DK, and from the point £, where EF' meets AB, let fall upon BC the perpendicular EL. The line BG is considered as known, as well as the perpendicular DK;; let therefore BG—=a, DK=b, and let BF, the base of the triangle BEF, be =x. We see that, from a certain point, the more BF increases, the greater will be the triangle. If, on the contrary, BF’ diminishes, the triangle will diminish also, but only to a certain point; for if B#' should become nearly equal to BG, the straight line EDE would be nearly parallel to AB, _ since it would nearly coincide with GD, in which case the triangle would be very great. There is-then a certain value of BF’, which gives the smallest triangle possible. In order to find this value, we must discover the general expression of the triangle BEF. Now the similar triangles BEF, GDF' give GBs: BE > DE : EE; and the similar triangles DKF and ELF, give TON Diags Ot aed 8Y Sd oy Pi therefore GF: BF':: DK: EL; that is, eesiah obi bb oe ? t—a, ph had aii Pie ed £1 therefore the surface of the triangle BEF, which is ae will bas 1 pes me | By 2 i : be SRE Oey aE We must therefore differentiate this quantity, and make the numerator or denominator equal to zero, or, since we may suppress the constant factor 3 6 (51), we need only 72 3 and, not to repeat an operation which has been already performed (50), we shall find «= 2 a; if, therefore, we take BF =2a—2 BG, the lne FDE, drawn through the point D, will give the triangle FBE for the minimum required. | 53. Let it be proposed to find, among all the parallelopipeds of the same surface and the same altitude, that which has the greatest capacity. differentiate the quantity Maxima and Minima. 51 Call / the altitude and c? the surface of the parallelopiped, x and y the two sides of the rectangle which serves as base. The whole surface is composed of six rectangles, of which two have each « for their base and fA for their altitude, two others have h for their altitude and ’y for their base, and the two last have « for base and y for altitude ; so that the whole surface is expressed by Zhe+2hy+2xry; ° that is, we have Q2he+2hyt Qey—c?. The capacity or solidity is hwy. Since then it must be te great- est of all those of the same surface, we have hady+thydr=0, or, what comes to the same thing, ecdytydx=—0d. But the equation 2hat2hy+2xy=c?, which shows that the surface of all these parallelopipeds is constant or the same, gives 2hdx+2hdy+2edy+2ydx=—0 Substituting, therefore, in this equation, the value of d «& found in the other, we have, after making the reductions, y=; the base therefore must be a square. To find the value of the side, we substitute for y its value w in the equation 2he+Rhy+-Qey=—c?, _ which thereby becomes 4h x + 2a? —=c?; which, being resolved, gives ©—= —h + 4/h? +1 c2, whose root @==— h — /h?+ 2 e?, being negative, does not belong to the present question; thus the true value of « is pT na ge 54. If it is now asked what must be the altitude A, in order that the parallelopiped may have the greatest solidity among all those of the same surface ; we observe that since, the altitude being A, the base must be a square, this solidity will be expressed by Aw? ; considering then A and «@ as variable, the differential of ha? must be equal to 0; we have therefore Qhaeadx«+ta2zdh=0,or2hdx+tecdx=0. But the equation 4ha+ 2a? = c?, which indicates that the sur- face is constant, gives for its differential, 4hdxt+4edh+4ede=—0; substituting, therefore, in this equation, for d/, its value found from the equation 2h d«-+ «dh =O, and making the reductions, we have h=w; the parallelopiped sought must therefore be a 52 Differential Calculus. cube, since its altitude fh is equal to the side «x of the square which serves as base. To find now the value of the side of this 1 cube, we must substitute for A its value x, in the equation Ahn - 2 070%, which thus becomes Age aeicetcasor Ow 4 eC", which gives 7 = Therefore, of all the parallelopipeds of the same surface, that which has the greatest solidity is the cube which has for its side a line equal to the square root of the sixth part of that surface. 55. Let it now be required to find, among all the triangles of the same perimeter and same base, that which has the greatest surface. Let a be the base AB, and c the perimeter of the triangle ABC (fig. 18). Let fall the perpendicular CP, and call AP, « ; CP, y; we shall have | PB=a—2; AC H=vAvW/2x2 4+ 42; and CB = \/j2tG@=0?2- Then the perimeter will be r a/ x2 +y2 + A/y2+ (a—2)2 TA=6, ay and the surface a a Now, by the conditions of the question, we must have d —* =0, or dy. But, if we differentiate the expression for the perimeter, we have xdx+ydy —dzx(a—z)+ydy nee Vaatye Ver ener which, d y being = 0, is reduced to ca A (a—a)dax ' V2ty Vytu—oe ” or, dividing by d x, and freeing from fractions, w/jF (a= as — (a—2) o/porae = 0. Squaring the two members, performing the operations indicated, suppressing the terms which cancel each other, and reducing, we arrive at this equation, w? =(a— x)? =a? —Qaxr+2?; whence « == +3 a, which shows that the triangle must be isosceles. Maxima and Minima. 53 To construct it, we raise a perpendicular from the middle of AB, and having described from the point B as a centre, with a radius equal to half the excess of the perimeter c over the base a, an are cutting that perpendicular in C, and drawn CB and CA, we have the triangle which has the greatest surface among all those of the same perimeter, and the same base. 56. If it is now required to find, generally, among all the trian- gles of the same perimeter, that which has the greatest surface, it must be observed, that whatever be the base, we see by the pre- ceding solution that w must always be equal to half of it, that is, that whatever be the values of a, vis always—=1a. This being the case, the equation: which expresses the perimeter will be reduced to / ia? + y? +/pa?+y? +a, or Q2/ia? +y2 =c—a; squaring and finding the value of y, we have Hin C4 — 128 C9 4 4A The surface a of the triangle will therefore be 2 & c2 —Qac IN aay Wael Since then it must be the maximum among all those of the same perimeter, whatever may be the base a, we must make equal to zero the differential of Oy Pen iy. areas, or of a a/c? —2 ac, a being considered as variable. We shall have then be ee 1 d.arn/c2@—2ac—d.a(c?—2ac)? 3b —(c? ac)? da—cada(c?—2 ac) cada = t2—2ac — ==. = 0. dd afc: rh: A/c2—2ac Making the denominator to disappear, ? ls da(c2? —2ac)—cada=0O; whence a = therefore the base a must be a third of the perimeter; and as we have already found the triangle’ to be isosceles, it follows that it must be equilateral. Therefore of all the triangles of the same perimeter, the equilateral is that which has the greatest surface. 54 Differential Calculus. 57. In these two solutions, we have not made the denominator © equal to zero, because, in the first, that would have given an imaginary value for x; and in the second, we should have found == ic, which would have no better satisfied the question, since, if the base were half of the perimeter, the two other sides would be confounded with the base, and the triangle would .be zero. In future, whenever the supposition of the numerator or denominator made equal to zero, would lead to no admissible solution, we shall, to avoid useless investigations, pass over it without notice. 58. In the last question but one, we were able to determine, among all the parallelopipeds of the same surface, that which has the greatest capacity, only by first considering parallelopipeds of the same altitude. In like manner, in the last question, we have found among all triangles of the same perimeter, that, which had the greatest surface, by first resolving the question for triangles of the same base. It is usually more simple to proceed in this way ; that is, to resolve the question by making the least possible number of quantities vary at once, and afterwards successively treating as variable each one of the quantities which have been considered constant. If, for example, it were required to divide a given number into three parts, in such a manner that the product of these three parts should be the greatest possible; calling x and y two of these parts, and a the given num- ber, the third part will be a— «—y, and the product of the three will be xy (a—a—y), of which the differential must be made equal to zero. But, instead of considering x and y as both variable at the same time, we shall differentiate, considering z only as varia- ble ; we have then, aydx«—2rydzx— y2 dx=0, whence we deduce t= (a—y). The product « y (a—x—y)is thuschanged into } y(a—y)?. We now differentiate, considering y as variable, and have tdy(a—y)?—Zydy(a—y), which we also make equal to zero, and have dy (a—y)? —2ydy (a—y)=0, whence we deduce Tatas” i i wherefore x, and a— x — y are each equal to 1 a. | 59. We may also, if we please, make all the variable quantities vary together, then, collecting all the terms which are multiplied by the differential of the same variable, make their sum equal to zero, and do the same thing with regard to the differential of each variable. Thus, in the last example, we should have axdy+aydz—2rydxr1— 12 dy —2x ydy—y? dx=0D, whence we obtain, by making equal to zero the sum of the terms affected by d x, and that of the terms affected by dy, aydi—2rydr—y2dr=0, and axrdy—2xrydy— x2 dy =0; Maxima and Minima. 55 or, dividing the first by y dz, and the second by zd y, a—2r1— y= 0 a—2y—r1=0), equations from which we immediately conclude that x= 1 a, and that y = } a, as before. It is easy to perceive the reason of this process, if we observe that the only condition to be answered is that the whole differential be equal to zero. Now this condition can be answered generally in one of two ways only, either by supposing each of the two differentials dx and dy equal to zero, which indeed satisfies the equation, but makes nothing known, or by supposing that the sum of the terms multiplying dz, and of the terms multiplying dy, are each zero, which is precisely what we have done. 60. When the conditions of the question are expressed by several equations, we must, before applying this rule to the differential equa- tion which is to determine the maximum or minimum, deduce from the other differentiated equations the values of the differentials of as many variables as there are equations besides that, and substitute them in that equation ; the rule is then to be applied as if there were that equation alone. Thus, in the example already given of the greatest parallelopiped, we had this equation, 2Zha+Qhytery=e?, and the condition that hx y was to be a maximum. If then h, z, and y are to be considered as varying at once, the first equation, being differentiated, gives 2Qhdtt2rdht+RBhdytRydh+2rdy+2ydit=O0, and the condition of the maximum gives hiadythydz+rydh=—0. From the first we find pupae dlr 8K tcl Sana eee d x zy | substituting this value in the second equation, we have, after the usual reductions, har®dythy?dx—ry?dxr—x2 ydy =0. We may now put equal to zero the sum of the terms multiplying d z, and that of the terms multiplying dy. We have hy2?—ay? = 0 or h=z, and hx? —7? y= Oorh=y; and since h = 2, we have alsoy—v2; the three dimensions 27, y, and h are therefore equal, which agrees with the former solution . and putting these values in the equation 3 2hitQhyt2e2yrt=C?, we have 6 h? = c2, whence as in the former solution. 61. We may not only make the quantities vary successively, or all at once, but we may take as constant any function whatever of 56 Differential Calculus. these quantities, provided that the number of these new arbitrary conditions, united to that of the conditions of the question, be not greater than the number of the variables 2, y, z, which enter into the question, ‘This remark may be of the greatest use In many ques- tions, especially when there are radical quantities. For example, let it be required to find among all the quadrilaterals of the same perimeter, that which has the greatest surface. If from the angles C and D (fig. 19) we let fall the perpendiculars CF’ and DE upon the side AB, and from the point D draw DK parallel to AB; then, call- ing AE,s; DE,t; AF,u; CF,x; and BF, y; we have by the property of right-angled triangles, DA = 24, DC =WN/64+024 6202, CB=N 23 F933 then if the perimeter be equal to a, we have V8tB+V 6+? Fe—p +Vetyetuty =4 Again, the surface ABCD is equivalent to the trapezoid : DEFC— the triangle DAE + the triangle CFB; that is, apep=(F") sty —F+ This being laid down, it would be necessary to differentiate the two preceding equations. But the radical quantities would render the subsequent operations very complicated. ‘To avoid these diffi- culties, we suppose, at first, that the three radical quantities are con- stant, which gives 1 any 1 d (Var pEa)= a (52-4 #2) = 3 (92-442) (9242) FEE A/52 + t2 which being =0, because the quantity was considered as constant, we have Ist, sds+tdt=0. We find also for the differential of the second radical 2d, (s+ u) (ds+du) + (%—?) (dt— dt) = 0; and for the third od, adut+ydy=0. The equation of the perimeter being differentiated on the same supposition, gives (since the differential of each of the radical quan- tities is zero), . Ath, du+dy=0. The expression of the surface being differentiated and put equal to zero on account of the condition of its being a maximum, gives, 5th, Saupe ae «)+(t-+-2)(dst+du)—tds—sdt+ady+ydxr=0, uHe Pade! udeudy ced semen ayy deme Co 1 The first gives d s = a the third d 2= ira and the fourth du==—dy. Substituting these values in the second and fifth, we have, after all reductions, —(édt+sdy)(u+s)r¢—(ydy+ xdt) («x —t)s=0, a Maxima and Minima. 57 and suxdt—suydy—s? ydy—tsrdy—x? tdt—sy2dy=0. If we deduce from this last the value of d ¢, we shall see that all the terms of its numerator are affected by s, and that, by substituting it in the preceding, all the terms will also be affected by s, wherefore s=0, which shows that the angle DAB must be a right angle. This being determined, the equation of the perimeter is reduced to t +-r/u2+ (@— H2 $r/e2+y2 f-uty=a; and the expression of the surface becomes Casta Let us now differentiate, supposing only the two radicals constant, we have udu-- (x —t) (dx —dt) = 0; tdxtydy=0; dttdutdy=0; u(dttdzr)+(¢+2)dutady+ydxr=0. The second equation gives xadx dy=— ; y the third, ryt sys tdx—ydu dt—=—du—dy = (substituting for d y) -—_————_; y these values, substituted.in the first and fourth equations, give yudu-+ («4 —?t) (ydx—adz+ydu)=0), and u(xdx—ydu+ydz)+ (t+72) ydu—2r2drty? dr= 0. Now, if, from one of these we find the value of x, and substitute it in the other, we shall have an equation of which all the terms will be affected by y, and which consequently gives y = 0, and shows that the angle CBA must also be a right angle. This being the case, the equation of the perimeter becomes t+ J/u2+(e@—p)2+¢+u=a, and that of the surface = (¢+ 12) X _ differentiating, therefore, supposing the radical constant, we find udu +(x —t) (dx—dt)=—0O, dttdu+dzr=—9, u(dt+-d2)+ (t+-7)du= 0. The second gives dit = —dzr—du, and substituting this value in the two others, udu--(#—t)(2dz+du)=0, and ; —udu-+(t-+27)du= v. This last gives du=0; whence the preceding becomes (x— t) 2dz=0, whence x= t. 58 Differential Calculus. This being determined, the equations of the perimeter and surface are reduced to 2¢-+-2u—a, and surface = ¢ u. The lines AB. AD, DC, CB, are all equal; and since the angle A must be a right angle, the other angles being such, the quadrilate- ral sought is a square. We might have arrived at this property more readily, but that was not our principal object. We wished to show how the liberty of treating certain quantities as constant, may, in many cases, much facilitate the operation, and this example was well adapted to the purpose, as, without this artifice, the calculation would have been very complicated ; similar methods may be applied to other polygons, and it will be found, that, in general, of all the polygons of the same number of sides, that is greatest which is a regular polygon. Whence it follows, that, of all figures of the same perimeter, the circle is that which contains the greatest space. Of Multiple Points. 62. We have examined what takes place when one of the two dif- ferentials dz or dy, or which is the same thing, when the numerator oie co or denominator of the fraction ae becomes zero; and we have found that one of these two cases always exists whenever there is a maximum or minimum. But it may be asked what takes place, if the ; dz denominator and numerator of the value Ty become zero at the Vy ; dx same time, and to what, in that case, is the value of rp reduced 2 J . In answer to these questions, we first observe, that when we differ- entiate the equation of a curve, as there are only terms multiplied by dz and those multiplied by dy, we may, calling A the sum of the former, and 6 the sum of the latter, represent the differential equa- , ee B tionby Adzx+ Bdy—O0. This equation gives Aa id 2 DOW, ag in order that A and B should become zero at the same time, they must have a common divisor, which, becoming zero when «x and y have certain values, renders B and A equal at the same time to zero. : : ‘ 1(a— 2)? For example, in the curve which has for its equation y2= a! ; a ay zay dy (a— 2x)? —2a(a—xz)’ or, substituting for y, its value, ie 2a(a—z) ae dy 3, Se 22) 2 ee (a— x)? —2x(a—xz)’ a quantity which becomes 8, when x==a; but we see, at the same time, that a— x is a common divisor of the numerator and ‘the de- we have aa nominator, and that the value of i is reduced to y Multiple Points. 59 Qa aa dx a = +_ ——— dy . a—s3x’ which, in the case of «== a, is reduced to + 1, that is, in this ex- ample, the value of @ = =F 1. We may indeed proceed thus; but this expedient is not always : dx : y sufficient when the value of as contains more than one variable, ¥ nor when it contains radical quantities, even if it have but one varia- ble. It is therefore necessary to give an easier and more general method. But it is first necessary to show the nature of the points of curved lines where this singular case occurs. It takes place, as we shall soon see, at multiple points, that is, at those points where several branches of the same curve meet. 63. Let us conceive that SOMZM'ON (fig. 20) is a curve of which two branches, at least, intersect at the point O. It is evident that to each value of AP, or x, within a certain interval, there cor- respond several values of y, as PM, PM’, and those belonging to the branches which intersect each other, become equal at the point of intersection O. In like manner, AZ being the axis of ordinates, to each ordinate AQ within a certain extent, there must correspond several abscissas QN, QN’, QN”; and those belonging to the branches which inter- sect each other, must become equal at the point of intersection. If, therefore, we represent by @ the value of z, and by 6 that of y, which belong to the multiple point, the equation of the curve must be such, that when we substitute in it @ for x, we shall find for y as many values equal to 6 as there are branches passing through the multiple point; and when we substitute b for y, we shall find a like number of values for x equal to a. It follows from this that the equation must be such as to admit of being reduced to this form, (1—a)" P-4(7-a)"(y-b) F(x)" yb)? F" (ra) yb). BP es 2 + T' (y—b)™ = 0. m indicating the degree of multiplicity of the point in question, and F, F' &c., T, designating quantities composed in any way of 2, y, and constants, or, as they are called for conciseness’ sake, functions of z, y, and constants. Indeed, it is evident, that if we make «=a, the equation which is then reduced to (y — 6b)" JT’ =O, will be divisible, m times, by y — b, and will consequently give m times the equation y — b= 0, or y=b. So, if we make y=), the equation which is then re- duced to (x —a)™ /’=9, will be divisible m times by x—a, and will consequently give so many times the equation x— a = 9, or a =a; which cannot happen unless the equation is reducible to the form above. ‘t Let us now conceive this equation to be differentiated m times in succession, making also dz and d y to vary, that it may be the more general. If we reflect on the principle of the differentiation, we shall readily perceive (as will presently be demonstrated by an example), by 60 Differential Calculus. Ist, that the last differential equation will be the only one, in which there are any terms not aflected by y—b,orx—a. Therefore, whenever there is a multiple point, the first, second, third, &c. differ- entials of the equation, must, when instead of « and y, are substitut- ed their values a@ and 6 answering to the multiple point, be all made to disappear, except those, the degree of whose differential is marked by m; 2d. It will also be perceived that, in this last differential equation, the terms affected by dd x, ddy, d® x, &c., and by all dif- ferentials of higher degrees, will each have as factor some power of a—a,orofy—6; and consequently, that these differentials will disappear at the multiple point. From these principles it follows, Ist. That at the multiple point we cannot have the value oft expressed otherwise than by 9, un- less by the last differential equation, since all the other differential equations being then rendered equal to nothing, the factor of dz, as well as that of dy, becomes equal to zero. 2d. That as this last differential equation contains neither dd, dd y, nor any higher dif- _ ferential, it might be derived immediately from the differentiation of the proposed equation, m times successively, supposing da and dy constant.. 3d. That in this last differential equation, dz and dy will be marked by the degree m3; and, that consequently if we divide by dy’ which will serve to find the tangents, which the different branches passing through the multiple point, have at that point. To illustrate and confirm this by an example, let us take the curve which has for its equation a(y — b)? —x(x— a)? = 0. If we differentiate this equation m times, that is, in this case, twice, we shall have, first, Zady (y —b) —dz (t— a)? —2adz(x—a)=0; secondly, 2Zaddy (y—b)+2ady?—ddz (x— a)? —2d x2 (x —a) —2addx(x—a) —2 dx? (x— a) —2rd2xz2?=0. In which we see that if we substitute a for z, and 6 for y, the first differential equation disappears, and in the second, the terms affected by ddz and dd y become nothing, so that it is reduced to 2ady?—2adzr?=—0. But if, instead of considering dx and dy as variable, in the second . differentiation, we had considered them as constant, we should have had, dy™, we shall have, by resolving the equation, m values of 2ady2—2dzx2 (2— a) —2 dx? (x—a)—2rdzxz2—0), which, on substituting a for 2, is also reduced to 2 ad y2—2 adx?2=0, and gives oP —=+ I, a result which indicates that there are two tangents at the point where x= a, and y==); this point is there- fore a double point, and the value Us of the subtangent, becoming dy Multiple Points. 61 then = -+ J, these two tangents make an angle of 45° with the or- dinate. ‘his will be confirmed by the description of the curve by means of its equation, which, giving y = b+ (t— a) JZ a? shows that the curve has two branches perfectly equal and similar, which interest each other at the point O, where =a, and y= b. Its figure is such as is represented (fig. 20). 64. It is easy to determine by these principles whether a curve, of which we know the equation, have multiple points or not, what they are and where. We must differentiate the equation ; put equal to zero the multiplier of da and that of dy. These two equations will determine the value or values of x and y, according as there are one or more multiple poimts; but to be assured of the existence of this multiple point, we must examine whether these values of « and y satisfy the proposed equation. Then, to ascertain the degree of multiplicity of the point or points found, we differentiate the equation anew, but, for the sake of greater simplicity, considering da and dy as constant. If, when the values found for x and y are substi- tuted in this second differential equation, all the terms do not disap- pear, then the point found is only double. In the contrary case it is more than double. We proceed therefore to a third differentia- tion, still considering dx and dy as constant: and having substituted the values of z and y, the point will be triple if all the terms do not disappear, otherwise it will be at least quadruple. We continue to differentiate and substitute, until we arrive at a differential, of which all the terms are not made to disappear by the substitution of the values of x and y. For example, if it is required to find the multiple points of the curve, which has for its equation y* —azry?+ br? —0, we differentiate this equation, and obtain Ay®dy—2arydy—ayrdr+3br- dx =0. Making the coefficient of dx and that of dy equal to zero, we have Ay? —2axy =D, and 8b2? — ay? = 0. The first of these two equations gives y =0, or 4y2? —2ax=d. The value of y= 0, being substituted in the equation 3622—a y2=0, gives 3b 7? = 0, or x=0; now the proposed equation is satisfied by substituting 0 for 2 and y; therefore the curve has a multiple point corresponding to += 0 and y = 0, that is, at the origin. ar As to the value 4 y2 —2az=0, or y? = oe if we substitute it bo in 3b c2 —a y2 = 0, we have 3 6b x? — Sree whence we de- ae , P fee. 0, Ob f= 64. but the first, viz. * = 0, gives y= 0, which 62 Differential Calculus. indicates the same point as before; from the second we deduce ae but these values of z and y? do not satisfy the equation Bie ee 12D. proposed. ‘There is therefore no other multiple point than that found at the origin. To ascertain its multiplicity, we differentiate a second time; we have 12 y? dy2—2ardy? —Raydudy—2aydzdy+6brdz?=0, of which all the terms disappear on substituting for x and y their values, zero. ‘Therefore the point is more than double. We pass then to a third differentiation ; we find 2 ydy? —Qadidy? —2Radrdy?—Radidy?+6bdxv=O0; or, substituting for 2 and y their values, zero, 6bd2z?—6adidy? =9; as this third differential does not disappear, the point in question is a triple point. To determine its tangents, we divide this equation by 6 6 andd y°, dx ad and have Pre ae, aa); which gives dx dx? a dx —_—_ = i Sea st | Pe a Ey 0, and dy P Ot + J¢. d x The first value, 2s = 0, indicates that one of the tangents to the y multiple point is parallel to the ordinates, that is, that one of the branches touches the axis of the ordinates; since the multiple point is also at the origin. The two values dx ; iy Toe dy b? show that the two other branches make with the axis of the ordinates each an angle of which the tangent is Jt and that they extend on different sides of that axis. We may know the figure of this curve by resolving its equation, which gives y= + \faive A/ a2 —4b2, taking for a and 6b two numbers at pleasure, and successively giving to x several values both positive and negative; it will be found to be such as is represented in figure 21. Finally, when we have determined a multiple point by the opera- tions given above, we must not always conclude that all the branches, which are considered as passing through that point, are visible. It may happenthat the equation which furnishes the tangents, has imagi- nary roots; and then there are so many invisible branches. ‘The points where this happens, are sometimes detached from the course of the curve to which they nevertheless belong; they are then called con- Multiple Points. 63 jugate points. But whether detached or not, they are not the less considered as having the number of branches indicated by the degree of their multiplicity: the curve to which they belong is an individual of a more extensive family, in which all these branches are visible; but they become invisible in this, because some one of the constant quantities which enter into the equation common to the whole family, becomes zero in the particular case of this individual curve. _ It is thus that in the curve, which has for its equation m (y — b)? — «(x —a)? =0, (and of which the curve in figure 20 is a particular case, and which it becomes when m = a) the leaf, which this curve has, no longer exists when @ is supposed = 0, which reduces the equation to m(y—b)? —2«? = 0. The two branches OMZ, OM'Z, which were above the point QO, will no longer exist in this last, or at least will not be visible ; for we may suppose that they are still there, regarding a@ not as absolutely zero, but only as infinitely small. There will nevertheless be two tangents, in fact; but this example enables us to conceive how cer- tain branches may disappear. ' 65. Several useful consequences may be drawn from what has just been said on the subject of multiple points. 66. Ist. When a fractional algebraical expression, into which there enter one or two variables, is such, that by substituting in it for each of those variables, certain determinate values, it becomes 8, we shall obtain the value which this expression ought then to have, by differ- entiating separately the numerator and the denominator, as many times in succession as is necessary, in order that they may not be- come zero at the same time ; and in this differentiation we may treat the first differences as constants. Indeed we may always consider Mai ai any fractional algebraical expression —- containing two variables, for A instance, as being the value of da (x and y being these two varia- fi) d " bles); that is, we may always suppose ts re and consequently dy Adx—Bdy=O0. But since, by the supposition, A and B become zero at the same time, when 2 and y have certain values, it follows from what precedes that to obtain the value of —-, we must differ- dy entiate this equation, considering dz and dy as constant, until we arrive at an equation which does not disappear by the substitution of the values of « and y. Now these successive differentiations give dAdzt—dBdy =0, ddAdzr—dd Bdy=0O, d? Adt—d? Bdy =—0, whence we deduce dt dBdzx ddBdx @B. dy dA’dy ddAdy dA’ oy “ ere ‘Fa weer 7: a ee . re oe . . le my N ae * 64 Differential Calculus. RPE that is, we must differentiate separately the numerator and. denomi- nator as has been directed ; and the last of these equations will give dx , the value of Fast % dy h he 67. 2d. When an equation, which contains several variables, is such, that to certain values of all those variables except one, there— shall correspond a certain number of equal values of that one, and this shall hold true of all the rest, we shall find these values by differ- . entiating successively so many times less one, the equation proposed ; considering, in these differentiations, dz, dy,dz, Sc. constant, if t, y,z, &c. are the variables; and putting equal to zero the multi- pliers of dz, dy, dz, &c., those of dx?,dudy,dzdy, &c., and so throughout. If all these equations correspond with each other and with the equation proposed, the values of 2, y, z, &c., which they give, will be the values required. 68. It may be observed with regard to multiple points, that since the values of x and y, found by putting equal to zero the coefficient of dx and dy, must satisfy the finite equatfon proposed, we cannot, unless the equation is given in finite terms, or may be reduced to them by the methods of the integral calculus, ascertain, by calcula- tion alone, the existence of those points. . Of the visible and invisible points of inflexion.t 69. It sometimes happens, that a branch of a curve, after having been concave towards a certain point, becomes afterwards, in its pro- gress, convex towards it (without discontinuing its course). Such is the curve represented in figure 22. The point O, where this change takes place, is called the point of inflexion. T’o determine these points it must be observed that the tangent to the point O must be a common tangent of the branch OB and the branch O 0, which meet at the point O; if, therefore, on each side of O, we take two arcs, equal or unequal, but infinitely small, the tangent must be the prolongation of each arc, so that the two little arcs must be in the same straight line. This. being laid down, we draw the ordinates MP, OQ, mp, and call AP, z, and PM, y. Wethen have Mr=dz, Or=dy, and supposing OM and O m to differ infinitely little from each other, we have | Or =d(¢+dt)=dxe+ddz,andmr=dy+ddy; since then MO and Om are in a straight line, the triangles Mr O and Or’ mare similar: and if we suppose, as we are at liberty to do, that the arcs MO and Om are equal, these triangles will be also equal, and we shall have Mr = Or’, and Or=~mr’; that is, dzx=dz+tddz,anddy,=dy--ddy, therefore | ddx=0, andddy=O. { These are sometimes called points of contrary flexure. oe aes Te Points of Inflexion. 65 In order, therefore, to ‘determine the simple points of inflexion, we must differentiate twice successively the equation of the curve, and consider in the second differential equation, dd z= 0, and ddy =0. Now it is evident that, in that case, this second equation is the same as if we had differentiated, considering d x and d y con- stant. If therefore, from the first differential equation we deduce the value of dz or dy and substitute it in the second, we shall have an equation, which, being divided by dy? or d x?, will contain only z, y and constants, and which, being compared with the equation of the curve, will give the values of z and y, which correspond to the sim- ple point of inflexion. Let us take, as an example, the curve which has for its equation 23 —b y2. = a8. We shall have 3272dx—2Rbydy=—0), differentiating again, treating d and d y as constant, we have 6x dx? —2bdy? =0; the first of these gives 342 dx dy=-5 ak substituting this value in the second, we have 4 qd 72 6 x Pt Pee ita tt Soi | b2 y2 or . Aby?—323= 0. 3 ‘ From this we deduce y? = pate which, being substituted in the equation of the curve, gives 28 —2 73 a3, or 1? = 443, 3 therefore BaD, and consequently 3 as 3a y ae i Fane oe eddy These are the values which determine the point of inflexion. Let us take, as a second example, the curve which has for its equation y=at (r— ays, we have dy —%(t—a)* d1, differentiating anew, considering d y and dz as constant, we find 6 $ gp dn? — fs (t—a) *d2? = =; (List Oh, therefore d x= 0; now the first differential equation ene. dy =2(t— a): °dz, becomes 2 SEED 0, (x — a)’ dy=2dz; (x—ayé 9 dy= 66 . Differential Calculus. 2 but since d z = 0, we have (x —a)° d y = 0, which gives either 2 . dy =0, or (tx —a) °=0; but as it is not possible that we should 2 have, at the same time dz and d y = 9, it follows that (rx — a) ° =0 is the true solution, which gives x = a, and consequently y = a. 70. We may here observe, Ist, that as we find d«—90, the tan- gent at the point of inflexion of this curve is parallel to the ordi- nates. Qd. If the curve should have several points of inflexion, the final equation would give several values of 2, that is, it would exceed the first degree. This takes place in curves which have a serpentine course, as in figure 23. 71. If we conceive that the two points of inflexion O and O' (fig. 23) approach each other continually, and are at last infinitely near each other ; then if we represent, as above, the two infinitely small arcs OM and Om, and the two other infinitely small arcs O'W’ and O'm’, on each side of the points of inflexion O and O’, the two sides O m and M’O’ will be, or may be supposed to be, one on the other; and since, at the point of inflexion, MO is in a straight line with O m, and MO’ with O' m’, there will then be three small con- secutive arcs in a straight line. This being laid down, let Mm, mm’, m'm" (fig. 24) be these three infinitely small arcs. Let fall the ordinates MP, mp, m’ p’, mp", and draw the Jines Mr, mr’, m'r’, parallel to AP. Call AP,z,and PM,y. We have Mr=dzi,rm=dy,mr'=dxzt+ddz,rm =dytddy, mr! —=dxtddz+d?2z,r'm' =dytddytd y. Now the three triangles Mr m, mr! m‘, m' rm’ are similar, since the sides Mm, mm’, m'm’' are in a straight line; if therefore we suppose these sides equal, which we are at liberty to do, the triangles will be equal. We shall have then dx=dzr+ddzdy=dy+ddy,dr+ddzt=dzr+ddzr+ d3z, dy +ddy = dy +ddy+d°y; dat Uddy =), da, y= 0, If, therefore, we differentiate the equation of the curve three times successively, considering as variable da, dy, ddz, ddy, and atfter- wards put 0 for each of the quantities ddz, dd y, d® x, d® y, each of the three equations resulting from those differentiations will hold true. Now it is evident that they then become the same as if we had differ- entiated three times in succession, supposing d x and d y constant. By a similar course of reasoning it may be shown that if three consecutive points of inflexion come to unite, there will be, at the point of union, four elements in a straight line, and it may be proved in the same way, that if we differentiate the equation four times in succession, supposing dx and d y constant, the four resulting equa- tions will hold true; and so of others. Therefore, if, from the first differential equation, we deduce the value of dz, and substitute it in all the others, we shall have, after that is, Points of Inflexion. 67 dividing the second by d y?, the third by d y?, and so forth, so many equations, which must hold true conjointly with the proposed equa- tion, in order that there may be one, two, three, &c. inflexions united, If, therefore, from two of these equations we deduce the values of x and y, these values, substituted in the other equations, must satisfy them. When there are only two inflexions united, the inflexion is invisi- bles it becomes visible when there are three ; in general, the inflex- ion is visible or invisible, according as the number of inflexions united is odd or even. ‘Therefore if £ = 0, represent generally the equation of a curve, it will be necessary, in order that there should be m points of inflexion united, that, differentiating KE, m- I times, on the supposition that d # and d y are constant, all the differentials ddz,ddy, d*x, d* y, &c. to that of the degree m + 1, inclusively, should be zero; and the inflexion will be visible or invisible according as m is odd or even. 72. Hitherto the ordinates have been supposed parallel. If they issue from a fixed point, the following is the method of determining the points of inflexion. Imagining the two infinitely small consecu- tive arcs, which, at the point of inflexion, must be in a straight line, we draw the ordinates (fig. 25) CM, Cm, Cm’, and describe the arcs Mr, mr’, which may be regarded as perpendicular to Cm and Cm’. This done, it is easy to see that the angle 7’ mm’ differs from the angle r Mm by the anglem Cr’. For we have Cmm'+rmM — |e Ly 3 Cmr' +r! mm! + 98° —r Mm = 180° ; therefore r’' mm —r Mm= 90° —Cmr'; but by the triangle Cr’ m right-angled at 7’, we have 90° = Cmr'+mCr'; therefore rmm—rMn=mCr'. If we draw the line mn, making the angle m' mn=m Cr’, the angle nmr’ will be equal to m Mr, and consequently the triangle tmr’ will be similar to the triangle m Mr. Calling CM, y, and Mr, dx, we have ' mr=dy,mr'=dx+tddz,andmr=dy+ddy; call Mm, ds, mm' will be =ds-+ dds. Describe from the point m with a radius mm, the arc mn: the sectors Cmr! and nm m' will be similar, and will give Canin 7) 3 ME in 5 that is, ytdy:dxr+ddzr::ds+dds:m'n, which, omitting the quantities which may be neglected, gives dsdx ee ; but the triangle m/ t n, which is similar to ¢ r’ m, will be so likewise to mr M; we have, therefore, Mr: Mm::m'n:m’t; that is, dsdu d s? 3 ¥y d s2 mt diz «ades.: : therefore rt=dy+ddy— 68 Differential Calculus. Now the similar triangles Mr m and m r’t give MH Tithe ee Pe bel : d 2 or dx:dy::dz4+ddx:dy+ddy—~——; therefore dxd s? ACA RASTA Gtr tiem ro ect eats oes ' 2 or Lary tet NS th ee ET this is the formula for finding the simple points of inflexion, when the ordinates issue from a fixed point; it becomes that for the paral- lel ordinates, when we suppose the point Cat an infinite distance, - dads? , and dz and d y constant; for then the term ———— must be reject- % y ed, as y is infinite. In the application of this formula, it will always be more simple to suppose d x constant, which reduces it to 2 se NS © 3 and care must be taken to treat d x as constant in the differentiation of the equation of the curve: bunt as the lines d x are arcs described with a variable radius, while, if the ordinates issue from a fixed point, they are referred to arcs described with a constant radius CO, as has been observed (39), we must be careful to substitute for d x its value, which will always be easily found, by observing that the sectors CS's and CMr are similar. It is not necessary to give the formulas for the other points of inflexion, on the same supposition, although it would not be difficult. Observations on the Maxima and Minima. 73. Let F’ be a function of one or more variables, and let it be susceptible of a maximum or minimum. If we imagine it to repre- sent the ordinate or abscissa z of a curve, that is, if we suppose «= F, it follows from what has been said on the subject of the points of inflexion, that if it be true that d F'or dx becomes zero, whenever there is a maximum or a minimum, the converse is not always true ; for we have seen a case (69) where d x was found = 0 at the point of inflexion, that is, the tangent became parallel to the ordinates ; but it is evident that neither the ordinate nor the abscissa is then either a maximum or minimum. What then must be done to ascer- tain the existence of a mazimum or minimum? We must differen- tiate the quantity several times in succession, considering the first differentials of each variable as constant; and, if the values which the variables have at the point of the mazimum or minimum sought, cause d F’, and dd f'to disappear, but not d? F, there is no mazi- mum in the curve which has for its equation x= F'; but there is a visible point of inflexion; so that the quantity # has no maximum or minimum. But if d° F disappears, and not d* F’, there will be a mms Ne, Cusps. 69 maximum or minimum. In general, there will be a maximum or minimum, if the last differential made to disappear is of a degree marked by an odd number. Of Cusps of different species, and of the different sorts of contact of the branches of the same curve. 74. When two branches of a curve come in contact, they may either have their convexities opposed to each other, as in figure 26, or have their concavities opposed, and thus one embrace the other, as in Sigure 27, and in both cases it may either happen that they continue their course on each side of the point of contact, or that they stop suddenly at that point, as we see in figures 28 and 29. In this last case, the point of contact is called a cusp; that represented in fig- ure 28 is called a simple cusp or cusp of the first species; that in figure 29 is called a beaked cusp or cusp of the second species. If more than two branches unite, their different varieties may be found at once at the same point, and there may also be found an infinite number of others; for example, the branches may, at the point of contact, undergo one or more inflexions. It may happen that an inflexion and a cusp occur united, so as to seem to form only a single cusp. In figure 30, if the branch LBD, which forms at /, with the branch EC, a cusp of the first species, had a point of inflexion at B, and if the point of inflexion B should be infinitely near the point E, we should only be presented with the appearance of a cusp of the second species. ‘These varieties may be infinite, especially if we consider that several branches ‘may touch at once. We shall not undertake to give the character of each; we shall only observe, that whenever several branches of a curve touch each other, it may be ascertained by the following facts. Ist. This point being multiple must have the conditions enumerated in art. 63. 2d. ‘The equation which serves to determine the tangents of the multiple points, must then have as many equal roots as there are branches which touch, since there must be so many tangents united. ‘Thus, for cusps of the first species, there ought to be the conditions common to double points, ; eas and the equation which gives the tangents. thereof, or which gives PL ought to have two equal roots. As the consideration of these points is not sufficiently useful to engage us in the details which their in- vestigation would require, the subject will be no farther pursued. On the Radii of Curvature and the development or evolute. 75. If upon each of the points M, m, m’, &c. of any curve line (fig. 31), we conceive the perpendiculars MN, m n, m'n’', &c. to be raised ; the consecutive intersections NV, n, n’, will form a curve line, to which has been given the name of evolute, because if we consider it as enveloped by a thread ABN, which touches it in its origin B, then, upon unwinding the thread, the extremity A traces the curve AM. ~ 70 Differential Calculus. In fact, in the development of WV n, for example, considering WV n as a small straight line, the thread MNn describes, about the point 7, as a centre, the arc Mm, to which it is necessarily perpendicular, since the radius of a circle is perpendicular to its circumference. 76. The curve AM being given, if we wish, for any point M, to determine the value of Mn, which is called the radius of curvature, we observe that Mn is determined by the concourse of two perpen- diculars infinitely near each other, MN and mx. We therefore im- agine ( fig. 32) two consecutive arcs Mm,mm’, infinitely near, and differing infinitely little from each other, which may be considered as two straight lines; we also imagine MN perpendicular to Mm at M, and m N perpendicular to mm’ at m. Then, in the triangle NM m, right-angled at M, we shall have 1:sin UNm::m Nor MN: Mm, or, because the angle MN m is infinitely small, 1: MNm:: MN: Mm; therefore MN = Pasi ; MN m but, if we produce Mm, we have Nmu=NMm+ MNm; be- cause Vmu = Nom +m mu= NMn+ MN mm, and taking away Nmm' = NM m, there remains m mu= MNm; therefore MNES m! mu If we draw Mr and mr’ parallel to AP, it is easy to see that the angle u mr’ being equal to m Myr, the angle umm’ is the quantity by which the angle m Mr is diminished, or the differential of the angle m Mr, which must be taken negatively here, where the curve is con- cave, and where it is convex, must have the sign plus; we have Mm therefore MN = d(r Mm)” We have therefore to find the ex- pression for d (r Mm). Now the tangent of r Mm =o, and dz -cosr Mm = re ds designating the arc Mm; but we have seen (23), that z being any arc, we had dz = cosz? d (tang z) ; therefore -2 d(r Mm) =F 4 (54); ds2 whence, d s8 MN = ps SEaeriiG EE pas 52 = ds? a(5") Tike a(5%) This is the formula for finding the radius of curvature when the ordi- nates are parallel. 77. If the ordinates are supposed to issue from a fixed point ( fig. 33), we have, as above, and for the same reasons, MIN = hes, mm U. Radu of Curvature. 71 but m/mu is no longer = -F d(r Mm), but the value of this angle is found as follows. Describing the ares Mr, mr’, we have mmu=rmu—r mn; but it was shown (72) that r/m wu differs from r Mm by the angular quantity m Cr’ or MCr, for the two last quantities differ from each other only by a quantity infinitely small compared with them; there- fore r’'mu= MC r+ r Mm; therefore, since mimu=r'mu—rimm, we have mmu= MCr +r Mm —r' nn! = MCr—d (r Mm) Now, calling Mr, dx, we have L:vsin MCnor. UMCr:: CM: Mr; that is, 1: MCr::y: dz; therefore MCr= “ and since dy HM) =o — aid 3 (5%). ade ax? dy we have 5 Steiger ELE a (3): and if the curve were convex, we should find, in like manner, 2 dt ws dy F therefore i 7 geste ke take de Mi pele icc ESS i a Career yds (3) TY 78. To give an application of these formulas, let us suppose that the curve AM (fig. 32) is a circle, having for its equation y2 = 2at—7r?, we shall have y= 4/2 ax—zx2; therefore adzx—xdx A/ 2 axr—xr2 ; dy and consequently, ad Ws = = te nT 29 Vat ot AJ? a ax— ( ) ‘ dy a—z at) —az dtr and — =e therefore'd ERA rt EN i = el a ereiore (2 (2ar—22)8 the formula which belongs to the present case, in which the curve is ds% concave and the ordinates parallel, is —___.., nara) which is changed into 72 Differential Calculus. a? dxz3 cy) eS 2 (oa ee a2 dx din le piandets 9) that is, the radius of curvature is always of the same magnitude and equal to the radius of the circle; so that the evolute is reduced toa point, which is the centre; and this agrees with what we know of the circle. : Let us take, as a second example, the parabola which has for its Ix]co 4 4 equation y2 =a 1%, or y==r/ac—=a" «x7, we shall have a —-1 dy=a? wes ans therefore ds VIF Fd p= VEP par de ad II + z 4Azta —3 d —3 = ad ¢ c=d7 “dx |4a-a,and= =}a°a *; 3 therefore d (Gynt ali Fhe d 58 the formula ———- becomes then a; 3 gu ?7dx8 (4a-+-a)? ne = - 38 reo | i= an Q eee as Q d22X4fatxr*dx a 79. The radii of curvature serve to measure the curvature of a curve at each point. Since, in the development of the element nn’ of the curve BN (fig. 31), the thread traces the small are m m’, this arc has the same curvature as the circle which has for its radius the line mn. ‘Thus when we have the expression for the radius of cur- vature, we know, for each point, the radius of the circle which has the same curvature as the curve has at that point. And as the cur- vature of a circle is greater in proportion as its radius is less, that is, as the curvatures of circles are in the inverse ratio of their radii, it will be easy to compare the curvature of a curve at any point with the curvature of the same or another curve at another point. Thus, if we wish to compare the curvature of the parabola at its origin, with that of the same curve when the ordinate passes through the focus, we observe that at the origin s = 0, and that the abscissa corresponding to the focus is } a (Ap.172). Putting therefore, successively, for the expression of the radius of curvature, 2 = 0, and t =a, we haved a, andaa/?e ; the radius of curvature is there- fore 4 a at the origin, and a@a/2 at the extremity of the ordinate which passes through the focus. Therefore the curvature at the first of these points is to the curvature at the second, as an/z® :fa,or:: 2/2 : 1. Radius of Curvature. + ae Since the radius of curvature AMZN is nothing but the thread which is supposed to have enveloped or been wrapped about the curve BN, it follows that it is equal in length to the arc BN, plus the part AB, by which the thread exceeded the curve when the development be- gan, that is, plus the radius of curvature at the origin A. Therefore the curve BW is rectifiable, that is, we may assign the length of each of its ares BN. Remark. 80. The points of inflexion were determined on the supposition (69) that the two elements of the curve, near the point of inflexion, were in a straight line. From this supposition it seems to follow, that at the point of inflexion the radius of curvature must always be infinite, because the two perpendiculars upon the two consecutive sides must be parallel. There are, nevertheless, several curves which, at the point of inflexion, have the radius of curvature equal to zero; the parabola, for example, which has for its equation But it must be observed that nothing in the supposition that was made, determines of what magnitude those two consecutive elements are. Now if they be each reduced to a point, they will be never- theless in the same straight line, and the two perpendiculars falling one on the other, will meet at the very point from which they issue. And this is what happens in those curves where the radius of curva- ture is zero at the point of inflexion. For the curvature then being infinite, the two consecutive elements are each confounded with the tangent infinitely less than in any other case, and must consequently be considered as two points united. The two elements may therefore be in a straight line, without the radius of curvature being infinite ; but we see from this, that the radius of curvature is, at the point of inflexion, either infinite or nothing. 10 *e ELEMENTS OF THE Ti Nye tY Go Reals 5 AE Cel. Ly Wie Explanations. 81. THe method known by the name of the Integral Calculus is the reverse of the Differential Calculus. Its object is to ascend from differential quantities to the functions from which they are derived. | There is no variable quantity expressed algebraically of which the differential may not be found; but there are many differential quantities} which cannot be integrated ; some, indeed, because they could not have resulted from any differentiation; such as the quantities rd y,x«dy—ady, &c., others, because means have not yet been discovered of integrating them, and among these last are some of which we may despair of ever finding the integral. As, however, great use may be made of those which we know how to integrate, we shall endeavour to show the methods, and shall afterwards show what is to be done with reference to those which refuse to be integrated. We shall begin by explaining cer- tain modes of expression which will hereafter be used. We call a function of one or more quantities, any expression, into which those quantities enter in any way whatever, whether mixed or not with other quantities which are considered as having determinate and invariable values, while the quantities of the function may have all possible values. Thus, in a function, we consider only the quantities which are supposed variable, without any regard to the constants which may occur in it. For example, U v,a + ba?, r/aan+ap, &e. are functions of x. + By Differential quantity is meant here, not only such as results from a differentiation, but in general, every quantity affected by the diflerentials dz, d y, &c. of one or more variables. Semple Differentials. 15 By algebraical quantities are meant those, of which the exact value may be assigned, by executing a determinate number of algebraical or arithmetical operations, other than those which depend on logarithms. On the contrary, we call non-algebraical or transcendental, those quantities, for which we can assign only proximate values, or values by means of approximations; loga- rithms are of this kind, and an infinite number of others. To indicate the integral of a differential, the letter f- is written before this quantity ; this letter is equivalent to the words sum of, because, to integrate, or take the integral, is nothing but to sum up all the infinitely small increments which the quantity must have received, to arrive at a determinate, finite state.* Of the Differentials with a single variable, which have an alge- braical integral ; and first, of simple differentrals. 82. Fundamental rule. 7 integrate a simple differential, we must, Ist, increase the exponent of the variable by unity ; 2d, di- vide by this exponent thus increased, and by the differential of the variable ; that 1s, divide by this new exponent multiplied by the differential of the variable. The reason of this rule is evident from the principle of differen- tiation (10). As the object is to find the quantity which must have been differentiated, it is evident, that we must make use of opera- tions the reverse of those employed for differentiating a quantity. This is made more clear by examples of the application of the rule. | Qritidz Ob Yi f ] co Bid Gomi) 2? hey oe re We see, now, that d (x?) is actually 2x7 da; and In like manner, 2 $41 5 = ar dx at i 5 aX dw == = —- = fan ; Y ($+ 1)dz 3 | ———- * In reading, it is convenient to call f, the integral. Thus J 22d x is read, the integral of 2x d x. 76 Integral Calculus. TE | —2 f=- =f a fii) papal iy d x == eee, 3+ 1) dx Q42 In general, m being an exponent, positive, negative, integer, or fractional, we have fra ee cammrid ay rae (m-+-1) da m +- 1° We have no need of this rule to find the intecral of da or a dx, which, we see at once, must be 2 for the first and a @ for the second. But, if we wish to apply the rule to them, we ob- serve that the exponent of x in these differentials is zero, and that they are the same thing as 2° d w and a x° d &, of which the integral, agreeably to the rule, is a eect diet dat OG (Of ljdz (0+ 1)dz “yO and a a. There is only one case which eludes this fundamental rule ; it is that in which the exponent m has the value —1 ; for then the : et il My see Ua t integral becomes — mS Fp S77 2, quantity unassignable because infinite; for, if we conceive the denominator, instead of being zero, to be an infinitely small quantity, it must evidently be contained an infinite number of times in the finite quantity a, and consequently the fraction must be infinite. We shall explain, hereafter, the reason why the calculus gives in this case an infinite quantity ; meanwhile we observe that the proposed differential, pg) aL , ; adz , ; ax” dx, which in this case is aa—'da, or nysesy i the differen- tial of a logarithm. It is the differential of ala or of la, as may be easily seen by differentiating (26). If there were a radical in the simple differential, we should substitute, for the radical, a fractional exponent. ‘Thus, 3 2 Arai fade Vv fardo=sac’. Remark. 83. We have seen, that when, in the quantities to be differen- tiated, there occur terms wholly constant, these terms do not appear in the differential. When, therefore, we go back to the integral, we must take care to add a constant quantity to the result of the integration. ‘This quantity will always have an indetermi- Complex Differentials. T7 nate value, as we have no other object than to find the integral, that is, to find a quantity, such that, by differentiating it, we repro- duce the differential proposed ; indeed, axe tb} aac} on-pily ae m—+-1 C being any constant quantity, have equally for their differential the quantity a wd x, whatever value is given to C. But when the integration is performed with a view of satisfying a given question, then this constant quantity has a value, determined by the state of the question. ‘This will be shown hereafter; but, in future, care will be taken to adda constant quantity to the result of each integration ; and that it may be known as such, it will be always designated by the letter C. + 6, Of Complex Differentials whose integration depends on the fundamental rule. 84. Ist. We may integrate by the preceding rule, any quantity, in which there occur no powers of complex quantities, and no complex divisors, except such as are constant quantities. Thus the entire integral of bx2dz +edzx, C being the sum of the integrals of each of the terms, will be axv* 6x3 —— a aa U Gi m ae 3; sary Mae av? dx + In like manner, Sf (a0 dx + jas (axtda tba de) ax eh ax b Th ah eager Gira Baap S 85. 2d. Even should there occur powers of complex quantities, they may still be mtegrated by the fundamental rule, provided they be not found in the denominator, and provided, also, their exponent be a positive whole number. For example, (a + 6x?)8 d x may be integrated by the preceding rule, by actually raising a-+ bx? to. the third power, which would give, (Algebra, 141), a? +30? ba? +306? ct + 63 x®, and consequently bdz a* 78 Integral Calculus. S(a4b02)3 deaf a®dax+3 a2bar2dx+3 ab? x4 dx+b2 x dwt = fardet$3o?ba7da-+f sab2n* du-Lf beasda = ate | oe he aT sd 86. As there is no complex quantity raised to a power indicated by a positive whole number, which may not, by the preceding rule (Alg. 141), be thus reduced to a finite series of simple quantities, we may always integrate any complex quantity which does not contain any other complex parts than powers whose ex- ponent is a positive whole number. ‘Thus, if we had to integrate gxda(a+ 6x7)? + avi dx(c+ex2+ fu), we should develope, by the rule already cited, the value of (a+ 6x?)?, and multiply each term of the result by g x? d x; we should in like manner develope the value of (c + ex? + fa*), and multiply each term of the result by a? «7 daw; we should then have only to integrate a series of simple quantities, by the fundamental rule. 87. We must however except the case, in which, some one of the exponents being negative, it should happen, after the develop- ment and multiplication, that the exponent of the variable in some of the terms became ==—1,; but we should then integrate by logarithms. or example, if we had li (a+ bw*)?, orax*dz(a+ ba?)?; we should bie it into ax da (a2+2abx2 + 6? x4), which becomes aa dx+2a? ba-t1detabl?rda, of which the two terms a? x-?da anda b? «da have, for their err 8 ees h2 integral, Se + A a, but the term 2 a2 6a-1d 2, which is the same as 2 a? b— , is (27) the logarithmic differential of 2 a2 bl x; so that fax dala + bx?)2 — — 88. 3d. If the differential Raha ee even contains a complex quantity, raised to any power (whether its exponent be positive or negative, whole or fractional), we may still mtegrate, o ap Binomial Differentials. 719 if the whole of the terms multiplied by this complex quantity, taken together, be the differential of the complex quantity con- sidered without its total exponent; or if it be this differential multiplied or divided by a constant number. We have only, in that case, to consider the complex quantity in question as a single variable, and apply, word for word, the fundamental rule. For example, ¢ dx (a-+ 6 x)? falls under this case, because g dw is the differential of a-++ 6 x, multiplied by S, which is a constant quantity ; so that, to integrate it, we write la(a+ba«ypr! ¢da(atbap ae rs 1 Nett i el nr Ce _edz(aporyt Gg _ g(abbnet ~@thbde * Fis + For, if we differentiate this quantity, we find again Lad 2 (ae Oe In like manner, if we examine the differential Of getich SAEs —=(a2dx+2anrd cx) (aw a2) 2 NV a2+22 we shall find that it is integrable, because a? rie xe dv is the differential of a x-+w«?, multiplied by a constant quantity a. By applying the rule, therefore, we have 1 f (a et li pd (av-+2?) * __ (@dr+2axrdz) (av 22)? fie, ane PMT TE LST T ye C=2a(an+a?)?+C. The only case to be excepted is that, in which the exponent of the complex quantity should be —1; when we should integrate by logarithms, as will be seen hereafter. Of Binomial Differentials which may be integrated algebraically. 89. We understand by a binomial differential, one in which the complex quantity, however complex, is some power of a binomial. Thus gv dx(a+b n2)? is a binomial differential. ‘The same may be said of g wvdu(a-+ 6a" )?, which may represent any binomial differential, because, by ¢, a, b, m,n, p, we understand any imaginable numbers, whether positive or negative. 80 Integral Calculus. There are no means of integrating generally every binomial differential. But it is apparent, from what has already been said, that we can integrate a binomial differential ¢ a d x (a+ 6 a” )?, in the two following cases. Ist. When p is any positive whole number, whatever may be the exponents mand n (85), with the exception of the case mentioned an art. 87. | Qd. When m, the exponent of x out of the binomial, is less by unity than n, the exponent of x in the binomial; that is, we may integrate generally, g a” —!da(a—+ 6a” )P , whatever value n and may have, except the case in which p—=—1. In fact, g a*"daxw is the differential of a+ 6a", multiplied by =, that is, by a con- stant quantity ; we fall then upon the case mentioned in art. 88 ; and may consequently integrate by the general rule, considering a-+ ba” asa single quantity. Besides these cases there are two others, which may be com- prehended in one, and which include the preceding ; they are the following. 90. Ist. We may integrate any binomial differential, in which the exponent of x out of the binomial, being increased by unity, may be exactly divided by the exponent of x in the binomial, and give for a quottent a positive whole number. The process to be followed in this case, to integrate, and also to show that the prin- ciple is general, consists in making the biomial quantity, without its total exponent, equal to a single variable, and expressing the proposed differential by means of this single variable and constants. ; which may always be done by proceeding as in the following ex- amples. Let it first be proposed to integrate g w* dw (a+ 6b x2 >, We see that this differential may be integrated, because the exponent of «x out of the binomial, viz. 3, being increased by unity, gives A, which, divided by 2, the exponent of x in the binomial, gives for a quotient the positive whole number 2. We make, therefore, a+ 6x? =z. From this equation we perma sO A 6 that «* dx, which precedes the binomial quantity, results, except- ing its constant multiplier, from the differentiation of «*, the square deduce the value of «x?, which is #2? = We _ observe Binomial Differentials. 81 — ada y L of x? ; we therefore square the equation 7? — ae and find Zimme OOO Le oo ( ) ; differentiating, we have b 4xt*dx—2 (F"\F b ie and consequently Ley pel —)5- (z—a) dz | b 25 2 b2 hi Substituting for «2 d xand (a+ 6 x2), their values in terms of z, in the expression g x? d xv (a+ 6 wx?) = we find g.(z—a) dz x s gotla, gaz dz PA Se Trak Tee obaae. a ates Then 44] 4 i z eat d z az dz cridax b 2)5 — Swat Pci N ee GA Js (a+ bx?) :f- 2 b2 9 h2 ~ (FF 2) 208 (F120? gx td Z a C me 2h a eee 4) uy gn il 5 5 Ci =o (Gr? 32) + & substituting then for z, its value a + 6 x?, we have Sex dx (a+b22)> — ae (a4bx2)>t? | > (ab 12) +C. 91. We proceed in a similar manner in every other case sub- ject to the same conditions. Let us take, for example, gadx(a+db x) 5, which must be integrable, since the exponent 8 increased by 1, that is, 9, being divided by 3, the exponent of x in the binomial, gives a positive whole number. We therefore make a +- 62° =z; and find x? == = en and as «° dx, which precedes the binomial, is, excepting its constant factor, the differential of x9, in order to —a } z-a\ ai and have x =( a : then differentiating in order to obtain x® d x, we have 11 . 6 ye have x® we cube the equation 7? = 82 Integral Calculus. 9r8dx=3 (<7): on —- 9), ; \gpebbeiiin and, consequently, ENON (oat ye wd. = ( ; ) 35 The differential g w® dx (a+6x°) ~ will be therefore changed zZ—a Lm test into g (ea ay ad * ==, by expanding poere fg lads z—a\? mates E dz 2eaz! 3 dx garz %dz b -) 3 b8 353. ee 32) oe of which the integral is 3— 2 2—2 > ,l— oz _ 2g az ais lr ch dh Zz prey 365(3—%) 369@—3) | Bd>(—1—8) . . Hie 1-2. which, separating the common multiplier Sn oe g 1—2/ <=? 2az (a? 357 2 (sS3—-s5+7os)+¢ 1s eS Behe aces Pees er +30) +6 or finally, substituting for z its value a i bx*, we find fe a dax(a+tb ny 2 Sa bas) date Ae cbae ee (a bas hacen q2 Cote = £.(apba%) $3 2 — 4 (a+b) 043.02 + 6. Such is the method to be pursued whenever the exponent of x out of the binomial, being increased by unity, and divided by the exponent of « in the binomial, will give as quotient a positive , 1s reduced to whole number. 92. 2d. Although a binomial differential quantity may not fall under the case of which we have just been speaking, it often hap- pens that it may be reduced to it by means of a very simple artifice, which consists in rendering negative the exponent of « in the binomial, when it is positive, and rendering it positive when it is negative. In order to this, we must divide the two terms of the binomial by the power of « in the binomial, and multiply the quantity out of the binomial by this same power of a raised to the power indicated by the total exponent of the binomial. For ex- ample, in order to render negative the exponent 2 of x, in the binomial gut du (atb«?), Binomial Differentials. 83 ae ’ . a 5 we divide a+ b wx? by w?, which gives g x‘ d x (4 +. b ) 7 Or guv*dx(ax-?-+56)*; but as the quantity 2, by which we have divided, is considered as raised to the fifth power, since it is under the total exponent 5 of the binomial, we must, in compensation, multiply the quantity out of the binomial by (#?)>, that is, by x19, which gives g w1*d x(a a—-? +6)?°. By this preparation, many binomial differentials which would not otherwise be comprehended in the preceding case, will be re- duced to it. For example, if it were required to integrate EE aa OF idan (0? mp2) aes (a? 422) we perceive that the exponent of x out of the binomial, that is to say, 0, being increased by 1, which makes it 1, cannot be ex- actly divided by 2, the exponent of xin the binomial; but we must not thence conclude that the proposed quantity is not integrable ; for, if we render negative the power of x in the binomial, by writing a? (x?) — 2 da (a2 0 i +1) 7 3 which is reduced to a2 a3 dha ae 4 hee we see that — 3 increased by 1, that is, — 3-1 or — 2, being divided by —2, the exponent of w in the binomial, gives as a quotient a positive whole number ; we suppose, therefore, z—l 2 2 GF aa2o- La z. whence,.o. = aia and as w—-*° dw is, excepting that it wants a constant multiplier, the differential of x—?, we differentiate, which gives —2x-8 d i fess dz 9 a? whence we deduce «2 dx = — ae . The differential a? x > dx (a? x-? + 1)-2, _ is therefore changed into —a?dz -2 — Pal oye aD gs ae tals of which the integral is Petre a 2g) Cor z 2+ C; or, substituting for z its value 84 Integral Calculus. ft =Wet1) tte (a2 +02)? 1 xr gerry il Mh Oayare a Ie Thus the process for integrating is the same in this case as in the preceding. 93. We have supposed hitherto that the power of « was found in only one of the terms of the binomial. If it should occur in each, we should reduce the quantity to a form in which it would appear in only one of the terms, by dividing the binomial by one of the two powers of x occurring in its terms, and multiplying the quantity without the binomial by the same power raised to the power indicated by the exponent of the binomial; and that for the reason just given (92), for rendering the exponent negative. Thus, if it were proposed to integrate 2 ae dx (an+22) 2, we should change it into a? x1 (x)? d 2 (a+ x), by dividing the binomial by x, and multiplying the quantity without by @ rais- ed to the power—4, which is that of the binomial. This quan- tity is reduced to a? eda (a+ nr) If we should apply to it the rule of the first case (90), we should not find it integrable, but by rendering negative the exponent of x in the binomial, we have a? a2 (w) 2 dx(ax a3 +1) v3, or a? x "da (az eRe which (92) is integrable. Making, therefore, a a-1 + 1 =z, we z—l_ .,. ae have a—1 —~-__-; differentiating, we have a dz nh — —2¢ =. OF it hep te : a a therefore arm dee (aa? A ayieant a dee gic d ia of which the integral is : + C,or—2a 23 + C, or, restoring the value of z, Quadrature of Curves. 85 dee =) = —2a(ar-} a as pct 8 =—2a |— 4140. If, upon the examination of a binomial differential, it is found to be comprehended in neither of the two cases above mentioned, it is useless to expect an integral purely algebraical. As to polynomial differentials, that is, those in which the com- plex quantity contains three or more terms, they are integrable in the cases mentioned in art. 85, &c. ‘There are also some other cases in which they admit an algebraic integral, but they are very few and rarely occur. We shall not, therefore, occupy ourselves with them at present. We shall hereafter show the method of discovering those which are integrable, and those whose integral may be referred to a given integral. Application of the preceding rules to the quadrature of curves. 94. In order to find the area or quadrature of curved lines, we consider these lines as polygons of an infinite number of sides; and from the extremities M and m of each side (fig. 34), we imagine the perpendiculars 17 P, m p, let fall upon the axis of the abscissas, dividing the surface into an infinite number of infinitely small tra- pezia. Then we consider each trapezium, as PpmM, as the differential of the finite space 4PM; because in fact P pm M = Apm— APM =d(APM) (6). It is therefore only re- quired to express algebraically the little trapezium PpmM, and then integrate this expression by means of the preceding rules. But, in considering P p m M as the differential of the surface, it must be observed, that it is no more the differential of the surface reckoned from A, the origin of the abscissas, than it is the differen- tial of any other surface KPML reckoned from a fixed and deter- minate point A’; since we have equally PpmM= K pm L— KPML=d (KPML). When we integrate, therefore, we have no right to refer the integral: given directly by the calculus to the space APM, rather than to any other space KPILM, which differs from it by a determinate and constant space KAJ. We must therefore add to the integral 86 Integral Calculus. found by the calculus, a constant quantity expressing that by which the space proposed to be determined differs from that given directly by the calculus. It will be seen in the following examples, how this constant quantity is determined. Let us, in the first place, find the expression for the space P p m M. Calling AP, «x; PM, y; we have P p=da,pm=y+dy. The surface of the trapezium P p m M is (Geo. 178) ly d Be Miu POY dey da api Bes But to indicate that d y and da are infinitely small, we must re- 3 dydzx —, which is infinitely small, compared with y dx; we ject have therefore y d x as the general expression of the differential of the element of the surface of any curve. In order to apply this formula to a proposed surface, whose equation is given, we must deduce from this equation the value of y, which we substitute in the formula y d a, we then have a quan- tity in terms of w and d «, which, when it can be integrated by the preceding rules, will give, with the addition of a constant quan- tity, the expression of the surface of this curve, reckoned from any point we please. We have then only to determine the con- stant quantity, which is done by expressing from what point we choose to estimate the surface. We shall now illustrate this theory by examples. Let us take, for the first example, the common parabola, which has for its equation y? = pax. We have y= Ma ceeiie x? ; : therefore yd x= pt of d x. But (88) fp? a? de? + C= 2 pe a C; this last expression then is that “ he surface of the parabola; so that, knowing the abscissa w, and the parameter p, we shall have the value of the space APM; or, of the space KPLM estimated from a determinate point K, if the constant C be determinate, that is, if this integral express actually from what point we estimate. — Suppose, then, that we wish to estimate the space from the pont A; in that case we have pt af dx ———— > A ee «eee ° phe SE 4 ie of - bd — ”* Quadrature of Curves. 87 In order to know what becomes of the constant on this hypothe- sis, we observe, that when w becomes 0, the space APM is also zero; in that case, the equation is reduced to 0 ==0-+ C; where- fore C=O; in order then that the integral may express the space estimated from the point A, the constant C must be zero; that is, we have no constant to add, and we have, generally, the indefinite space 4PM — 2p? 2. But if we wished to estimate the space from the point K, such that AK = 6, (6 being a known quantity); we should have KPLM= 2p? «2 + C; now this space APLM becomes a when AP or x = 6; we have therefore, in that case, 0 = 2 2 2 be + C; therefore and consequently KPLM= 2p' 0? — 4p? 02. We thus see what purpose is served by the constant, which is added in integrating, and that the conditions of the equation alone can determine it. We observe that 2 pe x2 — 2 2 pe: a but p2 py; a re SY therefore 2 p? x2, or 4p" «1° X & = 2 y 2; since: therefore 1 3 F . 2p? x2 i ies the anager APM, this space will also have for its expression 3 av y, that is, 2 4P X PM, or 2 of the rectangle APMO, whatever AP may be. In like manner 3 but, mien «== AK = 6, the equation y? =p 2x gives y2 =p b, apt 8 — 2p 4 b2 x b; and consequently y= p? be: that is, KL — yp be therefore 2 pe be or 2 pe bz xX 6=2 KL x AK; therefore since the space KPLM is saga by 2 pe ghee pe bz, it will have also for its expression 3 AP x PM—2 AK x KL, that is, 2 APMO — 32 AKLI. The parabola is the only one of the four conic sections susce pti- ble of being squared. : 88 Integral Calculus. Let us take, as a second example, parabolas of all kinds, whose general equation (29) is y+" a™ 2"; we have therefore m n gia, Vom ear yeaa then Bishi, AR Ai ydex=artn gnt+rd a; and hE) fries era C= mn 41 NOY m--2n m n Mf gmt m+n | =Pt® oats mr OW” MR eyt C. So that if we wish to estimate the space 4PM from v ti face of a sphere whose diameter is x; which agrees with what has already been found ( Geo. 535.) 99, To find the surface of the paraboloid, (which is the solid generated by the revolution of the parabola 4M (fig. 39) about its axis), we have the equation y2=p w; whence ye 2ydy > Ay? dy? t= —, pe TAD ay igi klar. CAMs p i) 3 whence Ay? a 2 2 Jae FTP = lay? ahi “aay pe} wherefore cydy 4 y2 1 /ie pd y Guo :. Ji+ A ; which, being inne (90) gives cyd ¥(1 4y?\3 3. Sy dy p2 Now, in order that this quantity should express the surface reck- oned from the vertex A, it must become ise mage Y =a One bot, 2 in hae case, it becomes 5 (1)? + C; orf — - ++ C; whence - pre aa, C = 0; that is are narnae wherefore, the surface of the indefinite alana: AMLA is PRC fy AH? yp 4 Ir pe I2r ; Application to the measure of solidity. 100. In order to measure the solidity of bodies, we may imagine them composed of infinitely thin parallel segments, or of an infinite number of pyramids, whose summits unite in the same point. In the first way of viewing the subject, the difference of the two opposite surfaces, which terminate each segment is infinitely small, and must consequently be omitted in the calculus, if we would indicate that the segment is infinitely thin. Thence: it follows that we must take, as the expression of the solidity of this segment, the product of one of its opposite bases by its infinitely 96 Integral Calculus. small altitude. If, for example, we consider the pyramid SABC, (fig. 41) as composed of infinitely thin segments, like abcde f; we may take, as the measure of this segment, the product of the surface a 6 c or def, by the thickness of this segment. In like manner, if we consider the solid generated by the revolu- tion of the curve AM about the straight line AP (fig. 39), as com- posed of infinitely thin parallel segments, we must take as the measure of each segment, the product of the surface of the circle, which has for its radius P/V, by the thickness P p. This principle being laid down, we thus ‘estimate the solidity of the whole body. We consider each segment as being the differen- tial of the solid, because the segment Mm UL is in fact = Aml A— AMLA=d (AMLA) ; and having determined the algebraical expression of this segment, we integrate. Let it be required, for example, to find the solidity of the pyra- mid SABC (fig. 41). Supposing the surface of the base ABC to be equal to the known quantity 6?, and its altitude S T'=h, we represent by x the distance S¢ of any section ; which gives d x for the thickness of this segment. ‘The surface a 6c is found by the following proportion (Geo. 409) ; : SS 6 4 Apes Oo es that is, h2 : 22:3: 6? : abc= thus the solidity of the segment will be b2 12.0 62 g2deesy 1b2 ¢8 b2 78 af oe ae tose if we reckon the solidity from the vertex 8. This quantity, which expresses the solidity of any pyramidal portion S a6, is the same 2 72 te M4 =) which is abc X at which agrees with what has b been demonstrated (Geo. 416). 101. As to the solids of revolution, we may find a general ex- pression for the elementary segment or differential. For, suppos- ing that r : c expresses the ratio of the radius to the circumference, we shall find the circumference, which has PM ( fig. 39) or y for its radius, by the following proportion, . | ee - eG Reels: a ae as Curved Surfaces. 97 if we multiply this value 2 of the circumference, which has PM cy? et for the sur- face, which, being multiplied by the thickness P p or d 2, gives cy2 dz 27 solid of revolution. ‘To employ it in any particular case, we have only to substitute instead of y its value in terms of w derived from the equation of the generating curve AWM, and integrate. 102. ‘Take, as an example, the spheroid generated by the revo- lution of the ellipse about its major axis (fig. 42). The equation of the ellipse is y2 = . (a v—vw*) calling AP, x, and PM, y, and the axes AB and D d, a and 6; we have then, Gy? 07. c b? Qr ~ @raP of which the integral &b2 fax? £8 € 64. [az us = gra (“3 — 3) + Cara (3): when the solidity is reckoned from the point A. In order to obtain the entire spheroid, we suppose x = AB =a, b2 3 3 i ‘ b2 and we have seats Xx (5 —): which is reduced to aiid a for its radius, by $ y or half of the radius, we have for the expression of the element of the solidity of every b2 (axv—ax?) da = sons (acd x—x2d x), Qra Q 3 12r 5 as c b2 c b2 c 62 Eee. 2 ° AG x 4a, or G7 <3 43 now 3; expresses the surface of the circle, which has 6 or D d for its di- Crm Bae te DR cb? ameter, since (Geom. 287,) r2 : Chis Rte and ao xa would consequently express the solidity of the cylinder circum- scribed about the ellipsoid. ‘Therefore, since the solidity of the which is the same thing as 4 ellipsoid is here a xX 2a, we conclude that it is two thirds of that of the circumscribed cylinder. And as the sphere is only an ellipsoid, of which the two axes are equal, the sphere is therefore 2 of the circumscribed cylinder, which agrees with what has been demonstrated (Geom. 549). 103. If we wish to determine the solidity, reckoning from a determinate point K such that AK =e; we take the general inte- cor (Pa2?t Puge 2.0%: 3(“> — =) + C; and as the solidity is to be reckoned 13 98 Integral Calculus. from K, this integral must become 0 at that point, that is, when - . : c 6? ae? e8 a= e: in this case it becomes ——— _ § ——- —-— be ( Qgraee ( 2 3 ) De whence and consequently c b2 ae? e® Crise gala ae thus the solidity, reckoned from the point K, has for its expres- sion CO! wer sag gb? -/ae2) 8 Q7 a2 (<--4) ~ Qra? (5-3 Such, therefore, is the value of the segment of an elliptical sphe- roid, comprehended between two parallel planes, perpendicular to the axis, and at the distance « — ¢ from each other. We may, by this formula, calculate the solidity and consequently the weight of the masts and yards of vessels, which are portions of elliptical spheroids. ‘The same formula serves also to measure the capacity of casks, whose external surface may be regarded as a portion of such a spheroid. 104. Let us take, asa second example, the paraboloid ( fig. 39). 2 The equation of the parabola is y? = pa, thus the formula sie 5 sda if becomes —/ = =, whose integral is c px? ge DEN & a Cs % ei 2p Dirge BUNA OR ir Boia 2 by substituting for p «x its value y?. If we reckon the solid from the point A, as it is zero when v0, the constant C must be C Usui cy? oF X 5a the surface of the circle, which has P/M for its radius, that is, the base of the paraboloid AMZL.A ; therefore the paraboloid is half the product of its base by its altitude; it is therefore half of the cylinder of the same base and altitude. If we wish to reckon the solidity from the known point A, such that 4K =e; then, as the solidity must be zero.at the point K, that is, when 2 =e, the general integral must be zero at the same ig : ie Pere ; oe cp 62 time ; that is, ae + C, becoming eT a C, equals zero, zero, and the solidity is reduced to expresses Measure of Solidity. 99 wherefore op e* ee Ar Tei and consequently PS UG 5 Ar? whence, the solidity of the segment of a paraboloid comprehended between two parallel planes, whose distances from the vertex Ope one pret AT AT the excavations of mines. 105. We may take, as another example, the hyperboloid or solid generated by the revolution of the hyperbola about one of its axes. We may also take the ellipsoid generated by the revo- lution of the ellipse about its lesser or minor axis, which is called the Flattened Ellipsoid (Oblate Spheroid). ‘That generated by its revolution about the greater or major axis is called Elongated Ellipsoid (Prolate Spheroid). We should, in like manner, find that the flattened ellipsoid is 2 of the circumscribed cylinder: that is, that a and 6, being the greater and less axis of the gene- are x and e, is This formula may serve to estimate b rating ellipse, the elongated spheroid has for its solidity oT , and 26 the flattened spheroid has for its solidity “ors thus the elongated ; b2 25 spheroid is to the flattened spheroid : : a ie ‘ oT >: 6:4, as the lesser axis to the greater. We shall now leave the solids of revolution. But in order to accustom beginners to extend the application of these methods, we shall give two additional examples. 106. In the first, it is proposed to find the solidity of a cylindri- cal ungula formed by cutting a cylinder by a plane oblique to its base, and which, for greater simplicity, we will suppose to pass through the centre. It is the solid ADBE represented (fig. 48). If we conceive this solid to be cut by parallel planes infinitely near each other, and perpendicular to the base AEB (fig. 44), the sections will be similar triangles, whose surfaces will conse- quently be as the squares of their homologous sides. ‘Thus, calling CE the radius of the base, r, the altitude DE, h, and PM, the base of the triangle PIT N, y, we have 100 Integral Calculus. CED eePALN i072 “sayy now therefore Qrr ar”? calling, therefore, AP, x, which gives d « for P p, the thickness of the segment comprehended between two contiguous planes, we hy2d 2 ar have for this segment. Now y is an ordinate of the circle which serves as base, and we have consequently y? = 2 r ©— 2?. hdx(2r2—z2z?) The elementary segment becomes then 5 z or ea (2r «©dx—x? dx), of which the integral, reckoning from the 3 > point A, is ie (- x? — > ) . Therefore, to obtain the whole so- r lidity, we have only to suppose «== 2 1, which gives = (47°—=>) =ghrea st x 4r—= CED x 4 AC = CED x 3 AB, that is, two thirds of the prism, which should have the triangle CED for its base, and the diameter 4B for its altitude. ‘This may serve to measure the solidity of fortifications. 107. As a second example, we shall investigate the solidity of a segment of an elongated ellipsoid, comprehended between two planes parallel to each other and to the greater axis. Before proceeding in this investigation, it must be demonstrated that the sections of an ellipsoid, made parallel to the greater axis, are ellipses similar to the generating ellipse of the solid, that is, that their axes have the same ratio to each other as the axes of that ellipse. To this end, we conceive the ellipsoid cut by a plane, which for the sake of preciseness, we suppose vertical, and passing through the greater axis AB (fig. 45). The section will be the ellipse A DBE equal to the generating ellipse. We also conceive the ellipse cut by three other planes, of which two are vertical and the third horizontal. Let the less axis of the ellipse DE and its parallel MN be the intersections of the two first with the plane Measure of Solidity. 101 ADBE, and ST that of the third with the same plane ADBE. This done, we say that the section of the ellipsoid by the plane represented by S’Z'is an ellipse similar to ADBE. We conceive perpendiculars to the plane ADBE to be raised from the points O and R, meeting the surface of the ellipsoid. These perpendiculars will be at once ordinates of the section made by ST’, and of the circular sections made by MN and DE. Now since they are ordinates of the circular sections, if we call the perpendicular at the point A, z, and the perpendicular at the point O, ¢, we shall have 29 DE ABB, foot) a UN. But, calling CD, 46; PM,y; CA, 4a; and CR = OP, u; we have DR=4b+u,RE=3b—u,MO=y+u, ON=y—u1, so that DR. RE= } 0? — wv? = 2?, MO. ON=y?—wW=?. But, by the nature of the ellipse (Ap. 123), we have b2 y? ai a2 (Z a? — v2), calling CP,«. And & representing the ordinate SR to the less axis, we have (Ap. 122) k? = 3 (4 62 —u?), whence we de- b2 k? duce u? = } 6? — ——-; substituting these values of u? and y? in those of z? and ¢?, we have b2 k2 oa =, and ¢? = a h2 2 b2 72 a2 ? whence it is evident that : =e ‘ ais —_ eee :1h?: k? —x22:: SR2 o SR. RT: SO. OT; _ that is, the square of the ordinate z, corresponding to the point RR, is to the square of the ordinate ¢, corresponding to the point O, as the product of the two abscissas to the first, is to the product of the abscissas to the second; the section made by S'T' is therefore an ellipse. 22 :3t?: b2 k2 a MM sae Cots a : ; Bika. Besides, the equation z? = , Or 2 == —, gives ) q ed, § 102° Integral Calculus. now 2, or the ordinate to the point R, is the semi-minor axis of this ellipse, and k& or SR is the semi-major axis; the two axes of this ellipse have therefore the same ratio as those of the generat- ing ellipse; and as nothing in the course of this'reasoning deter- mines at what distance CR this section is supposed to be made, the same thing takes place for every section parallel to AB. This being determined, if we wish to have the solidity of any segment of an ellipsoid, comprehended between the two parallel planes represented by AB and ST’, we represent by S the sur- face of the generating ellipse ; and since the ellipse, of which Sis the greater axis, is similar to this, we shall have the surface of this last by the proportion af 32: 9: 3 j dae’ ph multiplying this surface by the infinitely small thickness Ar or 2 a for the value of d u of the elementary segment, we have 4 this segment; but, according to what has just been said above, a2 we have Le te (3 ye — ar); whence the elementary Bae will be Sdu (ib? —u? cae eed a a (¢ 6? du—v? du), of which the integral is ee (1 62 u— 4u), if we reckon from 4 the centre C. But if we reckon from the point K, the integral : S will be ie (4 62 u—iw)+ C. In order to determine C, we call CK, e; then the integral must be zero at the a K where u= e¢; we shall have, therefore, is (4 6? e—1¢2)+ C=O, and consequently S OF eae (z 6? e— 68), wherefore every segment of an elongated ellipsoid, comprehended between two planes parallel to the greater axis, has for its ex- pression Savbidys Ss 4 Measure of Solidity. 103 S or The (3 6? u— fF 6? e— 3 u® + 4 3), rs Now 462 u—} 6? e= jf 6? (u—e). In like manner, 3 1 ay cee xe —_—7 Ue = 2+ eu--u?), Moreover, u— e represents the distance of the two parallel planes or the altitude of the segment comprehended between them; if, there- fore, we make u — e —A, calling this altitude A, and substitute for e its value u—/ drawn from this equation, we have, after all re- ' S P ductions, cpa( 22 h—hu? +h? u— 5) t Sh MERLE h i ot) + Fp ("— 3): 2 but we have had, above, k? = _ (i 6? — wu?), and consequently b2 k2 + 62 — yu? — era The value of the solid segment is therefore changed into - Shk?2 | Sh? (« 3) 1 2 1 f2 ga 46 But we found stele for the expression of the surface of the sec- tion made by S77, or the inferior section ; calling then this surface s, we shall have Sh? h finally, if we represent by # s! the surface of the section made by LK, and by / its semi-minor axis, we shall have, from the simi- ba%s's" pyar? larity of the sections, # 67: /?:: 8 :s'; whence S= } which gives, for a final expression, s' h? h That is, we must, Ist, Multeply the surface of the as section by the altitude of the segment; 2d, multiply the surface of the great- 2 a er section by the ratio - of the square of the altitude of the seg- ment to the square of the semi-minor axis of the superior section, 104 Integral Calculus. and by the distance from the centre to the inferior section minus a third of the altitude of the segment. This rule may be usefully applied to measuring the solidity of that part of the hull of a ship which the lading causes to sink below the surface, whenever the figure of this part may be compared toa portion of an ellipsoid. In this case, s will represent the section of the vessel without its lading, at the surface of the water; s’ the section with the lading; A the distance of the two sections; / the greatest breadth of s’, atid u the distance from the greatest horizon- tal section of the spheroid, to s. As to the mode of measuring s and s’, it may be observed, that one of these surfaces is determined by the other, since, belonging to similar ellipses, they must be to each other as the squares of their greater or less axes. It is therefore only necessary to know how to determine one of them. Now, we shall presently see, that the surface of an ellipse is to the surface of a circle which has for its diameter the greater axis of the ellipse, as the less axis is to the greater. As we know then how to estimate the surface of a circle, at least to’ as great exactness as may be desirable, we shall easily determine that of an ellipse whose axes are known. On the integration of quantities containing Sines and Cosines. 108. We found (21, 22) that d (sin z) =d z cos &, d (cos 2) = — dz sin 2; therefore, reciprocally, ff dz cos z = sin z + €, —dzsnz=cosz+C. It is required to find the integral of dz cos 3 z; we have fd 0083 = fe = SB Je In like manner fidesing cm [Pte 9: K eae, In general, m being any constant quantity, —mdzsinmz COS m Zz fi dz sinma= = —— + C. a—— (010 Let it. be proposed to,integrate (sin z)" dz cos z. pea (sin z)" dz cos x = (sin z)" d (sin z), Sines and Crosines. 105 we have } n-+-1 SA (sin z)" dz cos z= hat) ORE. C . n+] If the proposed differential were (sin m z)"dz cos mz, we should give it the form (sin m z)"mdzcosmz_ (sin mz)" d(sin mz) m ra (4 4 ee of which the integral is (sin mz)" +! 4, m (n-+ 1) In like manner Pico Kdecbinomi ds aes @ omen 716, __ (cosmz)"r! crs) Let it be proposed to integrate dz sin p z cos q z, p and q being con- stant quantities. By what has been demonstrated (7'r. 27), sin pz cosq z= sin (pz-+qz)-+4sin (pz—qz) =tsin (p+9)2-+4in (p—g) 2; we have therefore to integrate $dz sin (p+) 24+3dz sin (p—g) 2 — (P+) dzsin (p+ g)z , , (Pg) dzsin (p—g)z =} $2 POP Pa" Prd Pentel of which the integral is 5 CORD AD hh OP AD sd) Rae Gy Par? Row We should integrate in the same manner d z sin pz cos gz sin r z, by converting these products into the sines or cosines of the sum or difference of the arcs p z, q z, rz, Sc. (Tr. 27). If we wished to integrate d z (sin z)3, we should change this dif- ferential into d z sin z (sin z)?; now sin z? = sin z sin z= 4 cos (z —z) —4 cos (z+-z) —l SENS — J ° =i coso—icos2z=4—Acos2z; +o therefore dzsin z8? =4sin z< ae But we have just seen, in the preceding question, that the integral of d 1+ 5, was 31 ase we shall therefore have z= tanga.l pee ere! To determine the constant C, we observe that when z = 0, that is, at the point of departure, the latitude A/W becomes the latitude + As we have supposed the radius equal to unity, the logarithms of tangents must be diminished by the logarithm of the radius, which is 10,000000 ; that is, we must take away the characteristic 10. 1382 Integral Calculus. of departure BQ. Let ¢ represent the sine of this latitude. The constant C, then, must be such, that, substituting ¢ for 2, we may have z=0. We have therefore O—tanga.l eke, and consequently C=—— tang a hc Wherefore z=} tang a. 1 — 3 tang pees = tang a (375 part oe a 2 reed, ay ser l bah = tang « ( ae pee 7 fee) By reasoning precisely as in the preceding question, we shall find eee is reduced to cot (3 complement of AM); and for ive oer the same reason ae is reduced to cot (} complement of BQ), wherefore, the difference of longitude or z = tang a . (log. cot (1 co. lat. of point arrived at) — log. cot (4 co. lat. of point of departure) ; which furnishes a very simple rule for finding the difference of longitude, either by the tables of increasing latitudes, or by the reduced maps, whatever may be the course. By the Table of Increasing Latitudes. Find the increasing latitudes corresponding to the latitude of the point arrwed at, and of the point.of departure. Take the difference of the latitudes thus found, (or their sum, tf the lati- tudes are of different denominations), and multiply it by the tan- gent of the course; you will have the difference of longitude in degrees, minutes, and seconds. By the Reduced Maps. Find upon the meridian the latitude of the point arrived at, and that of the point of departure. Draw, through the extremity of each, a perpendicular meeting the course proposed. The difference of these two perpendiculars, applied to the scale of longitude, will give you, in degrees, minutes, &c. the difference of longitude. Loxodromic Curve. 133 Indeed, AQ, (fig. 53) being the meridian; OL the course; AQ, AP the two increasing latitudes; PQ or RT is their differ- ence ; now, in the right-angled triangle SRT’, we have tang) TRS :: BL: TS; therefore , TS= RT x tang TRS; now RT'is the difference of the two increasing latitudes, and the angle T'RS' is equal to the angle made by the course with the meridian. The curve line QM (fig. 52), which marks the course of the vessel on the surface of the globe, is called the Lowodromic. In whatever part of the course we suppose the point M, the triangle Mrm has always the same angles, since the angle mr M is always the same, and the angle m is a right angle. ‘There is therefore always the same ratio between M7 r and mr, as between the radius and cosine of the angle Mrm, which we have called a. Therefore the number of leagues in the distance QM, is to the number of leagues made in latitude, that is, the number of leagues in [//, as radius is to the cosine of the course. This serves to determine the difference of latitude, when we know the course and distance in leagues. ‘The same proportion serves to determine the course, when we know the difference of latitude and the distance in leagues. | On the manner of reducing, when it ts possible, the integration of a proposed differential, to that of a known differential, and distin- guishing in what cases this may be done. 128. We shall explain only the method for binomial differentials ; it will afterwards be easy to apply the principles to more complex differentials. Let us suppose, at first, that the proposed differential is heodz(a+tba), and that adz(a—+-b2™ )P is that on which it is to depend, (or that, to the form of which we would reduce it); that is, let us suppose that the two exponents of the binomial are the same. We shall suppose . fhe dx(atb x jp = (a+ ba pt (Ax + Baxktit Crk +24 aan Pxk+tt) 4 fRaomdx (at bx; kK and g being unknown exponents; ¢ a positive whole number ; 134 Integral Calculus. A, B, C, P, R, &c. constant coefficients, also unknown. Differen- tiating, we have has dx (a+ bx" )p =(p+1)nba*—ldz(a+ ber) (Avt+ Brkti4 Cakta,,, + Parktid 4+ (aptba ptl(kArk—lda+ (k+ 9) Bakti-ldx +t(k+29) Catt2I-lda.... + (k+ tq) Paktti—ldz) + Ramdx(a+ ba jy, or dividing the whole by (a-+ 6 2” )? dx, we have has =(p+1)nbar—1(A xt + Bok ti+ Cotta... .4+ Patra) + (a+b) (kAv'—14 (k+q) Bot+i-14 (k42q) CP! ee? RA + (k+tq) Paoti—-l)4+ Rx, In order that this equation may be still true, independently of any value of z, it is necessary, that after the multiplications and transpo- sitions are performed, the sum of the quantities which ‘multiply the same power of x be zero; it is by this condition that we determine the coefficients A, B, C, &c. But, that this may take place, the nnmber of powers of zx, which enter into this equation, must not exceed the number of these coefficients. Now the number of coefficients, as may be easily seen, is ¢-+2; let us therefore find the number of the powers of x. In order to that, we must determine & and gq. : They may be determined thus. &—1 is the least indeterminate exponent found in the equation; we make it equal to m or to s, ac- cording as mors is the least exponent. ‘The greatest determinate exponent to be found in the equation is, as may be easily seen, k-+tq-+-n—1; we make this equal to s, if we have made k—1l=™m; or to m, if we have made Kk —1 =s. Let us suppose £ — 1 =m, we shall therefore have k+tq+n—l=s This done, that the equation may not contain more powers of z, than there are indeterminate coefficients, the coefficients of z in this equation must form an arithmetical progression whose difference is q, which cannot fail to be the case, since we have supposed k—1l=m,kt+ttq+n—l=s, and ¢ a positive whole number. Now the greatest term of this pro- gression being k +-¢ g-++-n— I, and the least, k— 1, we easily find (Alg. 230) the number of terms of this progression to be SEIT tot a se therefore ° O40 eps and consequently g = 7 ; Becca for g and f, their value, in the equation k +-¢q-+-n—1=s, we havetx-+m oh Bees! and con- sequently (he Transformation of Differentials. 135 s—m—n s—m = ———__ - = —-—1; n n wherefore the reduction of one differential to another wili be possible, if the difference s — m of the exponents of x out of the two bino- mials, divided by the exponent of x in the binomial, gives a positive whole number, we then suppose, in the original equation, fe dx(a+tba yp = (a-+-6 yn yrds (A qm-+ 1 = Bymtntl +. Camt2n4+1 + ..... feta S12) + fR ada (aban yp ; and, in order to determine the coefficients, A, B, C, P, R, &c. after having differentiated, divided by (a-+-6 x" )? da, and performed the operations which are indicated, we transpose the whole to one side of the equation, and make the sum of the quantities, which multiply each power of x, equal to zero, which will give as many equations as there are undetermined coeflicients. 129. But if we pay attention, we shall find, that when ff rs dz (aba p depends on famdy (a-+ 5 x” )P, reciprocally, the latter depends on the former ; now, proceeding as above to reduce fordz(a+d an )p to fx dxu(a+tbx yp, we should find that m—s a — = a positive whole number, n and that we must suppose fimds (a+ bx" \p = (a+ b x ae: (Aastit Bastrntl +. &c. + Pym—n+1 ) 4 f Rede (a-ba P therefore, whether s be greater or less than m, provided that oS ai n eS or , gives a positive whole number, we may always reduce one of these differentials to the other by substituting for the first expo- nent of x in the series Av’ + Batt, &c. the least of the two exponents m and s, increased by unity, and taking for g the exponent of x in the binomial. For example, if we wish to reduce a4 dx (b2—22)2 to dx (b? — 22)2, which depends on the quadrature of the circle; we see that s —m is here 4 — 0, which, being divided by n, that is, by 2, gives 2, a whole number; the reduction is therefore possible; and since the formula s—m ee ——I gives t= 1, and as moreover the least exponent n m==(, we substitute 1 for £; we then make 136 Integral Calculus. Se dx (02 <22)8 (02 Ar + Be) + fRax(o2—277h, differentiating, dividing by (b? — 12)2 dx, and transposing, we have 0O=—=Ab2?— Ax? —3 Br4 + R +3 Bb? 22 —x4 — 3Az? —3 B24, whence we deduce —6B—1=0,—4A+3Bb?—0, Ab?-+R=0, wherefore, B=—}j4,A=—})?,R=}6; fn d x (b? — 422 1 == (b2 — x7)? (— $b? 4—JA43)+ bth dr / (62 — 22) + C. It is therefore easy, by this method, to find the differentials which are referred to a given differential, and consequently those which are referred to the quadrature of the circle, of the ellipse, and the hy- perbola, differentials of which it is easy to find the different expres- sions, by means of the different equations of these curves. 130. We may here take occasion to observe, that this method shows also the binomial differentials which are integrable; indeed, to find, among such binomial differentials, as h 2° dx (a+ 6b a” )p , those which are integrable, is to find those which depend on Rx—!du(a-b2y, which has been found (90) to be directly integrable; now it results therefore Rs s—n-+l1 es from what is laid down (128) that Areas must be a positive whole number, that is, that s+1 ne epethaaitny positive whole number, which agrees with what has been said (91). 131. Let us now suppose that the two binomials which enter into the differentials in question, have different exponents, so that the proposed differential is has dx (a+ 62 )", and that, to whose form we wish to reduce it, is 2™dx(a-+ 62” )P ,p» having a numerical value less than vr. If r is positive, we change the differential has dz (at ba” )" into ha dz (a+bx yr? X (a+b 2 \P- Then if r— p is a positive whole number, we may reduce has dz (a+ ba" )\"—? (a+b pp to a series of terms of this form (A'as + Baste Clast?rt &.)dx(a+ba yp, of which each may be reduced to the form 2” dx (a+ b 2” )p by the preceding method, if s—-m can be divided by n, and to reduce the whole to that form, we must follow exactly the method there given, taking for the quantity which was there called s the greatest expo- altel aon tat ; ; r ' , Transformation of Differentials. 137 nent of x in the expanded value of ha dz(a+b an )r—?. If we had, for example, 3 2 fe d x (b? — x7)? to reduce to fd t (b? — x?)?, ° 3, we should change fr dx (6? — x?) into ® a a . Js fi dx (b2 — 22) (b2 — 22)? or f (62 22 dz — x8dz) (62 — 22), then what we have to take for s, is 4. We suppose, therefore, con- formably to the method, fe 22 dx—x*t dz) (b2 —22)2 = (b2 — 22)? (At4+- B2®) 4 [Rac (o? — 2) If, on the contrary, the value of r is negative, the differential, to which the proposed differential is to be referred, must be prepared thus, @d2z(a+t ba” )P—” X (a+ be” pp then, if p—r is a whole number, as it will necessarily be positive (since we suppose r nega- tive and greater than p, whatever be the value of p) we may reduce amdxz(a+bx \p—"(a+ bx" )" to a finite series of terms of this form, (A’z + Bramtn + Cogmrent &c.) (a + ba)", we may then proceed as if it were required to reduce this last to the form a’ dz(a+-b2” )", that is, we may proceed in a manner precisely similar to that to be followed in the case when 7 is positive. If it were required, for example, to reduce gr? dz(a? +27)? to dx (a?-+ 27)! or ipa eiee which (110) is integrated by means of an arc of a circle, of which z is the tangent, and athe radius, we should change dz (a? +-2?)71 into (a? + 2?) dx (a? +-2x?)-?, and, as the least exponent out of the proposed binomial is 2, we should suppose Ra +22) dx (a4 22/2 = (a? + 2?)-1(A21+ Br) + fe a2 dx (a? + 1£2)-?, And we should proceed as above to determine the coefficients, A, B, and &. Then, by transposition, we should have the value of fe a2 dx (a? + £2)-2, in which we should afterwards reduce R (a? + 2?) du(a?+ x7)? to Rdzu(a? +27)-1 dz On Rational Fractions. 132. Every rational differential fraction is always integrable either algebraically, or by ares of a circle, or logarithms, or all ‘these means united, or by only two of them. They are always integrable algebraically, when they have no vari- ble denominator unless it be a simple quantity, excepting only the case 18 138 Integral Calculus. in which the denominator is raised no higher than to the first power, as we have already seen (82). ; It remains for us therefore to show the truth of our assertion, in the other cases; that is, when the proposed differential has a rational complex denominator. : We shall suppose that the variable in the numerator of the proposed differential fraction is of a lower degree than in the denominator. If this were not the case, we should make it so, by dividing the nu- merator by the denominator until the remaining power should be less than in the denominator. For example, if we had to integrate 3 ; ate we should begin by dividing x2 dz by a?-+3a aa? ; we should find «dz for a quotient and —3a22d%—a?xdzx for a remainder ; we should again divide this remainder by the same denominator, and find —38adzx for a quotient, and +8a2xdr+3aidz hog ses for a remainder; then instead of ———~——_—_. az + 3ar+ 22’ Sa2adx+3a% dx a?+3acr+22 © In order to discover by what means we may integrate rational dif- ferential fractions, we recollect that the differential of the logarithm of a quantity, being the differential of that quantity divided by the quantity itself, that is, being always a fraction, it is very natural to suspect that the integration of rational fractions often depends on logarithms. Let us take, for example, 2al(a+2)—2al(Qa+x); differentiating, we have we should take adx—dsadx+ Qadxz Q2adz atx Zata’ or, reducing to the same denominator, Zardz 2a2+3ax+ 2? f Now, it is evident that, in order to integrate this fraction, we have only to decompose it into two fractions, one of which shall have a-+x, and the other 2a-+-x for a denominator. The numerators will be constant numbers multiplied by dz; these two fractions would then be integrated by logarithms. 133. It is therefore very natural to endeavour, in order to integrate fractions of this kind, to decompose them into as many simple frac- tions as the denominator has factors, each one of which shall have for its denominator one of these factors. This, indeed, is the method which we may and must pursue, when all the factors of which the denominator may have been formed are unequal. 134. But when, among the factors of the denominator, there are any which are equal to each other, then we are not to expect that the method should be successful, because the integral cannot en- Rational Fractions. 139 tirely depend on logarithms. If, for example, we had, GiaF ; whose denominator has two equal factors a + x and a -b 2, we should find (88) that the integral of this quantity or of its equal dz(a+ x)~? is—(a+x)1-+C, which does not depend on logarithms. But we see, at the same time, that if we should differentiate such a quantity as a +2al(a+x)+2al(2a+7)—al(3a-+2%), we should have —arz dx Q2adz Q2adrz adx (a+)? ay a+ x Maasai ie (2azx+ a?) dx Q2adxz adz (a+ 2)? Qat2x 8a+2’ or, reducing the whole to a common denominator, 10 at*dx+ 26 ai xdtx+17 a? w2drt+3ax° ay (a-+ 2x)? (2a+-2) (8a+z) | : . a fraction which, in order to be integrated, would only require to be reduced to or 2ar-+ a? 2adz adt (a+ x)? Ae Ours teats Bate that is, to be decomposed into three fractions, of which the first should have for its denominator all the equal factors, and in its nu- merator all the powers of x less than ‘the highest power of the de- nominator. The other two fractions should have each, for its denomi- nator, one of the unequal factors, and no power of x in its numerator. In this manner every rational fraction may be integrated ; and we proceed in this manner, at least when there are no imaginary factors in the denominator ; which case will be examined hereafter. Th (atbeter?+....kx—!) dx (ME Not Bey.... Pa) ral, any rational fraction ; if we suppose the denominator to have a number m of factors, equal to z-++-g, a number p of factors, equal to xt h, &c. and any number of unequal factors, represented by x +-?, t+ q,x-+r, &c. the proposed fraction will be (atbatex?+....ka—!)dx 2 (Fey (eb X Be. (@-F A) (eq) (7) &e, In order to integrate this fraction, we must suppose it equal to Aw—ldz4+ Bam—2dz...+ hdr ; (Pay A'axr—ld¢4+ Ba—-2dxit... f'dx (-E hip hdr M da iN dia Vapitapg abr , representing, in gene- , &e, + 140 Integral Calculus. A, B, C, &c. being constant and undetermined coefficients, If, then, we can by any means detérmine these coefficients, it will be easy to find the integral. ‘This is evident in the case of the simple fractions Ldzx Mdzx Ndz ee pg! oer Lil («+i), Mi(«+q),N1(«+71r), &c. As to the fraction &c. of which the fet 18 — - we take, for the sake of greater simplicity, <+ ¢ =z, which gives %—=2z—g,anddzt—dz. By substituting these values, we reduce the whole to a series of simple quantities easily integrated, one only of which will have the form aS that is, be integrated by logarithms. In like manner, for the terms / mom 1 dey lap—2 dx ldo A! ar—Vda+ Ba i anal 98% wag (ap hp There thus remain only two things to examine; the first is how to find the factors of the denominator of the proposed differential frac- tion; the second, how to find the undetermined coefficients. 135. To find the factors of the denominator, we proceed as we should to resolve the equation produced by setting the denominator equal to zero, since (Alg. 184) to resolve an equation is to find the binomial factors, by the multiplication of which the equation was formed. Thus we must employ the methods given (Alg. 185, &c.) 136. As to the manner of finding the coefficients A, B, C, the way which offers itself as most natural is to reduce all the fractions in which they occur to the same denominator, then both members of the equation formed of the proposed fraction and these new fractions, having the same denominator, we may suppress this denominator, and having transposed the whole to one side of the equation, we shall find that, in order that the equation should be true, independently of any value of z, the sum of the factors, which multiply each power of 2, must be equal to zero. This condition will give as many equations as there are undetermined coeflicients, and by means of them the coefficients may be determined. The following are some examples. Let it be proposed to integrate ———__; we suppose a2 — 72’ Mares dx Bdz T5t es -, since the two factors of the denominator are a2 — 72 a+ and a—z, akde reducing to the same denominator, we have dx (Aa—Azv+ Ba+ Br) dz , suppressing the common a2 — 42 a? — x2 denominator, dividing by dz, and transposing, ,we have x —Aa—Brs=0; — Ba wherefore 1 -- Aa-— Ba=0, and Ax— Bx=0O; from the last of these we have A = B, wherefore, the first becomes Rational Fractions. ; 141 L— Aa—Aa=0,orl1—2Aa=—0O, 1 whence A = hs and B = — ; we have therefore DWF Qa 1 1 Hebets 4) Wat 4B ‘ Game B58 i Gite da of which the integral is dx 1 1 ; ol iata Sipe = aa! (+9 — gle t C= 5 miles a a——ZL Let us take, as a second example, the fraction WWatdzx+26a% cdxr+ lia? t2dx+3axedz Gt Gata Gets) which was found (134) by differentiating Race + 2al(a + z) +2al(2a-+x)—al(S8a+ x), we shall then suppose Wat dz+26aiede+ lia? xr-dx+3arrdz (aa)? @a+e) Bape) _ (Ac+ B) dz Cdz Ddz (apa? 1 Rafa! Bape reducing to the same denominator, suppressing the common denomi- nator, dividing by da, and transposing, we have 10a* 4+ 26a%x + 17a? 2? + 3a7° ) —6Ba?—S5Bar — Br? —Ax? —3Cai—6Aa?x —S5Aazr? — Cr? $=0, —2Da®—TCae x —5Cax? — Dre —5Darx—A4D ax? therefore sa—A—C— D=—0, lia?—B—5 Aa—i Ca—4 Da=0, 26 a® —5 Ba—6A a? —7 Ca2?—5 Da? =0, 10 a4 —6 Ba? —3Ca?—2 Dai=0, equations from which we deduce + A=2a,B=—a?, C=2a, D=—a; + These values are found in the following manner. A=3a—C—D; B= 17a? — 15a? 4+5Ca+5D a—sd Ca—4Da B=2a?+ Da o — 260 —5 Ba—6 Aa? —5 De? Pit 7a? ae 26 a®—10 a8—5 D a?—18 a? +6 Ca?+6 Da?—5 Da? | 7 a? 7C=—2a+6C0—4D C= —2a—4:-D __10a*—6 Ba? —d3 Cas 243 142 Integral Calculus. the proposed differential is therefore changed into (Raxz+a?)\dz , 2adz adx (a+ x)? ora Baa precisely as we found it above. The two last terms have evidently for their integral 2al (2a+-x)—al(3a-+ <2); with regard to the aa cys (a+2)? dz —dz; whence we have (2az—a?) dz ab 24029 er dz ze z ze ter —dx, we make ax =z, and have x = z — a, and > i ; the pp t2al(a+s)+2alQa-+z)—al(Ba+2), 2 of which the integral is 2a/z + — or 2al(a+zx)+ whole integral is as it should be. 137. This method is general. But there are several shorter ways of finding the coefficients. We may, for example, find the coeffi- cients of simple fractions, independently of each other, in the fol- lowing manner. Let Mi be the fraction proposed ; h «+a, one of the factors of the denominator ; let P represent the other factors, or be the quotient of M divided by hx —a. Conceive a ae to be Qdz ; we shall have Adz heal P Ndz tie Qdx N MM renee Mf? OP tes mat 3 therefore, by reducing to a common denominator, observing that M Sah ae a xX (hz+ a) = M. we shall have V= AP+ Q(hux+a). But if we differentiate the equation (hz -++- a) P = M, we haveh Pdx+(hr+a)dP=dM. Now as this equation and the equation VW = AP + Q (hx -+-a) must be true for every value of 7, they must be true, if we give to x any value whatever. We therefore give to x the value which gives the decomposed into most simple result, that is, the value —< obtained by supposing the _ 10a —12a4—6 Da? + 6at+12 Da? 243 = 5a—6a—3D+3a+6D..—2a=2D_ —=—a C =—2a—4D=—2a+4a=2a B=2a?+ Da= 2a? —a?=a? A=8a—C—D=3a—2a+a=—2a; Rational Fractions. | 143 denominator hx+-a—=0. We thenhaveh P dx —=dM, and N=AP. yr , given by the first, eee §, M Substituting in the second the value P= is, hNdzx dM one of the simple fractions, we must divide the numerator Nd x of the proposed fraction by dM the differential of its denominator, and having substituted for 2 the value obtained by making the denomina- tor of the simple fraction equal to zero, multiply the whole by the coefficient of 2 in this denominator. To obtain, for example, the value of the numerators A and B of 2 Bare ; and ————, into which we above resolved the atx a—z ; d x ; f fraction ———.,, we differentiate the denominator a? — x?, which a? —2 and we have A = ; that is, toobtain the numerator A of any the fractions gives — 2xdzx. We then divide the numerator dz of the proposed 1 57! in which successively sub- stituting for x, a and a, (which are the values obtained by making the denominators a+-x and a—x of the partial fractions succes- sively equal to zero), and multiplying by 1 and—1, the values of A, fraction by —2 xd x, which gives — we have —, and — for the values of A and B, as was found above. 2a 2a We might also find general rules for ascertaining the coefficients of the numerators of the partial fractions which have for their de- nominator the product of the equal roots; but shall not now stop to investigate them. 138. Although the rules just given for integrating rational frac- tions be general, yet when some of the factors of the denominator are imaginary, we have, for an integral, quantities composed of imaginary ones. Such an integral is not the less real, though it is sometimes with difficulty reduced to a real form. In this case, we first take out all the real factors of the denominator, and then decompose the re- mainder into factors not of the first but of the second degree, which are always real. ‘Then, for each factor of the second degree, which may always be represented by a z?-++ 6 xc, we form a fraction Azxdit+t Bdz ax?+bx+c : above. 139. If among the factors of the second degree, there are found any which are equal to each other, we form, for each group of these equal factors, a fraction of the form A w—lde-+ Br2—2dz+.... Qdz (ax? +ba+c)" ‘ n being the number of equal factors az? +-ba--e. 140. It only remains to show how these quantities may be inte- grated. With regard to the first, let us suppose, for the sake of making the operations more simple, that the partial fraction is reduced to the of this form and determine the coefficients as 144 Integral Calculus. A'idx+ Bidz 4 Tae terms by a. We then cause the second term of the denominator to disappear by making + 4 a@'’==z; which gives x==z—4a', and dt=dz; by substituting these values we obtain a quantity of the form form which may be always done by dividing both Czdz+Ddz bie eg AA? zd - of which, the first part — is integrated by logarithms (124) and the second by means of an arc of a circle, whose radius is g and tangent <. As to the quantities which have the form A.mr—lda+ Bae—2dz¢-+ . we Q dz (72+ 2axz-+b) we make the second term of the denominator disappear, and obtain a quantity of the form Mxn—ldz oa +... Pdz Ceeage ys “a which is integrated by reducing to the form “den by the method ? given (131), the integral of the sum of the terms, in which z has. even exponents. ‘I'hose whose exponent is odd may be integrated by i Nat : , ; zd z what is given in article 91, or by being reduced to -—__—_,_ accord- ing to the method given (1390). , Thus every rational fraction is either cateneatea exactly, or de- pends only on arcs of a circle or logarithms. On certain Transformations, by which the integration may be facilitated. 141. On this subject no general rules can be given. The inspec- tion of the quantities, experience, and practical address will dictate, on each occasion, what is best to be done. The object of the transformations here spoken of is to render the proposed differentials rational, as we then know how to integrate them. We subjoin, however, a few observations. 142. If there are no radical quantities but such as are simple quantities, we first give them fractional exponents, which we reduce k to the same denominator. Then, if 2! represent one of these quan- 1 tities so prepared, we make x! = z, which gives = 2, and dr —Iz'—1dz. We substitute these values, and obtain a quantity entirely rational. If we have, for example, dtrn/ x Bes Se dss fide hi ' Rational Fractions. 145 5 | Rc 06 lo which we change into ad t?dxtadz 4 < ~& +. 46 we give it the form Then making xe =z, we have z= 2z®, dr =625 dz, and conse- quently . 628dz+6azidz Si oati Reon ot 6z5dz+6az2dz z+1 if? and easily integrated by the rules already given for rational fractions. 143. Every quantity, in which there is only a complex radical not exceeding the second degree, and in which the variable under the radical sign does not exceed the second degree, may always be ren- dered rational by one or other of the two following methods: 1. Af- ter having freed from the radical sign the square of the variable under the radical sign, we make this radical equal to the same variable plus or minus another variable. 2. We decompose the quantity affected by the radical sign into its two factors, and make it, after being reduced to this form, equal to one of its factors multiplied by a new variable. which is reduced to ‘ dx : ALAR If we had, for example, --———., we might make a/z2—a2 = t—Z; r2—a2z 2 2 ee then + = — a2 -. Whence dz = MGs ther ot Sa and 22 wee NA dy erate cs EU MED re me d: d ueGe. Asli whence —_ =— =, which is easily integrated. We might also, in this same example, make /72—a2, OY /(@—a) (@+a) ==(%— a) z; then, squaring, and 2 dividing by x —a,t + a= (tx — a) 27; whenee 2 = “7 "* sialiogea” Zaz —Aazdz /22 —a2 = ae diz= @ane therefore OD een OZ A/ x2 — a2 Ke Po, es Ti which is integrated by the rules given above for rational fractions. These methods may be applied to the rectification of the parabola, meee 4 2 dy? etl BEY of which the element 4/¢z2+dy2 1s dayz + - 1g * 146 Integral Calculus. 4 y? dy Jit rc ; d 72 ha We first free y? of its factors by writing it act e+ y? ; and y2 then make AG +y2?=y+2. 144, When there is no radical but a square root, and no powers of x but even powers, we make the radical equal to a new variable dx multiplied by the given variable. If, for example, we. had ~——— » ¥ % /a2 — x2 we might make 4/a@2—a2—=2z. If there were a second term under the radical sign, we might, notwithstanding, make use of this trans- formation, having first made the second term disappear, at least when there is no power of x without the radical sign. 145. Finally, we may, with a view to making a quantity rational, put the variable or any fraction of the variable equal to a new variable or some fraction of one, in which we leave something indeterminate, which may serve to effect the object in view. For example, to ascer- tain in what case we might render the quantity amdaz(a-+-ba” )P rational, we should make (a+b )? = 21, q being indeterminate. L ZP— A 4 We should have a+ b2* =z? ; ior mata fl Lo Be aaa q etm ps ZP—a \” ; np b i ¢ ( b : therefore q GI ak. g wi ye ht eT m ff b Bay bees Zz : a dz(a-+bx) iyb P az( b ) ; which is integrable, whatever may be the value of g, when i n is a positive whole number or zero (85), and which may be rendered : ; m -+- 1 rational by making g = p, when MaDe 1 is a negative whole n . k number. And if the value of p is+-5, & being an odd whole num- ber, we may reduce it to the case mentioned (143), by making g=&, im +1 k' Tie has for its value + ot k’ being an odd whele number. 146. We shall extend transformations of this kind no farther. 7 Integration of Haponential Quantities. 147 We shall only observe that certain integrations are often facilitated ; J by making the variable equal to a fraction, such as —. For exam- z a15dxtadz 2 0 +- vl ey —2z>dz—az 8 dz rer yore which may be reduced by division to a series Adz 7 mEpyN which we f 1 ple, if we had ; by making asi we should have of simple quantities, and a quantity of the form already know the integral. On the Integration of Exponential Quantities. 147. There are no other rules to be given on the differentiation of these quantities, than that we should endeavour to decompose them into two factors, one of which shall be the differential of the logarithm of the other, or a constant part of it (27); and then divide by the differential of the logarithm of this second factor. ‘Thus we see that u (ayte oe LED ) is integrable, because the factor dy lz + yd : x is the differential of y / x, the logarithm of z¥; we therefore have, as dy lz" ) the integral, 2 +- sack (esabal ine + C; that is, (aytx+ 2) xy —— + C, PLETE ee - or 77 -+-C. By the same rule, we see that dx e** is integrable, be- cause dz is the differential of the logarithm of e¢? , divided by a con- stant quantity. We have therefore fa x ett —— CE ivobl = cane adtle nace When eis the number whose logarithm is 1, the rule is reduced to Se the proposed differential by the differential of the exponent of e. If we had xd zx e** to integrate, e being the number whose loga- rithm is t, we might do it, when m is a positive Whole number, by making findret = 02 (Am 4 Bm-14 Erm? +4 &e. +h), If, for example, we have x? dx e*” , we suppose frtdrex =e (Ar? + Br + EB). Differentiating (27), and then dividing by dze*” ,we have = * ne 148 Integral Calculus. a2? — Aaz?+aBrc+ak ' +2Az+B; whence Aa=1,aB+2A=0, aE+ B=0; that is, A= +, B= B65 Blas a? wherefore the integral J x? dx e?* is 2 lene ia )+e The number e, whose logarithm is 1, may be eriiced with ad- vantage for the integration of many quantities, especially when they contain logarithms. For example, if we had to integrate a” dx (1x)™, we should make /a=—2z—2zle; whereforez = e?; dta=dze'; and consequently 2” daz (lx)™= z™dze (+1) | which is integrated in the same case as the preceding and in the same manner. | On the Integ ration of Quantities with two or more Variables. 148. If we examine the rule given for differentiating quantities with several variables, we shall see, that in order to integrate differ- entials with several variables (when it is possible), we must collect all the terms affected by the differential of a single variable, and integrate them as if there were no other variable but that, that is, as if all the others were constant If we then differentiate this integral, making all the variables vary successively, and subtract the result from the proposed differential, the integral, thus found, is, after a constant quantity is added, the true integral, provided there be no remainder. If there be a remainder, it will not contain the variable, with reference to which the integration has been performed ; we pursue with the remainder the same process as before, and so on with each variable. If, for example, we had sar ydzr+ridy+torytdy+yodz; we should take the two terms affected by dz, viz. da? ydr+y* dz, and integrate them as if y were constant. ‘The integral is By fyea. | Now this quantity, being differentiated with reference to z and y, and the result subtracted from the proposed differential, nothing remains; we thence conclude that the integral is x? y 4 ys a+. If we had eedy+3r2ydz+22 dz+2r¢zdz+2dr+ y2 dy; by collecting all the terms affected by dx, and integrating, 2 and y 2 being regarded as constant, we should have z° y+ 2? z+ > Sub- tracting the differential of this quantity, considered as bavi three variablesy from the proposed differential, and there remains y? dy; 3 we therefore take the integral of y2 d y, which is a and adding it, together with a constant, to the integral already found, we have Integration of Quantities. 149 r2 3 BY peer a he TOC, for the integral. 149. But as it is not always possible to integrate a differential with several variables, it will be well to point out a character, by which it may be known when it is so. . 150. To this end, it must be observed, that if, in any quantity Q, composed in any manner of two other quantities x and y, we at first substitute for 2, a certain quantity p, and in the result substitute for y, we shall liave the same final result, as if we had first substituted q for y, and afterwards p for x. This is evident. 151. It thence follows, that if we differentiate a quantity Q com- posed of x, y, and constants, making first only x variable, and then differentiate the result, making only y variable, we shall have the same final result, as if we had first differentiated, making only y va- riable, and afterwards differentiated this result, making only z varia- ble. Indeed, let us conceive, that by substituting first «+ dz for x, Q becomes Q’; we have Q’— Q for the differential. If, by substituting y-+dy for y inthis quantity, Q’ becomes Q”, and Q becomes Q’”, so that Q’ — @ becomes Q” — Q", we shall have Q” — Q” — Q’ -++ Q for the second differential. Suppose now we substitute in the contrary order, and since, by substituting y + d y, instead of y, in Q, it becomes Q’”, we shall have Q’” — Q for the first differential, on the supposition that y is varia- ble. If we now substitute x + dz, instead of x, in this quanty, Q will become Q’, as above, and QQ’ will become Q” (150), so that Q'’— Q will hecome Q”’ — Q’, wherefore the second differential will be Q” — Q’ — Q’’+ Q, precisely the same as before. Let us now suppose, that A representing a quantity composed of x and y, i dy indicates the differential of A, taken by making y y A : . only variable ; and si dx that of A, making z alone variable. In like manner, Uuee dzdy will indicate that A was first differen- didy tiated, supposing only z variable, and the result was then differentia- ted, supposing only y variable. 152. These explanations being thus given, let Adx-++ Bdy be an exact differential, and JM its integral ; we shall therefore have dMdz, dMdy cp a =AdzrtBdy; therefore dM dM therefore also ddM dA ddM dB dagen Wed ois Talat aay ead 150 Integral Calculus. ddM dA ddM dB. dzdy dy’ e dyd dz’ but it has just been demonstrated (151) that ddMdxrdy ddMdydz. or dzdy . dydu herefe ca" Sodas wherefore al fh oP wherefore = —— dys dy Gon also as dat that is, if Ad«+ Bd y be a complete differential, the differential of A found by making only y vary and dividing by dy, must be equal to the differential of B found by making only x vary and dividing by dx, Thus we perceive that 1 y3 dz-+-xy?dy isa complete differen- ERAS) Ay?) dy UE de® y2 du a —. We perceive, on the contrary, tial, because ; in fact, the first member is reduced 2 to egy and the second to dy that z y dx+2 2d y is not integrable because eee is not equal to d (2%) dx 153. If more than two variables enter into the nee differential, that is, if it be of the form Adz+ Bdy+ Cdz, it is necessary, in order that it be integrable, that we have dA dBdA dWUdB _ adc. dy dz’? dz dz’ dz dy’ indeed, we may successively consider z, y, and x as constant; and the differential, which has then only two terms, (since this supposi- tion makes either dz — 0, dx = 0, or d y — 0), must be neverthe- less a complete differential, since the proposed one is so. It must therefore, in each of these cases, have the qualities of complete differ- entials with two variables. It is easy, on the same principles, to find the necessary conditions of a greater number of variables. Remark. 154. Let us suppose that Q is an unknown quantity composed of z, y, and constants, and that we know its differential A d x found by regarding y as constant. If we wish to find the total differential of Q, we suppose it to be Adz + Bdy; then B must be such that fhe Wiis | we should have ay — ne therefore d B = a d x; we integrate, considering x only as variable, since x alone was made to vary in B. een Differential Equations. 151 We have B = ees z; whence Bdy=dy bid dz. Now dy | dy 2 since A d x is supposed to be the differential of @, found by making % vary, we have Q =fA d x, the integration being performed, con- sidering x only as variable; therefore the complete differential of Q dA or off A dxisAdx+dy Taba d x, or the integration must be performed considering y as constant dA dy On Differential Equations. 155. When the proposed differential equation contains only two variables, 2 and y, and we have in one member the quantities 2 and d x, and in the other y and d y, the integration is reduced, for each member, to the rules given for differentials with a single variable. Thus, the equation a 7” y" dz —=by!1 2" dy, which may represent any differential equation with’ two terms, has its indeterminate quan- tities separated at once by dividing by y” and by 2 , and becomes GU 3 di byt" dy; of which the integral is evidently ayge—r+tli rays ae m—r m—r 1 a eer | 156. But as it may happen that either one or both of the members of the differential equation thus .separated may not be capable of in- tegration algebraically, while the equation is nevertheless algebraical, or may at least he reduced to an algebraical form, it will be well to examine such of these cases as most frequently occur. C. If, for example, in the preceding equation, we had m —r = — 1, and g—n—=—l, the differential equation would be reduced to adxt bdy = , and we can have the integral of each member only by means of logarithms; so that we may havealx= bly +I1C# But this equation may be rendered algebraical, by writing it batty? +i Cor ls? 1 Cy? . Now it is evident that if two logarithms are equal, the quantities to which they belong must be equal; wherefore 2*= Cy’, an alge- braical equation. 157. If we had only g —x = — I, the differential equation would bdy bea a*—" dz — , of which the integral is axrym™—r+l — empresa but we may give this equation an algebraical form by baleinlete the + It is allowable to suppose that the constant quantity is a logarithm. 152 Integral Calculus. first member by 7 e, e being the number whose logarithm is 1 (114) ; for we thereby produce no change in equation. We shall then azgm—r+l1 have a ANC ae Sy a xP or, (making m—r+l1=p),le ? =ICy’, and consequently, axP e P =Cy>. Hereafter we shall always indicate by e the quan- tity whose logarithm is 1. 158. Let us take as a second example the equation dz rndxz = ————.. JT — 22 The second member expresses (110), the element of that are of a cir- d z cle whose sine is z and radius 1. Whence z is the sine of rv, nie Raters fore that is, of fndxornx+ C. We have therefore, for the integral, z== sin (nz+C). In like manner, from the equation —dz S122’ we should infer z = cos (n x + oe ndz=—= 159. In the same way, since oe 3 (110) expresses the element of that arc of a circle whose radius is 1, and tangent z, if we had d ; A Dice ree we should conclude z = tang (nx-+C). And if bdz we had xdx = Pe ee ; we should, in order to reduce it to the form of the preceding, make z =m u, m being a constant coefficient. bmdu Ue Mea ae ag 2 — We should then have acaiima ua supposing therefore f m a, we should have m = Ve which would give Ae. du oe. f P nad2t—6 eee du n ivi z whence we deduce aetae = 5 d x Vass wherefore w, or ate > Joes tang ( e anlat+ c) Therefore z = le tang (5 tr/af+C ): om Differential Equations. 153 160. In the expression sin (wa +- C), tang (n x + C), which have just been found, nx«-+- C expresses the absolute length of the are in parts of the radius 1. But as it is more convenient to employ the number of degrees, than the lengths themselves, it will be best, when we meet with such expressions, to estimate the arcs in degrees, which is easily done by dividing by the number of parts of the radius contained in a degree, that is, by 0,0174533 (125) or, which comes to the same thing, by multiplying by 57,2974166. Thus the sine of the arc whose length is 6, and the sine of the are which has a num- ber of degrees expressed by 6 X 57,2974166, are the same thing. ndx dy 161. If we had ————— — ——-——., whose two members express Vi a2 Af 1—y? the elements of two arcs which are to each other : : 1 : 2, and whose sines are and y; then, in order to integrate, we should make each member rational, by putting in the first, 4/7 —x2 = a/—_1—z, and in the second, »/1—y2 =ya/—i—t. The equation would be changed ¢ (143) into a =e : —, the integral of which isnlz= It +/C, whence we he Ct= 2"; and by substituting for ¢ and z their values, C (y V1 — V1 2) = (ts =1—- V1 22)", which expresses generally the ratio of the sines x and y of two arcs which are multiples one of the other. But in order to employ this equation, we must first determine the constant C. Now, supposing, as we may, that the two arcs have a common origin, then « and y must become zero at the same time. But in this case, the equation becomes —COft = (—vVi)*, or — C= (— 1)"; now (—1)" is + 1 or —1, according as n is even or odd; we have therefore — C =— 1, and C—-F 1, the upper sign being for the case in which v is even, and the lower, when m is odd. ‘Therefore, finally Fy SSI WV 1g) = (tI VT) t By squaring, 1 — x? = — 7? —22z4/_] + 22, whence 2] = SW _ (+1) f=1dz | _ —1l—2z? . ag ORE 5 1 —«22 Sia Ree Foc) fel ge ndt& nra/—idz wherefore, Wee z d. SRW. In the same manner, om ase ate ee , whence A/ 1 — y2 Vpnaey NA/ iat Cae ea TEE pare) dt ——————_—_—. == —————__ or an ose Zz t z t 20 154 Integral Calculus. y In each particular case, we may always make the imaginary quan- tities disappear; but the simplest way will be to transpose the whole to one side of the-equation, and make some of the real quantities equal to zero; we shall then find the remaining equation to be divisi- ble by 4/—1, and that it will be the same as that formed by making the sum of the real quantities equal to zero. If, for example, we make n — 2, we shall have yf + ye =? 2 VHT. iat | 12%, or A/T —y2 +2207? —1 4+ 20/71. /1 a2 —yr/—1=— 9. Making then the sum of the real quantities equal to zero, we have 4/1 —y2 +207—1=0; and the whole equation is reduced to 22/1 -/1— 22 —ya/—1 = 0. which being divided by 4/—1, gives 2tr/1—x2 —y =0, or y=2tA/i— x2 ; now, if we square this equation, and the equation /1— y2 + 227 —1=0, or rather Visi 1 — 227, we shall have the same result. We may find in the same manner the cosines and cotangents of ndt dy ip I+y? (111), by decomposing 1+ 22 into (I+2z we) (1—ar/—T), and 1 +- y? into (1 +ya/—1) (lL—y*/—1 ); we should then finish the work according to the rules laid down for rational fractions (133 and 136). : 162. While we are upon this subject, we will make known a mode of expressing the sine and cosine of an arc, which may be of use. dy 5 Let'd a= ; ——~ be the equation which expresses the relation meat between an are x andits sine y. If wemakea/i—y2 = y /—i—z; —dz d z we shall have d x — I aa i a a dx a/—j, of which the integral is 7 z = —ara/—7 +1 Cor lz=—2r/_ 1 le+1C (157), —2A/_] multiple arcs. For then we should integrate which gives z= Ce ; and substituting for zits value, we have nae fear — ta/_] , y f—1— a/1—y2 = Ce v . With regard to the constant C, it may be determined by observing that the are x and its sine must become zero at the same time. We shall therefore have A/—1=—= C; Als ean a TEA wherefore y 4/—1 — 4/i—y2 = —e Vv ; and consequently / i — y2 Oi A/ ae Ye squaring and reducing, we have Differential Equations. 155 l—e tM gV—-1_ oti Y= OF FO Se — «rA/_ ae A/a OW ae id a JS 1 then, as y is the sine of x, we have oN __ jw tV-i 2/1 If, in the second member of the equation ely es 7 / 1 — y2 =yr/_—t+e Ss ’ we substitute for y the value just found, we shall have tra/—y ets Aen ee € —e — «rf — 8 RL SURES Be het. % + e Vv , 1—y2 OF COS % = 2 f 06 Maen sb gray ae pens plat Smt ia 2 therefore for the cosine we have trf—1 Thaw, 1 Hee +e Oe as GRIT ANIRE DAT AIL Gather But to return to the integration of equations. — 163. When the indeterminate quantities are not separate in the proposed differential equation, it is best, before undertaking to sepa- rate them, to ascertain whether the equation be not integrable in the state in which it is. This may be known by examining (152) whether oo =< es supposing Adz+ Bdy=0 to represent the equa- tion. If this condition exists, we may integrate by the rules of 148. 164. It may however happen that this condition does not exist, while the equation is nevertheless integrable; which it may be some- times rendered by multiplying it by some factor composed of 2, y, and constants. Let this factor be P. Then APdx+ BPdy =O will be a com- plete differential. It is necessary therefore that di AP). d( BE} TDS Bi Be The equation is thus reduced to finding for P a function of 2, y, and constants, which may satisfy this equation. But as this investigation would lead too far, we shall only find P in the case when it contains only z and constants, or only y and constants. Suppose therefore that P is to contain only x, we shall have only dA dP dB as) mee ae whence we deduce dB dy dt Spanien +s Secaaneae ye 156 Integral Calculus. dA dB | 1 we shall therefore easily find P, if a is reduced to a function of x, as is necessary in order that P should be, as we sup- pose it, a function of x only. We might also find the factor, if it were required that it should be composed of a function of x, multiplied or divided by a function of y of a known form. 165. By this means we integrate in general any equation of the form | Ayidy +X yitid«t A“ yr dxr=0), X, X’, X” being any functions of x ; g and r any exponents. We might investigate whether it would not be made integrable by being multiplied by a factor of the form P y” , P being a function of 2, and nm an indeterminate exponent, and should find that this may be done by supposing n == — r. But it is simpler to reduce immedi- ately the whole equation to the form yi—-tdy t+ Fyt—-"t+1dz+ Fi dz—D), ei) 714 by dividing by X and y”’, and representing the quotients cs and - by #’ and #”. In order then to integrate this, we suppose that P is the factor; P being a function of z. We shall then have Py dyt FP yt-"t1di+FPdc<=0. Now if P is a function of z, FP will be so likewise, and J PF'd x will be reduced to the the integration of quantities with a single va- riable. It is only required then to render Pyttdy FP oir as a complete differential, which requires that WW nae mit 9 Nite ke a ite NT dx dy ; that is, that dF Gay _— adap q—r She eT ad hig eee d whence ou = (q—r+1) Fdz; integrating, IP = f(q—r 41) Fdr=f(q—r 4) Fdz.le, whence Pel q—rt+)) Faz, Substituting this value of P in the equation Py 4—"dy + &c. and integrating, we have ES ges 2) Fa F +1daeVQ—r+l) Fa mn 7° x Hy qr" ° he x = SEER | € + Fy te +C=0. We have added, no constant quantity in the integration of the equation which gives P, because, there being no condition to deter- mine it, we are at liberty to suppose it nothing. Lit ee 4 Differential Equations. 157 Let us take an orem Suppose that we have to integrate fg phat a ia + (ba2+ter+f)dxr=0. Multiplying by P, we ante Pd: y SPURS ie yt 2, it is then necessary that d ay P 2 ewe v ) OCW et Ae a, dy Bee es dP 1A% whence uf —; whence /P = alzor P=7*. EE Thus the equation becomes ai dy t+axt—!deytbrttedrterttidzt fadz—O, of which the integral is 61ers CaS RO Bo a ula eS ALS aR Oi 166. The general equation just integrated frequently occurs; and the method we have given may be applied in many other cases. The following may be useful hereafter. If we had two equations, dz4t-ady+(bx+cy) Tdt=o, kdz+tady+(We+ecy) Tdt=o, v, y, and ¢ being variable; a, b, c, a’, &c. constant, and J any func- tion of ¢, we might reduce the integration of these two equations to the preceding method in the following manner. We multiply one of these, the first for example, by an indeterminate constant coeffi- cient g, and adding it to the second, multiply the whole by a factor P, which we suppose to be a function:of #; we have (gP+kP)di+(gaP+a P)dy +((gbP+ 8 P)x«+(geP+ecP)y) Tdt=0. We now suppose this equation to be an exact differential. We must have (153) st, CE PEEP) _ ACC OP +H P): ist: (gcP+c' P)y) T) dt Bynes Char a) ty AA (eee e! B) er ae (g¢P-e' P)y) F) | dt ay | dy } P vod 3 od. ne) aa Sek int 9 But P being a function of t, this last equation, in which ¢ is considered constant, and therefore dt—0, is reduced to0=0. And the two others give P (¢-+h)-—= (gb -+0) PT, and (¢ aa’) <= c+c')PT; dP gb+0’ dP Sucre — T dt, and —— = P gk P gaa Be eh putting these two values of aa equal to each other, and dividing by whence jude wherefore 158 Integral Calculus. b b/ o / T dt, we have ® a athe call dis Erk gaya of the second degree, and which, being resolved, will give two values of g Supposing then g known, we shall easily find P, since the equation / / = etn Dt Civet ere ee T dt. ek mista SN the equation (g P-+k P) dx - &c. being actually an exact differential, if we integrate it, we shall have (¢P+kP)e+(gaP+a'P)y~+C=0; therefore if g indicates the first value of g found from the above equation of the second degree, and we represent by g’ the second value of g, and by P’, the value which P takes by substituting g’ for g, we shall also have (g’P'+-4 P')x+(g/a P'+a' P')y4+C= C’ being a new constant. Indeed, there is no reason why we should employ one of these values of g rather than the other. And from these two equations it is easy to deduce the values of « and y, which will be expressed in terms of ¢ and constants. If the function J’ of ¢, which enters into the two equations, were different in each, we should proceed in the same manner, but should consider g as a function of ¢, and integrate as we should an equation of two variables, @ and ¢. If there were four variables, z, y, z, and ¢, expressed by three equations of the form adt+bdy+cdzt(ertfy+thz) Tdt=0, and the function 7’ were the same in each, we should integrate in the same manner, by multiplying the second and third by the indeter- minate constant quantities g and g’; we should then add these two ' products to the first equation and multiply the whole by the factor P supposed to be a factor of ¢ only. Then, supposing this new equation to be an exact differential, we should find (153) the equations from which to determine g, g’ and P. ‘The equation which gives g or that which gives g’ will be of the third degree; from which we have three values for g, three corresponding values for g’ and three for P. Changing the constant for each value of g, this will furnish three integrals, by means of which we may determine 2, y and z in terms of ¢. We should proceed in the same way, if there were a greater number of variables, provided the equations were of the preceding form. The method would be. the same, if there were one or more terms expressed in ¢, dt, and constants only. 167. And if we had in general any number m of equations com- prehending m +1 variables, combined together in any way whatever, we should multiply the second, third, &c. to the last, respectively by g, 2’, g, &c. supposed to be indeterminate functions of these varia- bles, we should add them to the first equation, and multiply the whole by a factor P supposed to be a function of these variables, and then suppose the whole equation to be a complete differential. If, for example, we had the two equations Adz+Bdy+Cdz=0, Aidt+ Bidy+Cdz=0, an equation in which g will be Differential Equations. ° 159 we should multiply the second by g; adding it to the first and multi- plying the whole by P, we should have P(A+A'g)dz+ P(B+ B'g)dy+P(C+C g)dz=0. Now, in order that this should be a complete differential, we must have (53) mmabieirys 415) 6s) OE (BEB g) dy | dx : ACs ea oe a Goals dz ee dx co Di Nn a st CUBES a PS dz Pra ae peed that is ; d P d(A- A’'g) —— (A+ A’ oa EAN ah Sah at ata ys LMG Sk d( B+ Biz)" Te he (B+ Big)+P BER ae aT es dP d(A- A'g) raph im eR al aos epee eee d(C+C'g). TTA eAbin eS ks Mae ae 5 dP d(B+ B'g) dp d(C+C€'g) ies B' ——— - : diy SES = dz BES) Ar iB dz i CiOe AP Cian If, from the two last equations, we deduce values of pie an he Fy dy’ and substitute them in the first, we shall have, after all reductions are made, UAL Ab) G(BAB gy / ic SEE CS ake cee (C+ C2) x (SF Te) +(A+ Ag) d(B+ Big) d(C+Cg) x (ones wea rarmd ale aa, d(C+Cg) d(A+A’g) Bes QE ie ete Pt Rg ree dx dz an equation not depending on P. We then find for 2 a function of x, y, and z, the most general possible and one which will satisfy this equation. Having found g, we find for P a function of z, y and z, which shall satisfy any two of the equations first found above, which, indeed, often requires a great deal of research, but which at least is always possible. It is to be observed that if we had only a single equation, that is, if A’ = 0, B'=0, and C’ — 0, the last equation just found would be reduced to | GA 'd B a BY tae GO a a A ee ee A ek a ae! ee eek (Fs 7z)+ & Ty)? (3; aa) mh which being an equation of the conditions between the coefficients een | Re 160 Integral Calculus. A, B, C, shows that in order that a differential equation with three variables, Adx+ Bdy+ Cdz= 0, may be integrable, even when multiplied by a factor, the coeflicients A, B C must have the relation indicated by the equation When this condition is fulfilled, we determine the factor P, in such a manner as to satisfy two of the three equations d(AP) d(BP) d(AP)\ | _a(CP) d( BP) __d(CP) Ly uo Ga eee naa) mea eh hae We thus see what is to be done with a greater number of equations and a greater number of variables; and we may determine, in the same manner, what are the equations, in which it would be sufficient that @ should be a constant, or a function of one or two of the varia- bles, "8c. 168 When the proposed differential equation does not come under any of the cases already given, we must endeavour to separate the indeterminate quantities. Sometimes the common rules of algebra are sufficient to effect this; at other times transformations are neces- sary. But there are many equations with regard to which it is diffi- cult to determine what transformations are best. The equation ax d2z-+-byladi=y'dy(e+f x)" is sepa- rated immediately by division, and is the same as (a+ byt) mdr=yrdyle+ fay, ad x ye dy | (fey apby’ is similar to that of a binomial quantity with a single variable. And if we have gtidzi=—artydy+2abr? y2dy+ab?y> dy; we easily see that it may be written grdu= (t+*4+2b 2? y? + 62 y*)aydy, which may be written gxrdx=(2?+by?)? Xaydy. Now with a little attention we see that the separation will succeed if we make 22 +b y? =z; indeed, we thus have x? = z— by? and “xde=bkdz—bydy; by substitution, 3gdz—bgydy=azx ydy,; ao 0% bg + az? easily integrated. 169. As no general rules can be given for making the transforma- tions, we shall confine ourselves to some very general cases in which the separation is known to succeed. In general, the separation succeeds in all those equations with two variables, which are homogeneous, that is, in which the indeterminate quantities x and y have, in each term, either when combined or sepa- rate, the same sum of dimensions. which becomes the integration of which an equation from which we deduce =ydy, which is Differential Equations. — 161 For, suppose Adzx+ Bdy =Oto be a homogeneous equation, and that we divide the whole by a power of z, whose exponent is equal to the number of the dimensions of the equation, it is easy to perceive that there will remain in A and B only powers of 2 and constants; so that the equation will be #dz+ F’'dy=0, F and F’ being functions of “ and constants. This done, since ey OD lll dy—ydz oe cr — 72 we shall have dz — d (2 )+eay; if therefore we make q Sah we have dz = have — = 207 4 OY, Pdy= 0, F' and F" being now func- AEM and substituting for dz, we tions of z and constants. Now this equation gives dy Fdz yo F2etk' 2?’ an equation entirely separated, since # and F” contain no other varia- ble than z. For example, if we had y?da+y22dy+623 dy =O, which is homogeneous, and the number of whose dimensions is 3, we should divide by z°, we obtain Ay vE et ties dy body. 0; making, therefore, t= %, OF % = a, we should have zdy—ydz substituting in the proposed equation, it becomes z2dy—yzdz+22dy+bdy=0, dy zadz whence 7% Es B22 1b? Ly=}1(222+ 6)+1C; whence y= C (222+ b)#, or y* = C* (222+ - 5B), 2 or finally y* = C4 (2 5 + 6b ), restoring in place of z its value - of which the integral is 170. It would be of great use to be able to render equations homo- geneous. But there is no general method for this purpose, and we are obliged to have recourse to transformations. ‘T’hose which prom- ise any success consist in making one of the variables, or a function of one, or even a function of two, equal to a function of a new vari- able with indeterminate exponents. These exponents are afterwards 21 162 ” Integral Calculus. determined by the condition that the transformed equation is to be homogeneous. If, for example, we wish to find the cases in which the equation aidztbyraidy+cy*dy=O0,to which form every equation of three terms may be reduced, may become homogeneous, we make 2 =z; we shall then have ahzmhrh—-ldz+byr2itdytcytdy =0. Now, in order that this be homogeneous, we must have k= qh--n, and k =mh-+h—1; whence | i BAAS nary, and k = REET m—gq-+1 m—q-+ l so that, if the exponents 4, qg, m, and x are such that this last equa- tion may be true, we may render the equation homogeneous and consequently effect the separation. 171. In general, from not having any direct methods, we endeavour to reduce the proposed equations to other equations whose integra- tion is known. We proceed thus, for example, with the particular equation dy+ay? dx = ba™dz, known by the name of the equa- tion of Reccati, and which can be integrated only for certain values of m. If m were zero, it would be dy +a y2dz= b dz, which is sepa- ; i we dx, which may be easily integrated. b— ay : But in order to integrate the equation when m has other values, we must endeavour to change it into another, in which ay? and b shall be multiplied by the same power of 2; it will then become sep- arable. The following is the method by which we find the values of m which allow of this transformation. We make y= Aap-—+<2?t; whence dy= pAw-—idz+qau—ttdz+2dt; by substituting, we have pAw—ldrt+tqu-lidz+udt+axr1t? dx+aA?2 22? dz Q2aArwtrividr=—birdx. rable, giving We suppose ) p—l=2p,pA+aA?=—0,p+q=q—1; ¢+2aA=0; 1 and we have p= —1, A= re Ban et 2; which changes the equa- tion into a? dt+ar*t?dx=bxdz, or dt+-az-2t2?dz=bw+2 dz, which will be separable if m = — 4. : odadle ye. If we make, in this last, ¢ = —, it will be changed into dz--bam+?72dz—azr-? dx. Making then z= A’ 2?’ ++ «t', and proceeding as before, we have pi A'aP—1 dx gat t da tel dt +b 22¢+m+24/2 J x + 5 A? Pmt 2dz4-2b Ala’ t+¢+m+2¢ dx—ar-2dt. Differential E'quations. . 163 If we suppose 2p'+m+.2=p’—1; p A’+bA2 —0; q +26 A’ =0, —1l=p'+q7+m+2; we shall have m—t-3 p=—m— 3, Aree teor 8 gq’ = — 2m —6, and e—2m—6 dt! t ba-8m—10 2 dy—a2z-2drz, or dt'+t-ba—-™-4t?2 dxr—=ar2+4 dz, which will be separable if —m —4—=2m-4, or if m== — Se 1 If we make ¢/ — — and afterwards z/= A” 2+ 29 ¢t’, and con- a tinue the same process as before, we shall find successively that the 12 16 20 —=—,m = apes —, &2C.; 5 waht aod —Ar ang Sa is being a positive whole num- equation is separable when m — — that is, in general, when m = ber. ) . Taking the above successive substitutions for ¢, ¢’, t’, &&c. we shall find that y has for its expression 1 ee Agel Nig oe yHAr +r (Alg-m-3 4 g-2m-6 1 (Aig-2n-5 fog Fm -10 1 (Alg-3m-7 + &c., continuing the processs until the exponent of z in the first term of the last denominator shall be—m+2.7—{—1; and then the second term of these denominators will be z-2™—47—® ¢: ¢ being a variable which, after the substitution of this value of y, is determined by the integration of the resulting equation, which is then separable. ‘The only exception is the case in which r = 1], in which we have only to make y = A 2z—1-+- 2-78. Let us resume the equation dy +a y?dz¢—=62™dzx and imagine that, instead of substituting at first y = A.2? —+-21t, as we have done 1 above, we first make y—=-, and then z= A2?tdat, and proceed as z above; we shall, in like manner, conclude, that we may effect the —A4r My separation wherever m = ol’ r being a positive whole number. r And the value of y will be 1 (Aig—2m—3 4 g—im—6 1 (A! ga 3m—s + &c. Continuing in the same manner until the first term in z in the last denominator shall be of the power —mr—2r-+ 1; and then the second term must be 2—-2™—47-+2 4. We may reduce to the same case the equation 164 Integral Calculus. aidy+ay?27*dt=bx"dz, by dividing by z?, and then making g7—I+l — zx, Such are the general methods to be employed when dz and dy do not exceed the first degree. As to equations containing different powers of dz and dy, as they cannot but be homogeneous with regard todz and dy, we divide the whole by dz raised to a power equal to the sum of the dimensions of dz and dy; we then resolve ed i atts eat ' the equation, considering oa as the unknown quantity. Then, as dx and dy will not be higher than the first degree, it will be per- ceived whether the preceding methods are applicable to the equation. On Differential Equations of the second, third, and higher orders. 171. The liberty we have (19) in a differentiation, of considering any one of the first differences as constant, will contribute in very many cases to facilitate the integration. But as it may happen that in a differentiation, we have considered as constant the differential not most proper to facilitate the integration; we must begin by show- ing how we may reduce a differential equation in which some one difference is supposed constant, to another in which there shall be no constant. We may then suppose what we please constant. Let therefore Adz?+ Bdzdy+ Cdy?+ Dddy= 0, be the equa- tion with two variables and second differences, in which the first difference d x of one of the variables has been supposed constant. After having divided this equation by d x, we write it C 2 Vp gay ep ae oe +Da{7!)=0, which is in fact the same, since, if we suppose d x constant, d (3*) L dd ! is equal to 2 . But if we do not consider dz as constant, then d dy\ dxddy—dyddz_. Uae ot ee dx? ; whence the equation is changed into 2 = gt Aa yee aT (Gee )= 0, dx d x2 in which there is no difference constant. Let Adz+ Bdz?dy+Cdy? dz+Ddy3+Edzrddy+Fdyddy + Gd — 0, be an equation with third differences, dz being always constant. We divide by dz?, and have Cdy? dy* Adz+Bdy-+ Fc! +D-, +E dt y ddy dy ddy dz a Figs dz __ A ae Ct ee ; a . Differential Equations. 165 oil which may_be written Adz+Bdy+ ese + Ba (5%) rife (i) 404((.) (2) = _and, considering every quantity as variable in the differentiations here indicated, we shall have an equation in which there will be no longer any constant differential. Let us apply these principles t to an example. Let dz?dy—dy%=adrddy+xudzrddy be an equation in which we have supposed dz constant. It cannot ‘be immediately seen how this equation can be integrated; but if we render d x variable, by writing it dudy — Bete =(adr+xrdz)d (5). we can, in the differentiation indicated in the second member, con- | sider dy as constant, and we shall have dy® dxd “Rippon le daca) ws which becomes, when reduced, dx? +xddzxtaddz—dy? = 0, of which the integral, as may be easily perceived, is tdxr+adxr—ydy+Cdy=0), adding a constant Cdy of the same order as the integral. This equation, integrated anew, gives gr? tar— 4 CytC=0. 172. Let us now examine equations with second differences and two variables. We give this name to those in which there is no dif- ference exceeding the second order, to whatever power dz and d y may be otherwise raised. We shall suppose one of the differences constant; but it will be easy to learn thence how to proceed if they were both variable. Let then A ddy+ B=0 be the general equation which may rep- resent any differential ‘equation of the second order, with two varia- bles and y, and in which d zis constant. A and B are functions of z, y, dz,d y, and constants. We write this equation in the form —k k Add vt (5 2 Jay + Fo ar=0, k being an unknown function of the same nature as A and B. We then multiply by P, supposed to be a function of z, y, dz, dy, and constants. We have se ’ AP ddy+P (~=*)ay t+ Fear=0, which we suppose to be a complete differential. 166. — Integral Calculus. This done, we have three differences, viz. ddy,dy, and dz. Considering these as the differences of so many different variables, we must have (153) d(AP) (Cal) (AP) _ (a) —— — dy. ddy ‘WhpdRwnaes ddvaie B—k Pk a(P( dy ))_ (7) dx Aish ThA ay From these three equations, we may, by the process employed in art. 167, deduce an equation in which P shall not occur, and which may serve to determine &, taking for & a function the most general possible of 2, y, dz, and d y, with indeterminate coefficients, which may be substituted in this equation. After which we may determine P, by taking, in like manner, a function of the same kind, and such as to satisfy two of these three equations. But we may simplify this ‘investigation, by confining it to finding for P a function of 2, y, dz, and d y, which shall satisfy two of the equations. The first two of the three equations just found, give ETAR\Ot wd 1. d(P(B—k)) Ay) ESE SN WY And ae eS bE or d(AP) | d(PB) 1 d(Pk) rae cary Sarees ddy dy ddy ’ ae d(AP)_ 1 d(Pk) dz *) idz ddy — d(P k) ddy’ Substituting, in the last equation but one, the value of , de- duced from the last, we have d(AP) _ a{PB)* We d(AP). dy 7a? B b+ 7 ddy dy dz’ whence we have es: d(PB) d (AP) , (AP) | whence it will be easy to find & when P is known. From this last equation we deduce d (PB) Pk=PB—dy TH | d(AP) ,a(AP). + dudy Syot ih od dy? substituting the value of P (B—k) and of P& in the equation d(AP) _ (Jana ) dy ddy ; and in the equation ‘ a Differential Equations. 167 se ae a( 22) nie, dy bin dx dx dy! and we shall have | d(PB) d (AP) d(AP) d(AP) ( gay) Lode it caren Oe ddy f and d (PB) d (AP) d(AP) d ( ———- — dz —__+ ( ddy ie dx wey dy ) Loy Mod PB dyd(PB) - A( AP)» dy? ad(AP) a a(7 dx ddy '°" da dz dy ) dy The question is therefore reduced to finding for P a function of z, y, dx, dy, and constants, which shall satisfy these two equations. But although this be always possible, it is not always easy; for which reason, we shall leave this general investigation, and examine some equations more limited, but still very much extended. We first observe however that it is easy, by the principles just given, to ascertain whether the equation is integrable in its present state, we have only to suppose P = 1, when, if the equation be integrable, it will satisfy the two following equations : dB dA dA aL Crea aed be bee ere = dy ddy ; dB aris d d A ddy hae ya) and ee d& ( dB dy dB dA ..dy? dA has (4 dx PEE Ucar ea es i) 2 aR hel i This is general, whatever may be the differential equation of the second order, d x being constant. 173. Let it now be proposed to integrate the equation Gdc?+ Hdidy+Kdy?+Lddy=0, in which the factor P which is necessary to render the equation inte- grable, need only be a function of x, y, and constants. It is sup- posed, moreover, that G, HT, K, and ZL, contain neither dz nor d y, but are only functions of 7, y, and constants. If we compare this equation with the general equation Addy+B=0, we have A= FL, and B= Gd22?+Hdzdyt+Kdy?._ Substi- tuting in the two equations found above, in order, to determine P, and observing that we have supposed P, G, H, K, and LZ to contain 168 Integral Calculus. d(PL) _ neither dz nor d y, we shall have : RA = KP; and a second : ; 5 (eye _ equation, which, after we have substituted in it for lone its value KP, is reduced to P d( PHar+ KPdy— an) a(PGds panes 7+) dx “a dy z But since catty ah 6 it dy we have (“53 dy d d(PL) 18 CS a dy ier czay. aaa CWS ae eas oe and consequently d(PL) d(KPdy) _ dya( dx ) iain hs, dy tinh he wherefore our second equation is reduced, after having divided each member by dz, to d(PH) _dd(PL) Ah d(PG) —— az dzdz dy d(PL) dy This equation and the equation == KP are the equations which we have now to operate on, in order to integrate the proposed equation. We observe now, that in this last equation, y only is to be consid- ered as variable. This being fixed, performing the differentiation indicated, and deducing the value of ed we have Pats. d dL. Betas 4 vid 3 taking therefore the integral, y only being considered as variable, since the differentiation was performed on that supposition, we shall have LPi= te dy ILE. We add the quantity 7 XY for a constant, by which we understand a dd(PL) didx to differentiate PZ, making x variable, and divide afterwards by d 2, anh differentiate the result, making z again variable, and divide y dx. t By the expression , 1s to be understood that we ought . Differential Equations. | 169 function of z and constants; because z has been supposed constant in the differentiation. | Eire Bk From this equation we deduce P = ra Sze y. If we substi- d (PH) tute this value of P in the equation “ — &c., and then divide by Wher oy we shall have an equation from which to determine ZX. But as Y must be a function of 2, it follows, that in order that the proposed equation be integrable by the multiplication of a factor composed only of x, y, and constants, all the y’s in this equation — must disappear. Let us suppose, for example, that we have the equation 2yd22+(21+3y2)drdy+2r2dy+2xyddy=0, which in its present form is not integrable. We have ; L=2y,G=2y, H=21+3y7,K=22?; therefore fife “2 y uy a? y Substituting this value of P, and those of LZ, G, H, &c. in the equa- d(P tion Rane bl &c., we shall have, after tr pape pier AXy 2ydX 2ZXy , sy2dX 3Xy2 yrddX a2 rdt 22 cde gee a atin Me making the sum of the terms affected by y equal to zero, and then dividing one of the equations by y and the other by y?, we shall have after making all the reductions, = ils \, Rady ar qip ah ee io g tak ie are ryt The first gives XY—=x*; and this value, substituted in the second, satisfies it; we have therefore Y= x®, and consequently rey If we now go back to the value of P/, found (172), we shall have Pk=2ry2dt+32? y2 dzdy, and P(B—kh)=2xrydrdy+t2r% ydy?; so that the equation, brought to the general form (172), becomes tsy2ddy+t(222yd7+2 23 y dy)dy + (2 ty?dt--3 t2y?2 dy) dz=0. In order to integrate, we follow the rule given (148); we first take a? y2 dd y, and consider d y only as variable, which gives 28 y? dy. Differentiating this quantity, considering all as variable, and subtract- ing from the equation, there remains (222 ydxr) dy+ (22ry? dr) dz. We integrate the first of these terms, considering y only as variable, 22 ae 170 Integral Calculus. and we shall have 2? y2 dz, the differential of which, taking 2 and y as variable, subtracted from the preceding remainder, leaves noth- ing; whence the integral is, when a constant is added, a y2dy+ 22 y2dz+Cdz=0 We may take as a second example, the equation 2da2+ (82+y+2)dydt+2c¢dy?+(22+2y)ddy =), which is integrated in the same manner. We shall find that XY must be equal to xz, and H= 2x y. 174. If, after the substitution of the value of P, in the equation d (PH) dx must give X is then a differential of the second order; whence it appears, that the method is, in this case, of no use. But it must be observed that the equation which will then be obtained, will be of the form , &c., all the y’s disappear of themselves, the equation which Adz2?+BXd224+CdXd1+EddX=0, A, B, C, E being functions of z and of constant quantities. Now, in order to integrate this equation, we must write it thus AP! dx?+BP' Xd 2?+-(C—k’) P'd xd P--k'P'd Pd2z+-EP'd dX=0. We now suppose that, P’ and 4’ being functions of 2 only, the last four terms taken together form an exact differential; then the first term, being a function of x, will be readily-integrated. The equations which result from this supposition, are d(EP') d(kP'dX+ BP'Xdzx) dx ddX ; d(EP') _d((C'—k’) P'dz) diskotea’ ddX : d(kiP'dX+BP'Xdr) d{(C—k) P'dX] DX Re an cae Mowe ahaa aes and d [(C— k’) Pi dx] bald d(BP'X dz) ae ae eee dx dX These four equations are reduced to the two following (from the consideration that k’, P’, A, B, &c. do not contain P), OE ond Bie tC eG dx dz d p' Deducing from each of these equation, the value of s : and putting one of these values equal to the other, we shall have, after making all reductions, Edk' + (C—k'’)d E—k' (C—k’)dz+ BEdt— Ed C=0, a differential equation of only the first order, and on which depends the value of XY, and consequently the integral of the proposed equa- tion. Supposing, therefore, that 4’ has been determined by means of this equation, we may easily obtain Pp’, by means of the equation Differential Equations. | 171 Wh otvesg eee 2%) aE chat Eau Bee wae H k' dz |g Pore fr ; HI being a constant quantity. When the values of k’ and P’ are found, we may find X, by substituting the values k/ and P’ in the equation AP dx? + BP'Xdx? +(C—k) PidrdX +k P'd Xdz + EP'dd X=0, and integrating. Now as this equation cannot fail of being a com- plete differential, we have for its integral dz JAP dz4 Xdcf BP'dz+aX fu P'dz+ Ldz=0, f, being a constant quantity. This is easily integrated by what has been already laid down (165). We may therefore find XY whenever we can find k’; whence it may be laid down as universally true, that whenever nothing is wanting to make the equation Gdz?+- Hdzdz+ Kdy?+ Lddy? =0, an exact differential, but a factor composed of 2, y, and constants, this equation will be always reducible to a differential equation of the first order, whatever may be the value of G, H, K, L. But if, after the substitution of the value of P, in the equation d( EP’) dx ki p' = and consequently neh?) @&c., the equation still contains y, which cannot be made x to disappéar, without subjecting the coefficients G, H, K, L, to cer- tain conditions, we conclude that the factor P must also contain z and dy; we must then have recourse to the general method (172). We might proceed in the same manner to ascertain in what:cases any other differential equation of the second order, of a known form, may be integrated by multiplication by a factor composed of 2, y, and constants, or of z, d y, d z, and constants, or of y, dz, and con- stants, Svc. 175. With regard to differential equations of the third order, if we suppose them to be represented generally by A d*y + B=0, A and B being functions of z, y, dt, dy, ddy, and constants ; if we sup- pose, moreover, that P is the factor composed of x, y, dz,ddy, and constants, which will render it integrable, we may write it thus B—k k—h Ve Then the following equations must be true. Cee ad(AP) _ _ ddy d(AP) _ dy i 9 ddy dey dy dy® Ph (B—k) ( “=*) asks S Pp d {| P—— say» 2 ae )iia( ddy i! dy ; Ciuc Go Y. dy ddy : 172 Integral Calculus. (BBY Ph a(Pray) CE) ys daw. er ee (k —h) Ph a(P dy Js «(7) ‘Ania ad Cle alba a) a. By means of these equations k, h, and P may be determined. But _ we shall carry this investigation no farther. The process would be similar for differential equations of still higher orders. 176. It may be observed in conclusion, 1°. That when one of the two finite variables is wanting in an equation, it may be always re- duced to an equation of a lower degree, by making dy = pdz, p being a new variable. 177. 2°. That the general equation d®y-+ad"—lyda+6a@—*yda2+&c...thyde+Xdx =0, a, b, &c. being constants, XY a function of z and constants, and dz being constant, may always be easily integrated by a method similar to that employed above, for the equation Adxv?+ BXdz24+CdXdt+hddX=0. To this end, it must be written + P(p—F’) a°—*yd22?+ Pk ge—2yd 272 +&c. crate ie + Phydat PXdx —0, P being the factor which will render the equation integrable, and which we suppose to be a function of x; and k, k’, &c. indeterminate constants. We shall suppose that the terms, taken two and two, beginning with the first, form an exact differential. This supposition will give the equations necessary for determining P, k, k’, &c. Having put ‘the values of equal to each other, we shall have equations in terms of k, k’, &c. by means of which & may be determined by an equation of the degree m. ‘The value of & being found, we may easily find that of k’, k’, &c. and that of P will be obtained by in- tegrating, which will be without difficulty done. Then for each value of k, we shall have a particular integral, observing to add to each a different constant. From n—1 of these equations we may deduce the values of d y"—1, d y®—?, &c. and by substituting them in the last, we shall obtain the value of y in terms of z. 178. 3°. If we should have several equations in which the differ- ences were not multiplied together, except that they were multiplied by the constant difference, and in which the variables should not exceed the first degree, nor be multiplied together, we might inte- grate them by multiplying the second, third, &c. each by a constant factor p, p’, &&c. adding them to the first, and multiplying the whole by a factor P, supposed to be a function of the variable whose difference is constant. We should then decompose the terms affected by the differences of the same variable, as in the preceding equation. » ae Differential Equations. 173 If, for example, we had Slade eG ak Se lag Le uh ane a aa an addz+b'ddyt(cedz+e dy)dz+(fz+e' y)dz?=0, by multiplying the second by p, adding it to the first and multiplying the whole by P, we should have P(afa)ddz+P(c+e'p)dzdz+P(f+fp)zdx? +P(b+0'p)ddy+P (e+ep)dydz+P (g+g'p)dz?=0. We should then decompose c+-c’p, into e+-c\p—kand k; and e-+e'p also, into e+-e’p—k’ and k’. Then supposing that the terms, taken two and two, form exact differentials, we should have the equations necessary to determine kf, £’, and P. The equation in terms of 2 will rise generally, to the degree 2x, which will furnish 2n integrals, by means of which we may eliminate all the differences and obtain the equations in terms of z and 2, of y and z, &c. 179. 4°. If the equations were still more general, p, p’, &c., as well as P, might be considered as functions of all the variables and their differences, and these functions might be determined by the condition of the total equation being a complete differential, ee. sean 6 ; ees Re | Hes : is on priv’ ot $8. Mia <. Wh hs ey ott ne o.9 Peeves wl ah Hire Abs ey oe beige bull eS CO thy 0 sbi #93 ay Bis ty a fab « ae pr it oY aetlearvaad Ade Ts! vont ow aye ind: awinogg batt ih way tt ae Uf iat ‘4 hy ott Oba oat ¥ ee a aks: ih bivosla ow pete Ap aie y 249” ehh 10h opie? gat BV) Sab a Nit totiaups one Boy: apie Pei ioe LO} yee gabait eworTi ips a Halevcatt i Wf Hout ose, Be Metastl a ott? 43 Bat ate ‘Hiv 2 0 din "? bag itt fats ort ip ainianits ¥8 th sp ae % 2p Hpi Ve bavi ee hye yet ess RD rs han, Gi 10-2 Bis Sid antetod | ii peor tal LS out, aiden hag. 1 BAA a yh ft dgisqey Math, tla, ond anoionpg' otk Th. ae Bye “hag aad Bitsy oth, td, Vor atk vijannsi) am, Nerrabin (lias schithgerte: SR ail Hea? 4 yore ue panigrest: yy ach. Hien aermdagiqud isi) | Flite AQ As aisilih ony Lise - hy Terns Alo! inte ih P gia at ‘Kalsahs Wit tas My) Ade guiginane My , , ly et hy) CDRs, P “0h ite . NOTES. Note referred to in Art. 95. SrvcE an equation to a conic section is always of the second de- gree, and since the most general equation of this degree may in every case be reduced to the form bt? + ¢ ut—teu2 +fttguth=—O, it follows that we may always make a conic section pass through five given points, provided that these points, taken three and three, are not in the same straight line, a conic section never meeting a straight line in more than two points. Suppose A, B, C, D, E (fig. 56) to be five given points, having this condition. If we refer these points to the line AD, which joins two of them, by drawing the lines BF’, CH, EG, at a given angle or perpendicular to AD, then the distances AF’, BF; AG, GE; AH, HC; AD, which are considered as known, may be regarded as the abscissas and ordinates of a curve line. Now we may always suppose that this curve line has for its equation b@2+cutteurtft+euth=O; for, let AF =n; BF =m; AG=n'; GE=m'; AH =n’; CH =m"; AD=n'"; then it is evident that, Ist. For the point A, we shall have « —0, and ¢ = 0, which reduces the equation to h = 0. 2d. For the point B we shall have uw =n, and t= m, which changes the equation into bm? +-cmn-—+en?-+ fm....gn=0, sinceh =O. 3d. For the point Ewe ‘shall have w =n’, and t= m’, and conse- quently 6 m'? + cm‘ n'--en'? fe fami} & n’=@. 4th. For the point C, we shall in the same maner find b m2 a ry mu! n!’ 1 ¢ n!2 + fm! +n = orn 0. 5th. For the point D, where t= 0, and wu =7n'", we shall have e nll? 1 gn! = 0, or enlL ¢ = 0. . Now as these four equations contain all the quantities c, e, f, g, of the first degree, it will be easy to find their values; then, by substituting them in the equation bi?+cut+eu?+fitgu+trz=—dO, or rather bt2?+cut+teur?+fi+gu=0, since 4 = 0, we shall have the value of c, e, f, g in quantities wholly known, and the equation will be divisible by 5. It will then be easy * , 176 Notes. — to construct the curve and to determine whether it be an ellipse, hyperbola, parabola, or circle. If only four points were given, one of the coefficients would be arbitrary; this would give the power of imposing, at pleasure, one condition; if only three points were given, two conditions might be imposed, and so on. © We distinguish lines by the degree of their equation. Thus the straight line, whose equation is of the first degree, is a line of the first. order. The conic sections are lines of the second order. It will be seen, therefore, that the above method may be used to deter- mine the equation of a line of the third order, which may be made to pass through as many points less one as the general equation of this order, with two indeterminates, has different terms. ‘The same may be said of the higher orders. The method under consideration will serve to connect, by an ap- proximate and simple law, several known quantities, the law of which is very compounded or unknown. Suppose, for example, that three quantities are known, which may be represented by the lines CB, ED, GF (fig. 57), and that thse quantities depend upon three others AB, AD, AF. It is proposed to find a quantity AT inter- mediate between the first, or situated near them, and which is de- rived from AH after the manner in which CB, ED, &c. is derived from AB, AD, &c. This question’ may be satisfied in an infinite number of ways by taking an equation with two indeterminates, w and ¢, having at least as many different terms as it contains such quantities as CB, ED, GF’. But among all these different ways, that which is the most readily applicable to the different purposes to be answered by this method, is to regard the line /H/ as the ordinate, and the line Af as the abscissa passing through the given points C, E, G, &c., and which has for this equation t=a-+butcu?+&e., by taking as many terms as there are quantities or points C, E, G; then by supposing as above, that w is equivalent to AB, ¢ will be equal to CB; and u being equivalent to AD, ¢ will be equal to DE; and w being equivalent to AF’, ¢ will be equal to GF’, and so on; we have thus as many equations for determining a, b, c, &c., as we have points. The values of a, b, c, &c., being determined, if we substi- tute them in the equation ¢ = a +- bu + cu? + &c., we shall have an. equation in which every thing will be known except u and £, accordingly if we put for uv the known distance AH, which answets to the quantity sought HJ, we shall have the corresponding value of t, or HI. If we would imitate the perimeter ABCDEF ( fig. 58), we should let fall perpendiculars from a certain number of Notes. 177 the points of this curve upon a determinate line HZ, since by the method just laid down, we can determine the equation of a curve that shall pass through all these points, and in which ¢ being of the first degree, u will be of the degree denoted by the number of these points less one; then this equation will serve to determine interme- diate perpendiculars, approaching so much the nearer to the true ones, according as we take in the first place a greater number of points A, B, C, D, &c. See Bézout’s Algebra, Art. 411. Nore 2. General Demonstration of the Binomial Formula. In Lacroix’s Algebra, (art. 136, & seq.) is given a demonstration of the binomial theorem for the case of positive integral exponents. The following demonstration of the same formula for the case of exponents of any value whatever, integral or fractional, positive or negative, is taken from the Elements of Algebra by Bourdon, (art. 202, & seg.) It may first be observed that the binomial +a may be put under the form 2 (1+<); whence it follows that (c+ a)™ =a” (: ae = x” (1-2), supposing “ eon b If then it can be shown that cn formula m— L m—2 (1-+-z)"=1 +mz+m— holds true, whatever be the ne of m, it vif follow, if we substitute © for z, and multiply by 2”, that x 2 (c-paynaem (1+me te eae ee a —I] —m@tLmam—itm— a a mala and that this last formula must be See as true. Now it has been shown that when m is any whole number (A/g. 136, p.) — 1 — 2 (1+ AV Re ao Nore 3. On the Method of Indeterminate Coefficients. From Bourdon’s Algebra, (Art. 208 and 209.) To give an idea of this method, let it be proposed to develope the expression EE aaa, in a series proceeding according to the ascend- ing powers of x The development may evidently take place 180 Notes. since pay > may be reduced to the form a (a’-++ b’ x)-1; and by applying the binomial formula we may find the development sought. _ Let us therefore suppose it performed, and that a Te A+ Br+C2r?+D2x3+ Ext +Fx+... (1) A, B, C, D, E,... being functions of a, a’, b’, but independent of x, and which, as their value is to be determined, are called indetermi- nate coefficients. In order to determine these coefficients, we multiply both mem- bers of the equation (1) by a’ + 6’ 2, arrange the terms according to the powers of x, transpose the term a, and we have Ada+Ba|,.+Ca' Da' Ea Ves eae oka Pio y(t pyle We now observe that, if we suppose suitable values to be given to A, B, C, D,... equation (1) must be verified, whatever value is given to x; the same is true of equation (2). Now if we suppose « = 9, the latter equation becomes 0 = A a’ ’ a : i —a; whence we obtain as the value of A, A= — and if Ais a a equal to —, when z= 0, it must preserve the same value, whatever a be the value of z, since by supposition, A is independent of 2: thus, whatever may be the value of x, equation (2) is reduced to Ba |x+Ca|z?+Da'\a3+... wi: Des iver Tae Tom ax or dividing by +Ab’+ Bo'| +COH'| Now as this equation must also be verified whatever value is given to x, we make x = Q, and the equation becomes B a’+ A b’/= 0, / / / from which we deduce B = — ca Oe Ose kd 6 — gee a a’ a’ a‘? As B must retain the same value, whatever may be that of z, we suppress in equations (3), the first term B a’ 4+- Ab’ which this value of B renders equal to 0, and divided by z, and we have o= | Ca + Da tt kKa'|2+... + Bb+ Cb'| + Dod! Again making « = 0, this equation becomes C a’ + Bb’ = 0, bh’ b/ / /2 whence we deduce C = — ey or C= — Bes POS: -— i ; a! a'2 a’ a's By the same process we should find D a’ + C b/ = 0, whence Ob! /2 / 13 Dotto org pie gees OO ee a qo a’ a’ Notes. 181 Here it is easy to perceive that any coefficient is formed from / ; apte L that which precedes it, by multiplying that coefficient by eee a 0 this way we have a a ab’ ab/2 a b/s a b/4 ee ey og ht a’ +b x a’ a‘? a‘s a'* als The fundamental principle of this method of indeterminate coefh- cients is this. Jf an equation of the form0 = M+ Nat P x? + Q2°+... (iM, N, P,... being coefficients independent of =), may be verified, whatever value is given to x, each separate coefficient | must necessarily be equal to 0. Indeed, since these coefficients are independent of 2, if we obtain their values, by making particular suppositions with regard to 2, the value obtained will still belong to them, whatever value is given to z. Now if we make x= 0, we find JZ = 0, and by dividing by a, the equation is reduced to 0=—=N+Pr1+ Q27?-+...; if we make again in this new equation 2 = 0, we find V = 0, and by dividing by x, the equation is reduced to O= P+ Qa+t...&e We have therefore separately M=0,N=0,P=0,Q=0...; and by this means we obtain as many equations as there are coefli- cients to be determined. The application of this method requires that one be previously acquainted with the mode of developing with reference to the expo- nents of z. In ordinary cases the development may proceed accord- ing to the different ascending powers of 2x, but sometimes the expression must be separated into factors, before it is developed. The expression 1 aire a eat Tae , and we must then Suppose “eran ~(A+Br+C24+D04.. iy 1 avant for example, must be put under the form 182 Notes. Nore 4. Of the Methods which preceded and in some Measure supplied the Place of the Infinitesimal Analysis. There are several methods of resolving questions analogous to that of the infinitesimal analysis ; and although there are none which unite the same advantages, it may not be the less curious to exam- ine the different points of view under which this theory may he regarded. On the Method of Exhaustions. This is the method which the ancients made use of in their diffi- cult researches, and especially in the theory of curved lines and curved surfaces, and in the estimatiom of the areas and solidities contained by them. As they admitted no demonstrations which were not per- fectly rigorous, they would not allow themselves to consider curves as polygons of a great number of sides. But when they wished to dis- cover the properties of a curve, they considered it as the fixed term or limit to which the inscribed and circumscribed polygons continu- ally approach, and as nearly as we please according as the number of sides is increased. In this way they exhausted, as it were, the space comprehended between these polygons and the curve; which cir- cumstance doubtless procured for this mode of proceeding the name of the method of exhaustions. As the polygons thus made use of were known figures, their con- tinual approximation to the curve, so as finally to differ from it by less than any given quantity, led to the knowledge of the proper- ties of the curves under examination. But geometricians were not satisfied with thus inferring or divin- ing, as it were, the properties of curves; they would have them verified incontestibly; this they effected by proving that any sup- position contrary to the existence of these properties led necessa- rily to some contradiction. This kind of demonstration was called reductio ad absurdum. By this means, having first ascertained that the areas of similar polygons are to each other as the squares of their homologous lines, they inferred that circles of different radii are to each other as the squares of their radii. ‘This is the second proposition of the 12th book of Euclid, and the 287th article of Legendre’s Geometry. Analogy led them to this conclusion, by imagining regular polygons of the same number of sides, to be inscribed in the given circles. For, as upon Notes. 183 increasing to any degree the number of these sides, their areas remain _as the squares of the radii of the circumscribed circles, they easily perceived that the same thing must hold of the circles, to which these polygons continually approached. But this was not enough. It was necessary rigorously to demonstrate that this is true in fact, and this they did by showing that every contrary supposition necessarily leads to an absurdity. In this manner the ancients demonstrated that the solidities of spheres are to each other as the cubes of their diameters, that a cone is the third part of the cylinder of the same base and altitude ; propositions which are contained in the fourth section of Legendre’s Geometry. By means of inscribed and circumscribed figures, they also dem- onstrated the properties of curved surfaces and of the solidities con- tained by them. The law of continuity led them to the conclusion, and the conclusion was verified by a reductio ad absurdum. In this manner Archimedes demonstrated that the convex surface of a right cone is equal to a circle which has for its radius the mean proportional between the side of the cone and the radius of the base, that the whole surface of a sphere is equal to that of four of its great circles, and that the surface of a spherical zone is equal to the circumference of a great circle multiplied by the altitude of the zone. It was also by a reductio ad absurdum, that the ancients extended to incommensurable quantities, the relations which they had discov- ered between commensurable quantities. ‘This method is certainly very beautiful, and of very great value. It carries with it the char- acter of the most perfect evidence, and never permits its object to be lost sight of; it was the method of invention among the ancients, and is to this day very useful, because it exercises the judyment, accustoms one to rigorous exactness in demonstrations, and includes the germ of the infinitesimal analysis. It is true that it requires some considerable exertion of mind; but is not the power of. pro- found meditation indispensable to all those who would penetrate into a knowledge of the laws of nature? And is it not necessary early to form the habit, provided we do not sacrifice too much time to its attainment ? On observing with attention the processes made use of in the method of exhaustions, we perceive that there is a great resem- blance between them and those used in the infinitesimal analysis. In each, auxiliary quantities are employed, always containing some thing arbitrary in their statement; from considering the properties of these quantities, inferences are drawn with regard to the un- 184 Notes. known properties of the curve or other quantity in question, ‘The auxiliary quantities are then omitted, and the desired result re- _mains freed from every thing uncertain or arbitrary. But “though few things more ingenious than this method have been devised, and though nothing could be more conclusive than the demonstrations resulting from it, yet it labored under two very considerable defects. In the first place, the process by which the demonstration was obtained was long and difficult; and, in the sec- ond place, it was indirect, giving no insight into the principle on which the investigation was founded. Of consequence, it did not enable one to find out similar demonstrations, nor increase one’s power of making more discoveries of the same kind. It was a dem- onstration purely synthetical, and required, as all indirect reasoning must do, that the conclusion should be known before the reasoning is begun.” * In the hands of Newton, this doctrine made great progress towards perfection. His prame and ultimate ratios, are precisely the ratios made known by the gradual approximation of the auxiliary * quantities to the quantities whose properties are sought. By this theory Newton extended the principles of the method of exhaustions, and he simplified its processes by freeing it from the necessity of having its results verified by a reductio ad absurdum, and by showing that these results are sufficiently established by the accuracy of the mode employed to obtain them. Newton thus expresses himself in the conclusion of the view he gives of his theory. ‘‘These lemmas are premised to avoid the tediousness of deducing perplexed demonstrations ad absurdum, ac- cording to the method of the ancient geometers.” t This great man advanced this doctrine far more considerably, by reducing this very method of prime and ultimate ratios to an algo- rithm, in his method of fluxions. By means of this calculus he in- troduced into algebraical analysis, not only these prime and ultimate ratios, but their terms taken separately, which was a modification of great importance, on account of the new means of transformation which it furnished. Newton, however, did not enjoy this glory alone; he shared it with Leibnitz, who had the advantage of pub- lishing his algorithm first, and who, being powerfully seconded by other celebrated geometricians, associated with him, advanced his method far more rapidly than the method of fluxions was brought forward during the same time. * Playfair. View of the Progress of Mathematical and Physical Science. t Scholium to Lemma XI. Sec 1, Book I. of the Principia. " att ee Notes: 185 On the Method of Indivisibles. Cavalieri was the forerunner of the inventers of the infinitesimal ‘analysis, and opened the way for them by his Geometry of Indivisibles. | He was led to this by a tract of Kepler, on the measure of solids, in which he introduced, for the first time, the consideration. of infinitely great and infinitely small quantities. In the method of indivisibles, solids are considered as composed of an infinite number of-parallel surfaces, surfaces as composed of an infinite number of lines, and lines of an infinite number of points. These suppositions are certainly absurd, and should be employed with caution. They are rather to be considered as means of abbreviation, by the help of which, we obtain readily and easily, in many cases, what would require long and. laborious processes by the method of exhaustions. For example. Let AB (fig. 59,) be the diameter of a semicircle AGB, ABF'D the circumscribed rectangle, CG the radius perpendicular to DF’; draw the two diagonals CD, CF, and, through any point m of the straight line AD draw the straight line m p g perpendicular to CG, cutting the circumference in the point 2, and the diagonal CD in the point p, Conceive the whole figure to turn about CG as an axis; the quadrant ACG will generate the solid volume of the hemisphere whose diameter is AB, the rectangle ADGC will generate the cir- cumscribed right cylinder, the right angled isosceles triangle CGD will generate a right cone, having for its altitude and for the radius of its base the equal lines CG’, DG; finally, the three straight lines mg,ng,pg, will each generate a circle, whose centre will be at the point g. Now the first of these three circles is the element of the cylinder, the second is ihe element of the hemisphere, and the third that of the cone. Moreover, the areas of these circles being as the squares of their radii, and these three radii being the hypothenuse and sides of a right angled triangle, (since Cg = pg, andmg=Cn), it is evident that the first of these circles is equal to the sum of the two others: that is, the element of the cylinder is equal. to the sum of the corresponding elements of the hemisphere and cone, and,. as it is the same with all the other elements, it follows. that the total solidity of the cylinder is equal to the sun of the total solidity of the hemisphere and the total solidity of the cone. But we know that the solidity of the cone is one third of that of the cylinder, therefore the solidity of the hemisphere is two thirds ; 24 186 Notes. that is, the solidity of the entire sphere is two thirds of the solidity of the circumscribed cylinder, as was discovered by Archimedes. Cavalieri professed to consider his method as only a corollary of the method of exhaustions, but confessed that he could not give a rigorous demonstration of it. The great geometers who succeeded him soon caught the spirit of this method, and it was in vogue with them, until the discovery of the new mode of calculation. It was to this that Pascal and Roberval owed the success of their profound researches on the cycloid. The former of these distinguished authors thus expresses himself in relation to this subject. ‘For this reason, I shall not hesitate hereafter to make use of the language of indivisibles —the sum of the lines, the sum of the planes ; I shall not hesitate to use the expression, the sum of the ordinates, which seems not to be geometrical, to those who do not understand the doctrine of indivisibles, and who think it is sinning against ge- ometry, to express a plane by an indefinite number of lines. But this comes of their not understanding it, since nothing is meant thereby, but the sum of an mdefinite number of rectangles, each formed by an ordinatesand one of the small equal portions of the diameter, the sum of which is certainly a plane. So that when we speak of the sam of an indefinite multitude of lines, we have’ always reference to a certain straight line, by the equal and indefinite por- tions of which they are multiplied.” This passage is remarkable, not only as it shows that these geom- eters knew how to appreciate rightly the merit of the method of indivisibles, but still more, as it proves that the notion of mathe- matical infinity, in the very sense which is at this day given it, was not unknown to them. For it is evident, from the passage just cited, that Pascal attached to the word zdefinite the same signification which we attach to the word infinite, that he called by the word small, what we understand by infinitely small, and that he neglected, without hesitation, these small quantities by the ‘side of finite quantities: for we see that he considered as simple rectangles the trapezoids or small portions of the area of the curve, which are comprehended between two consecutive ordinates, neglecting, consequently, the small mixtilineal triangles, which have for their bases the difference of these ordinates. No one, however, has dared to reproach Pascal with want of rigor. A | We shall conclude the notice of this method with one or two examples. Common algebra teaches us how to find the sum of any number of terms taken in the series of natural numbers, the sum of their ay Notes. 187 squares, that of their cubes, &c., and this knowledge furnishes to the geometry of indivisibles, the means of estimating the area of a great number of rectilineal and curvilinear figures, and the solidities of a great number of bodies. Let there be, for example, a triangle; from its vertex let falla perpendicular upon the base; divide this perpendicular into an infi- nite number of equal parts, and through each of the points of divis- ion draw a straight line parallel to the base, and terminating in the two other sides of the triangle. According to the principles of the geometry of indivisibles, we may consider the area of the triangle as the suin of all the parallels which are regarded as its elements; now, by a well known property of triangles, these straight lines are proportional to their distances from the vertex; therefore, the altitude being supposed to be divided into equal parts, these parallels increase in an arithmetical progres- sion, of which the first term is zero. . But in every ariththetical progression, whose first term is zero, the sum of all the terms is equal to the last multiplied by half the number of terms. Now, in this case, the sum of the terms is represented by the area of the triangle, the last term by the base, and the number of terms by the altitude. Therefore the area of every triangle is equal to the product of its base by half its altitude. Again; let there be a pyramid: From its vertex let fall a perpen- dicular upon the base, divide this perpendicular into an infinite number of equal parts, and through each point of division, let a plane ' pass parallel to the base of this pyramid. According to the principles of the geometry of indivisibles, the intersection of each of these planes, with the solidity of the pyramid, will be one of the elements of this solidity, which will be simply the sum of all these elements. But by the properties of the pyramid, these elements are ta each other as the squares of their distances from the vertex. Calling the base of the pyramid B, its altitude A, any one of the elements just mentioned 4, its distance from the vertex a, and the solidity of the pyramid S, we shall have B:b:: A: a, therefore B b = Az a; Therefore S, which is the sum of all these elements, is equal to B the constant quantity a2 multiplied by the sum of the squares a? ; 188 Notes. . and since the distances a@ increase in an arithmetical progréssion, (the first term of which is zero, and the last A,) that is, as the natural numbers from 0 to A, the quantities a? will represent the squares of these distances from 0 to A?. Now common algebra shows us that the sum of the squares of the natural numbers from 0 to A? inclusively is 2A%+3 A2+A Se TMT ET Wea But the number A in this case being infinite, all the terms which follow the first in the numerator disappear, by the side of this first term; therefore this sum of the squares is reduced to 1 A8. Multiplying therefore this value by the constant quantity =e found above, we shall have for the solidity sought Sea BAS that is, the solidity of a pyramid is the third part of the product of its base by its altitude. By a similar process it 1s proved that, generally, the area of any curve which has for its equation ay x", m ; , is ———-. XY; Y representing the last ordinate, XY the correspond- m +n ing abscissa, m,n, any exponents, whether integral, fractional, posi- tive, or negative. Thus the method of indivisibles supplies in some respects the place of the integral calculus; it may be regarded as corresponding to the integration of simple quantities, and this certainly was a great dis- covery for the time of Cavalieri. On the Method of Indeterminate Quantities. It seems to me that Descartes, by his method of indeterminates, approached very near to the infinitesimal analysis, or rather, that the infinitesimal analysis is only a fortunate application of the method of indeterminates. | Let there be an equation with only two terms A -|- Ba = 0; ® in which the first term is constant, and the second susceptible of being rendered as small as we please. According to what has been shown, (note 3,) this equation cannot hold unless the terms A and B x are each, separately, equal to zero. Therefore we may lay it down as a general principle, and as an immediate consequence of the method of indeterminates, that if the sum or difference of two pre- Notes. 189 tended quantities is equal to zero, and if one of the two may be supposed as small as we please, while the other contains nothing arbi- trary, these two pretended quantities will be each separately equal to zero. : | This principle alone is sufficient to resolve by common algebra all questions falling under the infinitesimal analysis. ‘The respective processes of the two methods, simplified as they may be, are absolutely ' the same. ‘T’he whole difference consists in the mode of considering the question. ‘The quantities which in the one are neglected as infi- nitely small, are wnderstood in the other, though considered as finite, since it 1s demonstrated that they must eliminate themselves, that is, that they must destroy each other in the result of the calculation. Indeed, it is easy to see that this result can only be an equation of two terms, of which each is separately equal to zero. We may therefore beforehand, suppose to be understood in the course of the calculation, all those quantities which belong to that one of these two terms, of which no use is to be made. Let us apply this theory of indeterminates to some examples. Let it be proposed to prove that the area of a etree is equal to the product of its circumference by half the radius; that is, that calling this radius #, the ratio of the circumference to radius, w, and consequently the circumference ma #, the surface of the circle S, In order to do this, we inscribe in the circle a regular polygon, then double successively the number of its sides, until the area of the polygon differs as little as we please from that of the circle. At the same time, the perimeter of the polygon will differ as little as we please from the circumference of the circle, and the straight line drawn from the centre to the middle of a side, as little as we please from the radius R. ‘Then the surface S, will differ as little as we please from $ a R?; consequently, if we make S = i @ KA +g, the quantity », if itis not zero, may at least be supposed as small as as we please. Now we put this equation under the form (S — 32 R?)—o= 0, an equation of two terms, the first of which contains nothing arbitra- ry, while the second, on the contrary, may be supposed as small as we please; then, by the theory of indeterminates, each of these terms separately is equal to 0; thus we have S—d2rh? =0,orS=472 R?; which was to be demonstrated. 190 Notes. Let it be proposed to find the Solidity of a Pyramid whose Base is B, and Altitude A. Conceive this pyramid to be divided into an infinite number of horizontal segments of the same thickness. Hach of these segments may evidently be regarded as composed of two parts, of which one isa prism having for its base the smaller of the two bases of the segment, and the other is a kind of ungula surrounding this prism. | Let « be the distance of any one of these segments from the vertex of the pyramid, and 2’ the thickness of the segment. In the pyramid, the areas of the sections made parallel to the base are as the squares of their distances from the vertex; consequently the superior or smaller base of the segment, at the distance x from the vertex, is ae t Accordingly the solidity of this segment, without its un- gula, is an “2 x'; therefore the total solidity of the pyramid, without the ungulas, is the sum of all these elements. And since 2’ may be supposed as small as we please, each ungula may consequently be supposed as small as we please, relatively to the solidity of the seg- ment; whence the sum of all the elements differs as little as we please from the solidity of the pyramid sought. Calling, therefore, this solidity S, we have S = sum ( v2 x/ + 9, g designating a quantity which may be supposed as small as we please. But since B, A, and 2’ are constant quantities, that is, the same for all the segments, it is evident that B 7/8 me sum -——— x2 2’ 1s the same as ———sum @ —} . A2 A2 a! ae ; Now — is evidently the number of segments comprehended be- x cA ' tween the vertex and x, therefore sum (=) for the entire pyramid, & ‘ A is the sum of the squares of the natural numbers from 0 to a But we know that this series of the squares of the natural numbers is. (- As, 98 AZ. 0A ) ie ROE» ? 1 6 x's a5 x2 Tika substituting this sum in the equation found above, we shall have , Brlt({2A% 8A? A Sm oa peal a +5-)+» Notes. 191 or, by transforming in order to separate the arbitrary terms from those which are not so, OPA A (ReR eT yies (Sas(Get+s)+e )=0, a perfectly exact equation of two terms, the first of which contains only definite quantities, or such as are not arbitrary, and the second may be made as small as we please. ‘Therefcre, each of these terms taken separately is equal to zero; whence we have from the first. S—iBA=0,orS=iBA, which was to be found. The solution here given is analogous to the method of indivisibles,, or rather it is the method of indivisibles rendered rigorous by certain slight modifications, derived from the method of indeterminates. We will now apply this method to the same question, making use of the notation of the infinitesimal analysis, to show how these methods. are related to each other, or rather that they are only one and the same method considered under different points of view. - Retaining the denominations made use of above, we have d S for the element of this pyramid. We have moreover as the value of the | same element, neglecting the ungula, | Ar a2 dx; we have therefore exactly os as a2 dx 4g, g representing a quantity which may be supposed as small as we please, relatively to each of the other terms. Taking, in each member, the exact sum of the elements, we have the rigorous equation S = sum = a2 fs +sumg.....-. (1). , B Now the common integral if ae 22 dx of the first term of the second member is B Sep eta Cs | 3 A? jaw? C indicating a constant quantity; but the exact differential of this integral is not B At. | te “ar COL it Is Be tg) eee dat a (32d? + dzx3), 192 Notes. \ that is, we have exactly B B et Be Nh lla @( sp 40) aay det Fy (32dx?2 + dx); taking then the exact sum of each member, we have B (sa: ae +C )=sum ap de} sum aa eo eee ae or, transposing, B B sum a Pi A re (: Az oe + ( )—sum We (32d x3 + a2). Substituting in equation (1), we shall have exactly So (sa ze +c)— (sum a (8 ad22 +d x3) — sum 9 ) an equation in which the last term only contains arbitrary quantities, and may be supposed as small as we please. For the sake of con- ciseness, make this term 9’; the equation will become, by transposi- tion, B (s— (yaa: oh ©) A fafa an equation of which, by the principles of the method of indeter- minates, each term taken separately is equal to zero, whence R 3 = 53 tat. G. 3 A2 oh In order to determine C, we have only to make 2 =O, then we have S= 0, whence C = 0; wherefore the equation is reduced to S == ——~ a x3, 3 A? that is, the solidity of the pyramid from the vertex to the altitude t 1S one in order therefore to obtain the whole solidity of the pyra- mid, we have only to suppose x = A, which will give S= 4 BA. This solution, as may be caailts seen, is no other than ‘that which would be obtained by the processes of the infinitesimal analysis, by neglecting nothing, and the common infinitesimal analysis is only an abbreviation of these processes, since we neglect only the quantities y, y', Which, in the result of the calculation, fall only on that one of the two equations into which the quantities are decomposed, of which no use is made. Now, what the infinitesimal analysis neglects may, by a simple fiction under the name of quantities infinitely small, be understood, in order to preserve the rigor of geometry during the whole course of the calculation. We thus see that the method of indeterminates furnishes a rigorous demonstration of um-nlé = Notes. 193 the infinitesimal calculus, and that it affords at the same time the means of supplying the place of it, if we choose, by common algebra. It were to ’be wished, perhaps, that this course had been pursued in arriving at the differential and integral calculus ; it would have been as natural as the method that was actually taken, and would have prevented all difficulties. | Of the Method of Prime and Ultimate Ratios, or of Limits. The method of prime and ultimate ratios, or of limits, has also its origin in the method of exhaustions, of which it is, properly speak- ing, only a development and simplification. We owe this useful im- provement to Newton, and it is in his book of the Principia that it is to be studied. It will be sufficient for our purpose to give here a succinct idea of it. When two quantities are supposed to approach each other con- tinually, so that their ratio or the quotient arising from dividing the one by the other, differs less and less, and finally as little as we please, from unity, these two quantities are said to have for their ultimate ratio a ratio of equality. In all cases, when we suppose different quantities to approach respectively and simultaneously other quantities which are consid- ered as fixed, until they differ respectively and at the same time, as little as we please, the ratios which these fixed quantities have to each other are the ultimate ratios of those which are supposed to approach them respectively and simultaneously, and these fixed quantities themselves are called limits or ultimate values of the quan- titles so approaching. These values and ratios are called ultimate values and ultimate ratios respectively, or prime values and prime ratios, of the quanti- ties to which they are referred, according as the variables are con- sidered as approaching to or receding from the quantities, considered as fixed, which serve as their limits. These limits, or quantities considered as fixed, may, however, be variable, as would, for example, be the coordinates of a curve; that is, they may not be given by the conditions of the question, but be only determined by the subsequent: hypotheses on which the calculation rests. ‘Thus, for example, though the coordinates of a curve are comprehended among the quantities called variable, be- cause they are not of the number of the data; yet, if I propose a problem to be resolvsd respecting any particular curve, as that of drawing a tangent to it, it will be necessary, in order to establish my reasonings, and calculation, that I should begin by assigning deter- 29 194 Notes. minate values to these codrdinates, and that I should continue to the end of the process to regard them as fixed. Now these quantities, considered as fixed, are comprehended, as well as the data of the problem, among the quantities called limits. These limits are the quantities whose ratio is sought. Those which are supposed gradually to approach them are only auxiliary quantities, which are interposed to facilitate the expression of the conditions of the problem, but which must necessarily be eliminated in order to obtain the result sought. We thus see the analogy which must exist between the theory of prime and ultimate ratios and the infinitesimal method. What, in the latter, are called infinitely small quantities, are evidently the same as the difference between any quantity and its limits; and those quantities whose ultimate ratio is a ratio of equality, are those which, in the infinitesimal analysis, are said to differ infinitely little from each other. | The principal difference between that which is called the method of limits and the infinitesimal method, consists in this, that in the former, we can admit into the process of calculation only the limits themselves, which are always definite quantities, while in the latter we may also employ the variable quantities, which are supposed to approach them continually, as well as the difference between them and their limits. ‘This gives the infinitesimal method more means of varying its expressions and its algebraical transformations, with- out introducing the least difference in the rigor of the processes. The property thus obtained by the infinitesimal method renders it susceptible of a new degree of perfection still more important, which is the power of being reduced to a particular algorithm. For these differences between the variable quantities and their lim. its, are what we distinguish by the name of differentials of their limits, and the simplification to which the admission of these quan- tities into the calculation gives occasion, are precisely what gives the infinitesimal analysis its importance. The method of limits or of prime and ultimate ratios, is neverthe- less far superior, for the facility of its processes, to the simple method of exhaustions ; since it is at least freed from the necessi- ty of a reductio ad absurdum for each particular case, by far the most difficult operation in the method of exhaustions; while, by the other method, it is sufficient, in order to prove the equality of any two quantities, to show that they are both limits of the same third quantity. 7 Notes. 195 If this method were always as easy in its application as the com- mon infinitesimal analysis, it might seem preferable, for it would have the advantage of leading to the same results by a process direct and always clear. But it must be confessed that the method of limits is subject to a considerable difficulty which does not be- long to the common infinitesimal analysis. It is this, that as the infinitely smal] quantities are always connected by pairs, and cannot be separated from each other, we cannot introduce into the combinations, properties which belong to each of them separately, nor subject the equations in which they occur to all the transfor- mations which would be necessary in order to eliminate them. sertihey ps Bt Sas hinds a oon Ah til A O3 ae ma! oe as a 7 1 Ay is A Adaed *. ho Hy! p ie ise ee Gi; ‘ 5) rt “ts Ltt. & Tnteg. Calculus Pl. Cambridge Mathematics moe tine Seis Nu Cambridge Mathematics LNt7- a Tnteg. Cateulus PLM. : ‘ >» as : 5 = - GS que - a ae - J J -_ a ¥ ‘ - a a a - a : — = - —_— = rr - — i = = = 4. > a » - 7 = = 7 7 al = ee = wi iene i ; ty - | 5 va ; ‘ , : i Wy : Teta AD , } 4 Mh a) va vay bits } ? ti be Yee ats UNIVERSITY OF ILLINOIS-URBANA 515B461P:E1836 C001 . FIRST PRINCIPLES OF THE DIFFERENTIAL AND 3 CAA 2 017224079 3