f 
 
 UC-NRLF 
 
 B M 5Mfl T3D 
 
 i- 
 
 
 «*, 
 
 ^ \ 
 
 A- . 
 
 «»•«?• V'7«S 
 
 - ■ %n^H 
 
 
 J&£,i$*s*- 
 
 ^3k 
 

 f///ff/'// ' s f>////// 
 
 
TREATISE 
 
 ON 
 
 ISOPERIMETRICAL PROBLEMS, 
 
 AND THE 
 
 CALCULUS 'OF VARIATIONS* 
 
 BY ROBERT WOODHOUSE, A.M. F.R.S. 
 
 FELLOW OF CAWS COLLEGE, CAMBRIDGE, 
 
 CAMBRIDGE: 
 
 Printed by J. Smith, Printer to the University ; 
 
 AND SOLD BY DEIGHTON, CAMBRIDGE; BLACK, PARRY, AND KINGSBURY, 
 
 LEADENHALL STREET; AND MAWMAN, TOULTRY, LONDON. 
 
 1810. 
 

* 
 
 £ 
 
 3 
 
 PREFACE. 
 
 There needs, perhaps, no other apology 
 for the present Work, than the mere statement 
 of the fact, that there is, on the same subject, 
 no English, and only one Foreign Treatise *, of 
 which the celebrated Euler is the Author. 
 
 The copies of this last, a work of uncommon 
 merit, are in this country very rare. But, not- 
 withstanding its various excellencies, a mere 
 translation of it would not have rendered unne- 
 cessary the present treatise, since, independently 
 of any other consideration, several of Euler's 
 most important processes are, by more concise 
 ones, now superseded. 
 
 But although there is only one separate Trea- 
 tise, the subject itself has been discussed in 
 several analytical works, English as well as 
 
 * Method us inveniendi Lineas curvas proprietate maximi 
 mininiive gaudentes, Lausanne, 1744. 
 
 3772*0 
 
11 PREFACE. 
 
 Foreign. Our countrymen are accustomed to 
 assign, in their treatises on fluxions, a chapter to 
 the maxima and minima of curves ; and, the 
 foreign mathematicians consider the same subject 
 under the head of " Calcul des Variations." 
 
 Maclaurin, Simpson, Emerson, especially 
 the two latter, have not explained the principles 
 of the subject with sufficient perspicuity and 
 precision; and, in point of depth of research, 
 there are various problems, to the solution of 
 which, their formulae are totally inadequate. 
 Indeed, which is remarkable, there is, I believe, 
 no English treatise that mentions Euler's 
 formula of solution ; which, for the simplicity 
 of its construction, and facility of application, is 
 exceeded by none in the whole compass of 
 analytical science. 
 
 The foreign mathematicians, Euler, Le Seur, 
 Bossut, and Lacroix, have furnished their treatises 
 on the differential and integral calculi, with 
 formula? of solution adequate to any case that can 
 be proposed. There is no deficiency on that 
 head. But, these Authors too suddenly carry 
 the reader into the middle of the business and 
 immerse him in calculations; and, if they soon 
 provide him with the instruments of solution. 
 
PREFACE, 111 
 
 they instruct him neither in the object nor the 
 principle of their construction. 
 
 This is not said in the spirit of an Author 
 preparing, by the censure of preceding Works, a 
 reception for his own. For the truth of the 
 statement, I would appeal to the experience of 
 all who have consulted the above cited Authors ; 
 besides, it is not difficult to assign the reason 
 of the alledged defects. 
 
 When Lagrange in ] 760, published his new 
 method of solving problems of maxima and 
 minima, he composed his memoir for mathema- 
 ticians, familiar with its subject, and well versed 
 in the researches of the Bernoullis and of Euler. 
 Accordingly, he very briefly states the principles 
 of his calculus, and enters into no explanation 
 of the nature of the subject. His compendious 
 method of computation, however, has been 
 adopted ; and subsequent Authors have com- 
 posed their treatises very much on the plan of 
 Lagrange's memoir, with some, but slight and 
 imperfect, preliminary explanation. These 
 Treatises, however, the student is expected to 
 understand ; that is, if the matter be fairly stated, 
 he is expected to understand an intricate sub- 
 ject, with advantages much less than consura- 
 
IV PREFACE. 
 
 mate mathematicians before him enjoyed ; since, 
 there is neither proper explanations presented to 
 him, nor is he directed, by way of preparation, 
 previously to consult the Works of Euler and 
 the Bernoullis. 
 
 Such are, in my opinion, the defects of ex- 
 isting methods ; still, however, I have not com- 
 posed a treatise on the subject, by merely 
 remedying them ; that is, by inserting formulae 
 of sufficient extent, and by more fully explain- 
 ing and illustrating their principles. But, on 
 a novel plan, I have combined the historical 
 progress with the scientific developement of the 
 subject ; and endeavoured to lay down and 
 inculcate the principles of the Calculus, whilst 
 I traced its gradual and successive improvements. 
 
 If this has been effected, which I think it 
 has, in a compass not very wide of that which 
 a strictly scientific treatise would have required, 
 the only serious objection against the present plan 
 is, in part, obviated. For, there is little doubt, 
 the student's curiosity and attention will be more 
 excited and sustained, when he finds history 
 blended with science, and the demonstration of 
 formulae accompanied with the object and the 
 causes of their invention, than by a mere ana- 
 lytical exposition of the principles of the subject. 
 
PREFACE. V 
 
 The plan, perhaps, would not suit any other 
 department of science, so well as it does this ; 
 which is limited in its extent, and has had but 
 few, although eminent cultivators. 
 
 Other advantages, besides that of an excited 
 attention, may accrue to the student from the 
 present plan. He will have an opportunity of 
 observing how a calculus, from simple beginnings, 
 by easy steps, and seemingly the slightest im- 
 provements, is advanced to perfection ; his 
 curiosity too, may be stimulated to an exami- 
 nation of the works of the contemporaries of 
 Newton ; works once read and celebrated : 
 3 r et the writings of the Bernoullis are not anti- 
 quated from loss of beauty, nor deserve neglect, 
 either for obscurity, or clumsiness of calculation, 
 or shallowness of research. Their processes 
 indeed are occasionally somewhat long, and 
 want the trim form of modern solution. They 
 are not, however, therefore the less adapted to 
 the student, who is solicitous for just and full 
 views of science, rather than for neat novelties 
 and mere store of results. Indeed, the Authors 
 who write near the beginnings of science, are, 
 in general the most instructive: they take the 
 reader more along with them, shew him the 
 real difficulties, and, which is a main point, teach 
 
VI PREFACE. 
 
 him the subject, the way by which they them- 
 selves learned it. 
 
 In a former Work *, I adopted the foreign 
 notation, and the present occasion furnishes 
 some proof of the propriety of that adoption. 
 In the Calculus of Variations, it is necessary to 
 have symbols denoting operations, similar to 
 those that take place in the Differential Calculus : 
 now, d being the symbol for the latter, 3 is a 
 most convenient one for the former : analogous 
 to S there is no symbol in the English system of 
 notation. If then I had used the fluxionary 
 notation with points or dots, I must have in- 
 vented symbols corresponding to h and the 
 characters formed by means of it. But, the 
 invention of merely new symbols is in itself an 
 evil. M. Lagrange indeed, whose power over 
 symbols is so unbounded that the possession of 
 it seems to have made him capricious, has treated 
 the subject of Variations without the foreign 
 notation; this he rejects altogether ; and, which 
 is strange, has employed the English notation, 
 but not adopted its signification. Thus, with 
 him, x is not the fluxion, but the variation ofx: 
 the fluxions or differentials of quantities « are not 
 
 * Principles of Analytical Calculation, 1803. 
 
PREFACE. Vll 
 
 expressed by him, but solely the fluxionary or 
 differential coefficients ; thus, if u be a function 
 
 of x, it ( = -r- or = - ) is the differential co- 
 v ax x' 
 
 efficient. What advantages are to arise from 
 these alterations it is not easy to perceive : yet 
 they ought to be great, to balance the plain and 
 palpable evils of a confusion in the signification 
 of symbols, and of the invention of a system of 
 notation to represent what already was repre- 
 sented with sufficient precision. No authority 
 can even sanction so capricious an innovation. 
 
 There is another point towards which I am 
 not unwilling to draw the attention of the 
 reader; and that is, the method of demon- 
 stration by geometrical figures. In the first 
 solution of Isoperi metrical problems, the Ber- 
 noullis use diagrams and their properties. Euler, 
 in his early essays, does the same ; then, as he 
 improves the calculus he gets rid of construc- 
 tions. In his Treatise # , he introduces geome- 
 trical figures, but almost entirely, for the 
 purpose of illustration : and finally, in the tenth 
 volume of the Novi Comm. Petrop. as Lagrange 
 had done in the Miscellanea Taurinensea, he 
 
 Met hod us inveniendi, &c. 
 
Vlll PREFACE. 
 
 expounds the calculus, in its most refined state, 
 entirely without the aid of diagrams and their 
 properties. A similar history will belong to 
 every other method of calculation, that has been 
 advanced to any degree of perfection. 
 
 The plan of the Work has been slightly de- 
 scribed; and indeed it scarcely requires any ex- 
 planation. On that, however, I chiefly rely, as 
 the means of rendering easy a subject, which is 
 acknowledged to be one of the most difficult. 
 But, although I am not aware of having omitted 
 any thing that is requisite to the full explanation 
 of the subject, yet I cannot flatter myself that it 
 will be thoroughly understood from this Work 
 alone. For, in general it may be laid down as 
 true, that no doctrine, of novelty and intricacy, 
 can be completely taught by a single Treatise. 
 It seems to be indispensably necessary for the 
 student, that the subject should be put under 
 several points of view : that if not apprehended 
 under one, it may be under another. For this 
 reason, though not wanting an Author's par- 
 tiality for his own performance, I recommend 
 the perusal of those works, to which frequent 
 reference is made in the following pages. 
 
PREFACE. IX 
 
 I return my sincere thanks to the Syndics 
 of the University Press, who, very liberally, 
 have defrayed two-thirds of the expense of the 
 present publication. 
 
 Cuius College, 
 Nov. 17, 1810. 
 
Table of the Foreign, and the corresponding 
 English Notation. 
 
 F. 
 
 d I 
 
 d*x 
 
 d'x 
 
 d n x 
 
 da ~ 
 
 dx 3 
 
 E. 
 
 i' 
 
 X 
 
 X 
 
 a 
 
 x z 
 
 i 3 
 
 F. 
 
 du 
 dx 
 
 d~u 
 dF 
 
 d 3 U 
 
 dx 3 
 
 d 2 u 
 dx.dy 
 
 dhi ■ 
 dx z . dj/ 
 
 *<£) 
 
 E. 
 
 u 
 
 X 
 
 it 
 x~* 
 
 u 
 
 ii 
 
 u 
 x:i) 
 
 (!)" 
 
 EuLER, 
 
 List of Foreign Authors that treat of' the Subject of the 
 present Work'. 
 
 Bernoulli, James and John. — Works. 
 
 "Coram. Acad. Petrop. torn. VI. VIII. 1733, 
 
 1736. 
 Methodus Inveniendi Lineas curvas proprie- 
 tate maximi minimive gaudentes, Lausanne. 
 Geneva, 1744. 
 Novi Comm. Petrop. torn. X. 1766. 
 .Calcul Integral, 1768. 
 
 f Miscellanea Taurinensia, torn. II. IV. 
 Lagrange. < Theorie des Fonctions Analytiques, 1797. 
 (. Lecons sur le Calcul des Fonctions, 1806. 
 
 Borda, et Fontaine. — Acad, des Sciences, Paris, 1767. 
 Le Seur et Jacouier. — Elemens de Calcul Integral, 1768. 
 Sauri. — Cours de Mathematiques, torn. V. 1774. 
 Legendre. — Acad, des Sciences, Paris, 1786. 
 La< uoix. — Calc. Diff. et Integral, 1797. 
 Bossut.— Calc. Diff. et Integral, 1798. 
 
ON 
 
 ISOPERIMETRICAL PROBLEMS, 
 
 AND THE 
 
 CALCULUS of VARIATIONS 
 
 CHAP. I. 
 
 The Problem of the Curve of Quickest Descent proposed. — James 
 Bernoulli's Solution of it. — Principle of that Solution. 
 
 The ordinary questions of maxima and minima were 
 amongst the first that engaged the attention of mathe- 
 maticians at the time of the invention of the Differential 
 or Fluxionary Calculus. This invention took place about 
 the year 1684, three years before the memorable sera 
 of the publication of the Principia. But, the principles 
 of the differential calculus, were not, like those of phy- 
 sical astronomy, given to the world, at once, and as it 
 were, on a sudden, in a formal treatise. They were 
 communicated gradually and by piecemeal ; in letters 
 between men of science, in small essays and tractates, in 
 solutions of particular problems published chiefly in 
 
*2 Neivton's Problem of the Solid of least Resistance. 
 
 a work intitled the Acta Erudltorum edited at Leipsick * ; 
 which, at that time, was the common vehicle of commu- 
 nication between men of science and literature. There 
 is scarcely a formula or process, which we now find 
 compactly stated in our modern treatises, that was not 
 in those Acts the subject of some correspondence and 
 discussion. The methods of drawing tangents, radii of 
 curvature, of determining the points of inflexion, the 
 maxima and minima of quantities, &c. were treated as 
 subjects of great importance by Leibnitz, and by the 
 two Bernoullis. 
 
 The first problem relative to a species of maxima and 
 minima distinct from the ordinary, was proposed by 
 Newton in the Principia: it was, that of the solid of least 
 resistance. But, the subject and doctrine became not 
 matter of discussion and controversy, till John Bernoulli, 
 Professor of mathematics at Groningen, required of ma- 
 thematicians the determination of the curve of quickest 
 descent. This he did in the Leipsick Acts for June, 
 l6a6t, p. 269, under the following form : 
 
 * Problems proposed and solved, became an amazing means of pro- 
 moting the methods of calculation, or the calculus, to which they 
 belonged. Of this John Bernoulli was aware, when, in his famous 
 Programma relative to the line of quickest descent, he says, " Cum 
 compertum h abeam us, vix quicquam esse quod magis excitet generosa 
 ingenia, ad moliendum quod conducit augendis scientiis, quam 
 diflkilium pariter, et utilium quaestionum propositionem." Opera, 
 torn. I. p. 166. 
 
 t Opera, torn. I. p. 155. 
 
John Bernoulli's of the Curve of Quickest Descent. 3 
 
 " Problema Novum, 
 ad cujus Solutionem Mathematici invitantur. 
 
 " Datis in piano vertical! duobus punctis A et B, 
 assignare mobili M viam AMB, per quam gravitate 
 
 " sua descendens, et moveri incipiens a puncto A, bre- 
 ee vissimo tempore perveniat ad alterum punctum B." 
 
 Six months, the time allotted for its solution, being 
 elapsed, and no solution appearing, John Bernoulli at 
 the request of Leibnitz (who intimated that he had 
 solved the problem) prorogued the term, and again 
 anounced the problem in a programma * edited at Gro- 
 ningen in January 1697. In the May following, his 
 brother, James Bernoulli, professor of mathematics at 
 Basle, published its solution in the Acta Enid. Lips. 
 p. 21 If ; and after the following manner: 
 
 * Opera, torn. I. p. 166. 
 
 t See also his Works, torn. II. p. 768. 
 
4 
 
 James "Bernoulli's Solution. 
 
 Let OGD be the curve ; conceive a portion of it CGD, 
 to be divided into two parts CG, GD ; and take another 
 
 o 
 
 [ 
 
 
 
 Vg ] 
 
 
 C 
 
 E 
 
 L N^^ 
 
 
 F 
 
 D 
 
 element to the curve CLD, divided also into two parts 
 CL, LD, and indefinitely near to CGD : then since by- 
 hypothesis, the time through CG + GD, is to be a 
 minimum, and since quantities at or near their state 
 of minimum may be considered constant (for their 
 increments or decrements are very small), we have 
 
 t.CG + t.GD = t.CL + t.LD 
 
 [t . CG, abridgedly representing the time through CG.~\ 
 
 and .-. t.CG-t. CL = t.LD-t. GD 
 
 again CE : CG :: t . CE : t . CG 
 
 \_CG considered as an inclined plane.] 
 and CE : CL:: t.CE.t. CL 
 
 consequently, CE : CG- CL [MO] ::t. CE: t.CG-t.CL 
 but MG : LG :: EG : CG 
 
 [by similar A* LMG, GCE.] 
 
 :. CE.LG:: EG x (t.CE) : CGx(t.CG- t. CL) 
 similarly EF:LG :: G/x (t . EF):GD x (t . LD -t .GD.) 
 
Curve of Quickest Descent a Cycloid. 3 
 
 Hence, equating the two values of LG, we have 
 EGx t.CExEFx GD-=GIxt.EFxCE x CG 
 
 or EG x -^7,x EFx GD=GIx fj? r - x CE x CG 
 
 [substituting for £. C£, f.UF.] 
 
 or 
 
 which is a property of the cycloid * : hence the curve of 
 quickest descent, or the brachystoclLVone, is a cycloid. 
 
 The proportion [l] may be thus expressed : 
 ~|, : ^ :: x/HC : y/ HE, or 
 
 sin. L.ECG : sin. L. GDI:: \/ HC : \/ HE ::veP\ at C: 
 
 vel y . at G, or the sine of the angle made by a vertical and 
 an element of the curve is proportional to the velocity, f 
 
 * This will easily appear by constructing the figure with its 
 
 generating circle ; or, the known equation of the cycloid may be thus 
 
 , . , EG GI dx dx' 
 
 deduced: since ..., — — — -— 77777-777; we have 
 
 </HC.CG »/HE.GD Jy.dz -J y' .dz' 
 
 and since dx', y ', dz', are the consecutive values of dx, y, dz, it 
 
 d x 
 
 follows from the above equation, that — 7- is every where the same, 
 
 y/y .dz J 
 
 or is constant, and therefore may be nut r: — — ■• Hence, adx^zz 
 
 1/ a 
 
 *J v 
 y . dx^+y.dy" 1 and dx — •— ^ — rdy, the equation to a cycloid. 
 v(«— y) y 
 
 t Lagrange, Cul. des Fond. 8vo. p. 425, says that this was the form 
 of the result obtained by Bernoulli. It is not however the form given 
 in the first solution that occurs in Bernoulli's Works, 
 
6 Principle of Bernoulli's Solution. 
 
 In the preceding demonstration two principles are in- 
 volved : one * borrowed from the doctrine of the ordinary 
 maxima and minima of quantities : the other, at the time 
 of the first discussion of these questions entirely new, and 
 in fact assuming ; That the property appertaining to the 
 whole curve, belongs also to any element of the curve : for 
 of the preceding problem, the condition was, that the time 
 down the whole curve OCD should be a minimum, 
 that is, less than down any other curve, passing through 
 the points O, D : but in the demonstration, an element 
 CGD is taken, and it is assumed that the time down 
 this element is also a minimum : so that, the whole 
 curve is determined, by determining a portion or element 
 of it. The principle undoubtedly is true, both in the 
 preceding and in other problems, but it is not univer- 
 sally true. Of the exceptions, and of their reasons, 
 we shall hereafter speak, and here only briefly notice, 
 that the Bernoullis and Brook Taylor, preface their small 
 tracts on this subject, with this principle, establishing the 
 same nearly after the following manner : 
 
 If the curve APB (see fig. p. 3.) has a certain pro- 
 perty of minimum, a portion of it PQp, has the same 
 property : for suppose AP, Bp, to be determined, then 
 if PQp possesses not the property of maximum or mi- 
 nimum, suppose another small portion Pqp to possess 
 it ; therefore P qp added to AP, and Bp, will form a 
 curve, that in the case of maximum will contain more, 
 
 " See page 4. line 5, &c. 
 
Explanation of the Principle. 'J 
 
 or of minimum, less than ^lPClpB; which is contrary 
 to the hypothesis*. 
 
 The first principle is, as it has been remarked, that 
 which is employed in the ordinary questions of maxima 
 and minima. If, for instance, we wish to determine 
 when an ordinate y is a minimum, dy its differential is 
 put equal to nothing. Hence, the contiguous or con- 
 secutive ordinate which generally is, 
 
 y+^.A.r + -^(A*) l ±&c.t 
 J ~~ dx 2.dx* v 
 
 is in this case reduced to 
 
 J 2.dx v ' 
 which, if we make Ax infinitely small, will differ from 
 the ordinate y only by infinitely small quantities of the 
 second order ; and if we neglect them, may be said to be 
 equal to the ordinate y. Of this nature, and requiring 
 this explanation, is the equality in James Bernoulli's 
 demonstration, when the time through CG + the time 
 through GD — the time through CL -f the time 
 through LD. 
 
 * P. 226. torn. II. John Bern. Leip. Act. 1701, p. 213. James 
 Bernoulli, the author of the essay, as if suspicious of the truth of his 
 principle, says, " Sensus theorematis vel demonstrations ejus videtur 
 paulo obscurior, nee satis determinatus : sed planior et infra ex appli- 
 catione, &c." John Bernoulli also, Acad, des Sciences, 1706, p. 236. 
 lays down the same principle : " Parceque toute courbe qui doit 
 donner un maximum, conserve aussi dans toutes ses parties les loix de 
 ce meme maximum, &c." 
 
 f Woodhouse, Prin. Analy. Calc. pp. 44, 163. 
 
S Object of the common Doctrine of Maxima and Minima. 
 
 The use of the principle, however, in Bernoulli's pro- 
 blem, and its illustration, will enable us to point out 
 the real and essential distinction between the ordinary 
 calculus of maxima and minima, and that which is the 
 subject of the present treatise. In the former the relation 
 of y to x is supposed to be given, or, in other words, the 
 function/' in the equat". y=fx is supposed to be known ; 
 
 dy 
 
 and the process of equating y and y 
 
 dx 
 
 . A# + &c., or 
 
 of making dy = O, enables us to find a particular value 
 of x which substituted in y—fx gives the greatest or 
 least value of y. Thus in the equation y = \/(2ax — x 2 
 + b), fx = \/(2ax — x*-\-b) and if the greatest ordinate 
 
 L M N 
 
 be required, (putting 31 Q =y, and MN= A.r), NR =y + 
 
 d*y ,. , ,di/ a- x 
 
 ^-(A t r 2 )+&c. and-p- = 
 
 2.dx lK J dx 
 
 dy 
 
 ■f. Aa?.+ 
 
 dx 
 
 y 
 
 = 0, conse- 
 
 quently x = a, and the maximum value of y = \/ [a 1 + b). 
 In the latter calculus, on the contrary, the relation of y 
 to x is not given but sought, or in other words the form 
 of the function fx [that is, whether it is y/(2ax — x*) 
 or e ax or (ax n - bx m ) r > &c] is the object of investigation. 
 
Object of the Doctrine of the Maxima and Minima of Curves. p, 
 
 For this reason, the augmented value of ?/ = MQ is not, 
 as before, its consecutive value NR=y + Ay, but Mq — 
 y + Qq=y + v, which must be substituted for y in 
 expanding any function of y. Thus, if the equation 
 between y and x, or in other words, if the function fx 
 
 is required, such that J*(ax - y~) y. dx shall be a 
 maximum, we have, changing?/ into y -f v, the consecutive 
 value of the expression for the maximum, (axy — y 3 ) 
 + (ax — 3y*)v + &c. and since this expression must be 
 
 a maximum, the second term, analogous to ~r-;A.r in the 
 
 dy 
 
 dx 
 
 /ax 
 former instance, must = 0, or ax — 3 y 1 = 0, and y = \J -—- . 
 
 Here it is plain, the result gives the form of the 
 equation, that must subsist between y and x, so that 
 generalizing, it may be said, that it is the form of the 
 function fx in y=fov, which is the object of search. 
 The two preceding instances plainly shew the points of 
 agreement and disagreement of the two calculi. 
 
 Before we quit this part of our subject we will 
 shew how Bernoulli's method of solution, without any 
 alteration of principle, may be abridged. Since the 
 time varies as the space divided by the velocity, and 
 the velocity as the square root of the height, we have, 
 
 CL LD CG , GD c ,, 
 
 <7HC+ s/HE = JHC* <7HE> from the P rmc, P Ie 
 
 c LC-(CL + MG) (LD-LN)-LD 
 
 of minimum, /. A___ L^ _^ , 
 
10 Cycloid of Quickest Descent from a Point to a Vertical Line, 
 
 MG LN . .. . . 
 
 or , = ? or from similar triangle 
 
 es, 
 
 £L 
 
 LG = 
 
 L/ 
 
 LDy/HE 
 
 CL>/HC 
 UW~HE as before ' P- 5 
 
 LG, or ^7 
 
 EL 
 
 CLV HC 
 
 In Bernoulli's solution the curve between ^ and B is 
 shewn to be a cycloid ; but nothing farther is deter- 
 mined ; the result is independent of the relative position 
 
 of the points A and B. Hence, the same result, as Un- 
 as it regards the nature of the curve, will obtain for A and 
 B\ A and B' &c. that is, the curves AB, AB\ AB" 9 &c. 
 are all cycloids. A question then naturally arose out of 
 
intersects the Vertical Line at Right Angles. \ \ 
 
 the original one ; Which is the cycloid, down whose 
 length a body will descend in the least time ? Or, in other 
 words, required the curve, a cycloid it must be, such that 
 a body shall descend from A to the vertical line BB'B" 
 in the least time. This, after the solution of the first 
 problem by James Bernoulli, was proposed by him 
 to his brother, and solved by the latter, not only 
 for the particular case in which BB'B" is vertical, 
 but for any inclination of that line, and for any 
 form of that line ; that is, John Bernoulli determined 
 the condition to which the cycloid must be subject, when 
 the time down it from A to a curve b V b" shall be 
 a minimum. The condition, which hereafter will be 
 deduced, is, that the cycloid shall cut the curve at right 
 angles*. 
 
 We may now, without either haste or abruptness, 
 proceed in the historical and scientific development of 
 the new calculus. And, the next step will be to the 
 famous Programma of James Bernoulli, which contained 
 the problem, whence the title of Isoperimetrical, since 
 applied to all problems of a like kind, is derived. 
 
 * This Bernoulli determined by an ingenious, but peculiar method. 
 Indeed, there appeared no general method of solving isoperimetrical 
 problems, with all their circumstances, till Lagrange in 1760 gave 
 his formula of solution, consisting of two parts ; the one containing 
 integrals, the other a definite expression. See the subsequent parts of 
 this Work. 
 
CHAP. II. 
 
 Isoperimetrical Problems proposed by James Bernoulli — Solved by 
 John Bernoulli — His fundamental and specific Equations — Appli- 
 cation of them to the Curve of Quickest Descent, and of a given 
 Length — Solution of Isoperimetrical Problems, by Brook Taylor — 
 Imperfections of his and the Bernoullis's Methods. 
 
 In 1697, James Bcrnoullredited a Programma, which 
 proposed this problem * : 
 
 Of all Isoperimetrical curves described on the common 
 base B1Y, to rind BFX such, that another curve BZN 
 
 shall contain the greatest space, the ordinate of which 
 PZ, is in any multiplicate or submultiplicate ratio cf 
 the ordinate PF, or of the arc BF. 
 
 * " Quseritur ex omnibus isoperimetris, super communi basi BX 
 constitutis, ilia BFX, qua; nou ipsa quidem maximum comprehendat 
 spatium, sed faciat, ut aliud curva £2>2Vcomprehensum sit maximum, 
 cujus applicata PZ ponitur esse in ratione quavis multlplicata, vel 
 submultiplicata, recta? PF, vel arcus BF, hoc est, quae sit quotacunque 
 proportionalis ad datam Ag rectam PF, curvamve BF." Acta Enid. 
 1697. Mai. p. 214. 
 
First Solutions of Isoperim. Prob. by John Bernoulli, erroneous. 13 
 
 Iii the terms of modern analysis, the ordinate PZ 
 is said to be a function of PF, or a function of the arc 
 BR 
 
 It is the last case of the problem, when PZ is a func- 
 tion of the arc BF, that baffled the sagacity and skill of 
 John Bernoulli, and caused between him and his brother, 
 a long and acrimonious controversy. All the solutions 
 of the former were erroneous *, till the publication of his 
 brother's. And, even when John Bernoulli had amended 
 his solution, he would not frankly and plainly acknow- 
 ledge his error, but slurred it over, with a faint and 
 half-confession of having previously been guilty of some 
 slight inadvertencies. The last solution, however, of 
 John Bernoulli, published in the Academy of Sciences for 
 the year 17 18, is, considering what was then the state of 
 analytical science, very admirable, and merits the eu- 
 logium which he himself has conferred on it, that of being 
 equally exempt from the tediousncss of his brother's, 
 and the obscurity of Taylor's calculations t. It is, how- 
 
 * Acad, des Sciences, 1706, p. 235. or Opera, torn. I. p. 424. In 
 this memoir he solved the first case of his brother's problem, making 
 two elements only of the curve to vary, but making besides the 
 elements of the abscissa to vary. On these grounds the solution of 
 the first case is right, but that of the second erroneous. 
 
 f " Une voye courte, claire et facile, suivant laquelle un geoinetre 
 d'habilete et d'esprit tnediocres puisse arriver jusqu' avoir, non sur la 
 foi d'autrui, mais de ses propres yeux, ces veritez abstruses, sans 
 s'engager dans la longueur du calcul de mon frere, ni dans l'obscurite 
 de celui de M. Taylor." Opera, torn. II. p. 238, also Mem. de I'Acad. 
 de Paris, 1718. p. 100. 
 
14 John Bernoulli's last Solution. 
 
 ever, obviously borrowed from his brother's, which was 
 entitled " Analysis magni Problematis Isoperimetrici," 
 and published in the Acta Eruditorum Lips, for May, 
 1701, p. 213. Previously to the Analysis, a solution, 
 as it was called, but in which results alone were stated, 
 had been published in the above mentioned Acts, for 
 June, 1700, p. 20*1. 
 
 As this treatise is not intended to be strictly his- 
 torical, the problem of the isoperimetricals will not be 
 traced through all its variety of change and emendation. 
 To history we shall adhere no farther, than is sufficient 
 to preserve an unbroken series of methods gradually 
 becoming more exact and extensive ; the series begin- 
 ning with the first rude, though perfectly just, method 
 of James Bernoulli, and ending with Lagrange's ex- 
 quisite and refined Calculus of Variations. For this 
 reason we proceed to notice, and shall give the substance 
 of the last solution of John Bernoulli, published in the 
 Memoirs of the Academy of Sciences for 1718. The 
 separation between that, and the first solution of the 
 hraclii/stoclirone, which has been already given, being 
 neither too wide, nor too abrupt, for our present 
 purpose. 
 
 Bae is a curve, and an element of it a bee is com- 
 posed of three portions ab,bc,ce; another element 
 aghe, indefinitely near the former, is composed of three 
 portions, ag, gi, ie, and since the solution is to extend 
 to isoperimetrical curves, we have, 
 
Fundamental Equation. — Elements of the Abscissa constant. \5 
 
 ab + be + ce = a g + gi + i e, 
 and consequently, (ng — ab) + (gi — be) + (« e — ce) = 
 
 B N_ P R S 
 
 or, gm-bn — oc-\-ih = [l] 
 
 A 
 
 By similar triangles, < 
 
 bmg, abf, g™ = '^ •% 
 
 °' &c ° 
 
 /oc, 6 eft. co=7~.c/. 
 oc 
 
 zc/z, c/e, ih = — . c£. 
 ce 
 
 Hence, substituting in the equation [l] 
 
 (fb hc\ 7 (he le\ . ^ 
 K ab be / ° y bc ee / 
 and this Bernoulli calls his fundamental equation, on 
 account of the uniformity * subsisting in the coefficients 
 
 * P. 104. Mem. Acad. Paris, 1718. Bernoulli, after noting this 
 uniformity and shewing what it consists in, adds, "On verra clans la 
 suite que cette uniformite contribue merveilleusement a reconnoitre 
 comme d'un seul coup d'oeil et non en consequence d'aucune analyse, 
 les equations qui conviennent a chacun des problemes que nous allons 
 resoudre." 
 
dy"\ • 
 
 ,) cu 
 
 16 Uniformity of Coefficients in the Fundamental Equations. 
 
 of bg and ci ; which uniformity, if we run a little before 
 our matter, and borrow the symbols of the differential 
 calculus, may be easily made manifest. Thus, fb, he, 
 le, being the differentials of the ordinates Na, Pb, Re 
 [?/, ?/, //'] are to be noted, dy, dy', dy"; and if dz, dz f , 
 dz" denote the differentials of the three arcs Ba, Bb, Be, 
 we have, 
 
 \lz dz' ° S K dz dz 
 In the foregoing demonstration, the elements of the 
 abscissa are supposed to be constant, or the points b, c, 
 by whose motion the curve is supposed to be changed, 
 move along the lines J^g, pc; but, instead of the elements 
 of the abscissa, those of the curve may be supposed 
 invariable, and this case Bernoulli includes in a second 
 fundamental equation, like the former, and demonstrated 
 by the subjoined process. 
 
 Since the elements a b, be, ec, are supposed to be 
 B N P R S 
 
 constant, from a, e, as centers describe circular arcs 
 
Second Fundamental Equation. — Elements of the Curve constant. ] J 
 
 bg, ci, in which, the points b, c, when we pass from 
 curve to curve, are supposed to move : now, 
 be 2 = gi z 
 
 :. bk x + he- = {bk+gn + oiy + {kc-co-b»Y 
 or, (bk + gn + oi) 2 — b k* = kc 2 - (kc — co- bn)* 
 or, by virtue of the formula, 
 x>-y- = (x +y) {x-y) 
 (2bk+gn + oi) (gn + oi) = [2kc-(co + bn)~\ (co + bn) 
 or, 2bk (gn + oi) — 2kc (co + bn) neglecting (g n + o/) 2 , 
 
 (co + bn) 2 ; hence, gn+ oi — (co + bn) 7-7 • • • [ 2 ] 
 
 u K 
 
 afb, bng, gn —~<> bn, 
 
 now, by similar triangles, / ^ 
 
 , le 
 
 cle, coi, 01 = — . co. 
 
 Hence, substituting in equation [2], and transposing 
 (fb kc\ . (kc le\ 
 
 ^-bk> bn -\Tk--7l)' cQ = > 
 
 which Bernoulli calls also a fundamental equation, and 
 in which is to be noted an uniformity similar to that 
 pointed out in the former fundamental equation. 
 
 If we express the equation symbolically, we shall 
 have, 
 
 (g-g>>-«-g)-=o . 
 
 These equations appeared in Bernoulli's last solution 
 of the Isoperimetrical Problems. And to be satisfied of 
 the skill and previous labour expended in their con- 
 struction, we need only remark, that they have no 
 
 D 
 
18 Application of Bernoulli's Method 
 
 dependancc on any property of maximum, or minimum, 
 and seem therefore to have no relation to the problems, 
 for which they were specially designed. The relation, 
 however, which they are made to have, cannot be better 
 understood than from the following instances. The 
 first of which, is the Br achy 'Stochrone, not the common 
 case*, but that, in which, to the minimum property, 
 the Isoperimetrical is added. In other words, when, out 
 of all curves of a given length, the curve of quickest 
 descent is required. 
 
 In the subjoined figure let B a .bee be the curve of 
 
 B N 
 
 quickest descent, of which a portion ah ce is formed of 
 three elements, ab,bc, ce; take a portion of another 
 curve formed of a g, gi, ie, indefinitely near the former, 
 then, by the principle of the maxima and minima of 
 quantities, the time down aft, be, ce, = time down an, 
 gi, ie. But the time varies as the space divided bv 
 the velocity varies ; therefore as the element of the 
 
 * See pages 4-. and 9. 
 
to the Brachystochrone a7?iongst Isoperimetrical Curves. \ 0, 
 
 curve divided by the square root of the vertical height 
 from which the body has fallen : hence, 
 
 ah I _JiL ce a g gi ie 
 
 but ab, ag, be, &c. are all equal ; 
 
 x/Pb + </R c ~ x /Pn + -^nTo> 
 1 1 1 1 
 
 or, — TTTf — = =^ _ 
 
 </Pb </p n s/Ro x/Rc> 
 
 or _ J L___ 1 1 
 
 y/Pb y/(Pb + bn)' ~x/{Rc-co) N /\ftP 
 
 _ ( 1 bn \_ _1_ 
 
 ' VPb WPb 2 pb- J ~~^~R~c + -ZR c i "VTCc 
 
 [expanding and neglecting the terms 
 involving bit, co*, &c.l 
 
 b n P b r 
 
 Hence, — = r ; and this is an equation expressing 
 
 the proportion of b n to co, obtained from the property 
 of the proposed minimum ; but the second fundamental 
 equation, involves bn, co, the proportion of bn to co, 
 there arising from the Isoperimetrical condition. Hence, 
 eliminating bn, co, there results the specific equation* 
 
 * " Specifique ; parceque d'elle resulte l'equation difFerentielle, qui 
 determine finalement l'espece de la combe cherchee." Acad, des 
 Sciences, 1718, p. 108. 
 
20 Equation to the Brachystochront, 
 
 Now if we observe this equation, it will appear per- 
 fectly uniform in its two members, since the right hand 
 member differs from the left, only inasmuch as the 
 points e, c, are used instead of the c, b. Hence, either 
 member is always the same, throughout the curve 
 required: hence, symbolically, 
 
 /dv di/\ 3 
 
 y-f- — -7-7/ y* = a constant quantity ; 
 
 or, — d \~ ) y\ — \/ a . dz 
 
 [d 1 is constant and multiplied into \/a y in order to 
 make the equation uniform.] 
 
 , ■ . d 2 x . dtf - d 2 y . dx . dz 
 that is, -2—, '- = k/ a . — . . . a 
 
 but dz* = dx* + dy z ; and since dz is invariable, = 
 dx . dx -f- dy d\i/ ; hence, substituting in [a] 
 
 d\v (dx*+dy x )_ dz . dif 
 
 Ct Jb 
 
 or, ~ dz -V*-jL, or d (£) .dz-2 Ja.d (L) 
 
 and integrating, -p = 2\/- ± - - being the correc"*.] 
 
 TJ dz' dx* + dy* a b fa ff 
 
 Hence, -— :, or j—-*- = 4 - ± -l- V - +-- 
 
 dx 1 dr y c v y ( 
 
 and, dx - - Tr< *£■' f 
 
 y/ (4 a c z + b l y — c-y dt4bc\/ ay) ' 
 
 See Emerson's Fluxions, p. 189. Third Edition. 
 
In Isoperim 1 . Problems, the Variation of three Elements requisite. 21 
 
 If in the correction - , b = , dz = 2(Lr \J - — 
 
 c v y 
 
 2*/ a. — , , r, and the curve is the common 
 
 cycloid ; which now is the curve of the quickest 
 descent, not amongst all curves of the same length 
 between B and C, but amongst all curves whatever that 
 can be drawn between these two points. This is, 
 however, only a particular case, for if b be real, the 
 curve is not a cycloid, nor is it any known curve, but 
 merely that which the differential equation determines 
 it to be. 
 
 If we now examine the preceding solution, we shall 
 perceive the office of the fundamental equations in 
 solving problems of maxima and minima. Analytically 
 speaking, they half resolve such problems ; and furnish 
 one equation involving two arbitrary quantities, [bn, co~] 
 whilst the maximum or minimum property furnishes 
 another. That which results from elimination is the 
 equation of solution. 
 
 We may perceive also, in the preceding solution, 
 the necessity of making three elements of the curve to 
 vary. Two elements are sufficient when merely the 
 brachystochrone without any restriction is required. The 
 minimum property furnishes one equation, containing 
 an arbitrary quantity [LG~], which is also the equation 
 of solution. If a second property, isoperimetrical, or 
 other, be introduced, it may be incompatible with the 
 curve determined by the equation resulting from the 
 minimum property. 
 
22 Second Instance of the Application of Bernoulli's Method 
 
 For instance, CGD may be determined such, that it 
 II 
 
 is the curve of quickest descent, and then taking CLD 
 indefinitely near, the time down CG+ the time down 
 GD = the time down CL -f- the time down LD ; but 
 CG+GZ) cannot equal CL-\-LD, for the former sum 
 is less, than that of any two lines, drawn from the 
 extremities C, D, externally to CG, GD. 
 
 This does not shew, beyond controversy, the neces- 
 sity of the variation of three elements. For, the points 
 L, G, need not lie in the same line ELG, since the 
 elements of the abscissa as well as those of the curve 
 may be made to vary : in which case, two equations, 
 derived from two properties may be obtained ; and this 
 method, which was used by John Bernoulli *, in 
 certain cases, but not generally, leads to right results. 
 
 If problems involving merely one property, the 
 
 * The method is right in the first case of the problem proposed by 
 James Bernoulli, but not, in the second. See p. 13. 
 
to the first Case of Problem proposed in p. 12. 23 
 
 maximum, require the variation of two, and those in- 
 volving two properties, the variation of three elements ; 
 problems involving three properties, would, it should 
 seem, require the variation of four elements : and this 
 is the case. Each property furnishes an equation con- 
 taining three arbitrary quantities (such as bn, co), and 
 the solution depends on the equation resulting from 
 their elimination. To such problems, however, the 
 Bernoullis did not extend their researches. 
 
 We shall give a second instance of Bernoulli's 
 method in solving the first I soperi metrical problem that 
 was proposed by his brother ; and proceed as in the 
 former case ; that is, use two fundamental equations 
 involving b g, ic, one derived from the Isoperimetrical 
 condition, the other from the common property belong- 
 ing to quantities, at their state of maximum. Now the 
 problem requires the curve to be determined, such, 
 that if?/ be the ordinate, and Y= f{y)*, fY dx shall 
 be a maximum. 
 
 Hence, since the elements of the abscissa are sup- 
 posed constant, [see fig. p. 15.] 
 
 f(Pb) +f(Rc) =f(Pg) + f(Ri), 
 
 by the property of maximum, 
 or, f(Pb) -f(Pg) =f(Ri) -f(Rc) 
 or, f(Pb) -f(Pb + hg)=f{Rc-cx) -f(Rc) 
 
 * / (y) is the method of noting the function of a quantity y. The 
 name and symbol were used by John Bernoulli in the Memoirs of 
 the Academy for 1706, and 17 IS. See the latter Vol. p. 106. 
 
. of the C 
 
 from which, expanding f{Pb + - r Re— ci)*, and 
 electing the terms involving the squares, cube-, -ke. 
 
 of /'i. r . ci. we have. 
 
 <~ - 
 
 d . - . _ = d .^/ AV • 
 
 [d . f{Pb) bein_ efficient of the second term, 
 
 in the expans: P 
 
 If we now substitute for 6ff, ci, in the fundamental 
 equation l] p. 15. there results, 
 
 Ni5 ~ be' u.j Pb)~^bc cc o.f. Re)' 
 
 In which equation, i specific Bernoulli calls it) the 
 
 law of uniformity being observable in the t tion 
 
 of its members, it follows, that. 
 
 (fh l-c\ l . . 
 
 V — - — / rr^- • is even- where the same. 
 
 (dn di/\ 1 i(dn\ 1 dx 
 
 or. \-jr — -td) et • or - " \nr > — F~ = — • 
 
 V- d^ 7 D.J K dz D.i (/ 
 
 — beinc from the invariability of the elements of the 
 
 abscissa, constant : 
 
 d'- z dy - d\u . dz n . J'. dx 
 
 hence, 
 
 . 
 
 * Pb—y, and bgrz lor's theorem, [See Princ. 
 
 . 44.] 
 
 /v- = - -— : +^~r - 
 
 .. Anal. Ci'-'. Pref. p. xxix ; and p. 43. 
 
The Circle, amongst hop 1 . Curves, contains the greatest Area. 25 
 
 but dz 1 = dx t + dy"- ; :. dz . d 2 z = dy . d y ; since dx is 
 constant, consequently, by substitution, 
 
 d 2 z (dif- -dz 1 ) d . Y . (/// ■ dx 
 
 or, 
 
 d*z 
 lz- 
 
 dz~ 
 
 dx = 
 
 '±:C 
 
 ~aT 
 
 dz 7 - 
 
 d . Y . dy 
 
 a 
 
 since 
 
 , and integrating 
 
 Y=fD.Y.dy=f^dy = 
 
 fdY, and - is the correction ; hence. 
 a 
 
 dx? 
 
 (Y±c) 
 
 dx*-\-dy* 
 
 dx = 
 
 and 
 
 (Y±c) . dy 
 v/<a*-[Jr±c]r 
 
 lfY=f(y)=y,thenD.Y=^ = ^=, 1, and 
 
 fdx=f 
 
 (y±c) . dy 
 
 v/(a*-[>±c]T 
 
 or x ---- C - V (a z - [y ± c] 2 ) 
 
 an equation to a circle. Hence, of all curves, of a given 
 length, that can be drawn between two points, the circle 
 is the curve that contains the greatest area. 
 
 If we apply Bernoulli's fundamental equations to the 
 B N P R _ _S_ 
 
 second case of the Isoperimetrical problem, [see p. 13. J 
 
 E 
 
26 Solution of the second Case of James Bernoulli's Problem. 
 
 that is, when the ordinates are not functions of the 
 ordinates Pb, Re, &c, but of the arcs Bab, Babe, &c, 
 the process will be somewhat different from the pre- 
 ceding. For then we have, instead of 
 
 f(Pb)+f(Rc)=f(Pg)+f(Rl) 
 f(Bab) +f(Babv) =f(Bag) +f{Bagi) 
 or, f{Bab)-f{Bab+gm)=f{Babc-co)-f{Babc) 
 and, as before, d .J'(Bab) . gm = n ./ (Babe) . co 
 
 But the fundamental equation involves bg, ei\ therefore, 
 
 gm } co, must be expressed in terms of them. Now, 
 
 by similar triangles, 
 
 f b le 
 
 gm = --—r . bg, co = — . ci ; hence 
 e ab fr ' ce 
 
 n.f(Ba 1 o):-^.bg = D.f{Babc).~.ci; 
 
 and substituting in the fundamental equation [p. 15.1. 9.] 
 
 ('fb Jic\ ab she I e\ ce 
 
 .ab~~bc) ' fb .d .f {Bab) = \Tc~7e) 'le.u.J {Babe)' 
 
 Now, Bernoulli remarks, that in this equation, as it stands, 
 there is not that uniformity, which enables us to pass from 
 one element ab to another be affected in the same manner, 
 and so on. There would have been the requisite unifor- 
 mity, if the factor in the right hand side of the equation, 
 
 ce be 
 
 instead of 7-, had been — . We shall cause therefore 
 le kc 
 
 .be 
 the requisite uniformity, if we multiply each side by —, 
 
 KC 
 
 and there results the specific * equation, 
 * See Note, p. 19. 
 
Equation to the Curve. 27 
 
 (fb hc\ ab be 1 /he le\ 
 
 ab be* ' fb' Tc' d f(Bab) '' ~~ ^Tc ~ 7e* ' 
 
 be 
 
 fb he ' nf(Bab) ^bc ce y 'he' 
 
 or 
 
 7-' — 7v» — r~\ = a constant quantity; hence, since 777 = 
 le d.j (Babe) ^ J fb 
 
 be 
 
 j—, and calling D.f(Bab) = S, we have, 
 
 h c 
 
 rl (dy\ dz % _dx 
 " \Tz)'ay7s~~a~' 
 
 d 2 z.dy - d*y .dz S dy* , . _ _ 
 
 f -7— T -1 = "" • -T~l • "X ; but. d z y . dy = d 2 £ . dz ; 
 
 «% 3 a dz* J * 
 
 d*y.dx* S dy 1 dy , S.rfs 
 
 Hence, g = i (/S.^+c) =:5±£, if Z=fS.dz, 
 
 or, a.dx = dy (Z + c), the equation to the curve. 
 
 This, as we have before remarked, is, in the history 
 of I soperi metrical Problems and their connected 
 Calculus, the most important problem ; since, as long as 
 John Bernoulli made two elements of the curve only to 
 vary, he constantly, by his erroneous solutions, afforded 
 matter of triumph to his brother James. The above 
 solution is John Bernoulli's last solution, and an exact 
 one; but, it was not published till 1^18, in the 
 Memoirs of the Academy of Paris for that year, six- 
 teen years after the appearance of James Bernoulli's *. 
 
 * This solution appeared in the Acta Enid. Lips. Mai, 1701, 
 p. 213. John Bernoulli's solution (which was faulty) was commu- 
 nicated to Leibnitz in 1698, to the Academy of Sciences in 1701, and 
 published in their Memoirs for 1706. 
 
l 28 John Bernoulli's Method distinguished by its Specific Equations, 
 
 On this solution, that of John Bernoulli's is essen- 
 tially founded ; and indeed on the very principle, the 
 neglect of which had vitiated all his former solutions ; con- 
 taining also, under the form of fundamental equations*, 
 James Bernoulli's proportions f. The latter are, with 
 the slightest trouble, transmutable into the former. Yet 
 John Bernoulli, with a total disregard of justice, wishes 
 to confer superiority on his method, by the reverse 
 operation ; that is, by transforming the former into the 
 latter. This, however, is not the sole nor the least trait 
 of his unfairness.^ 
 
 John Bernoulli's solution possesses greater elegance 
 and compactness than his brother's. Yet these are 
 qualities which we may attribute, almost with equal 
 justice, as well to time as to genius. It possesses, 
 however, a characteristic excellence in the uniformitv of 
 its spetjfic equations. A considerable advancement was 
 thereby made in the calculus of Isoperimetrical problems ; 
 and it is on a like principle of uniformity, that cer- 
 
 * Mem. Acad. 17 18, p. 103, &c. 
 
 f Acta Erud, Lips. 1701, p. 213, or James Bernoulli's Works, 
 torn. II. p. SOP, or John Bernoulli's Works, torn. II. p. 220, &c. 
 
 J The palpable error ofhis Corner solution he wishes to reduce to a 
 mere fault of inadvertence. " Pour reparer cette f'aute d'inadvertence," 
 he says, "Je vas donner ici une nouvelle maniere de resoudre, &c. w 
 Besides, he is ever seeking occasion of aspersing his brother's method, 
 [see Mem. Acid. (718, pp. 102, 103, 131.] and this he does sixteen 
 years after James Bernoulli's death : when, that event, the lapse of 
 (iui", the recollection of his brother's kindness, a zeal for a brother's 
 lame, ought to have assuaged and laid to sleep all angry passions. 
 
Brook Taylor's Solution of Isoperimetrical Problems. 29 
 
 tain methods of solution, of more recent date, are 
 founded*. 
 
 Some of the latter assertions the Student must be 
 content to take on trust : for their proof would lead into 
 too wide a digression. Yet, his curiosity will be amply 
 rewarded, if he will search for, what may be found, the 
 identity of the methods employed by James Bernoulli 
 in his "Solutio magni Problematis Isoperimetrici, " 
 and by his brother in his last essay, inserted in the 
 Memoirs of the Academy of Sciences. 
 
 With the above-mentioned methods, the researches of 
 the Bernoullis on these subjects terminated. Towards 
 the period of their close, in 1715, Brook Taylor, in his 
 "Methodus Incrementorum," solved the problem of the 
 Isoperimetricals, on principles not different from those of 
 the Bernoullis, but with some alteration of symbolical 
 notation. The most material alteration, or rather im- 
 provement, consisted in representing the fluxion of V 
 
 * In Lyons's Fluxions (1758), at page 99, the author, in solving the 
 Brachystochrone, arrives at this equation, 
 
 y _ _ *w 
 
 in which y, w, are the fluxions of two contiguous ordinates, and V, v, 
 the velocities in the arcs. Hence, since there is in this equation a law 
 of uniformity, precisely of the same nature as that which Bernoulli 
 
 pointed out ; the author infers, that-^-j (/ the fluxion of the arc) is a 
 
 given quantity. 
 
 The same principle of uniformity serves also, in part, as the basis of 
 Mr. Vince's solutions, Fluxions, p. 195, and of some of Emerson's, 
 pp. 1S2, 183, &c. Third edition. 
 
SO Imperfections of preceding Methods. 
 
 when fVx is the analytical expression of the maximum 
 property ; thus, 
 
 V = Mx + Ny 4- Ls 
 
 which mode of expression, Euler, as we shall hereafter 
 see, skilfully availed himself of. 
 
 The methods of the Bernoullis and of Taylor, were 
 held, at the time of their invention, to be most complete 
 and exact. Several imperfections, however, belong to 
 them. They do not apply to problems involving three 
 or more properties ; nor do they extend to cases in- 
 volving differentials of a higher order than the first : 
 for instance, they will not solve the problem, in which 
 a curve is required, that, with its radius of cur- 
 vature and evolute shall contain the least area. 
 Secondly, they do not extend to cases, in which 
 the analytical expression contains, besides x, y, and 
 their differentials, integral expressions ; for instance, 
 they will not solve the second case proposed in James 
 Bernoulli's Programma [see p. 12.] if the Isoperi- 
 mctrical condition be excluded; for then the arc s, an 
 
 integral, since it = fdx \/(^+~t~z)> i s not given. 
 
 Thirdly, they do not extend to cases, in which the dif- 
 ferential function expressing the maximum should 
 depend on a quantity, not given except under the form 
 of a differential equation, and that not integrable ; for 
 instance, they will not solve the case of the curve of the 
 quickest descent, in a resisting medium, the descending 
 body being solicited by any forces whatever. 
 
Obscurity of Taylor's Method. 3 1 
 
 There are no other peculiarities in Taylor's method, 
 than those we have mentioned, that demand our atten- 
 tion. It has no recommendation from its neatness and 
 perspicuity, but is justly censured by John Bernoulli 
 for its obscure conciseness *. 
 
 We must now direct our attention to a period of 
 greater interest, during which, the great Euler, who 
 left no part of science untouched or unadorned, directed 
 his attention to the calculus of Isoperimetrical problems. 
 
 * "■ M. Taylor, homrae d'esprit, et Geometre tres habile, qui a 
 heureusement penetre jusque dans ce que nous avons de plus profond, 
 comme il paroit par son livre de Methodo Incrementorum : sentant 
 bien la longueur embarassante de l'analyse de mon frere, et voulant 
 la rendre plus courte et un peu claire, a repandu lui-menie tant 
 d'obscurite sur cette matiere (aussi bien sur d'autres ou il a voulu etre 
 court) qu'il semble y prendre plaisir, et que je doute qu'il y ait 
 quelqu'un, quelque penetrant qu'il soit, qui Pentende partout, quand 
 nieme la matiere lui seroit deja connue d'ailleurs," &c. Mem. Acad. 
 1718, p. 103. 
 
CHAP. III. 
 
 Euler's first Memoir on Isoperimetrical Problems — Table of 
 Formula; — Solution of Problems by it — Metbods of Thomas 
 Simpson, Emerson, and Mac Laurin. 
 
 An interval of fifteen years had elapsed, before the 
 subject of Isoperimetrical problems was resumed by 
 Euler. He took it up where the Bernoullis had left it, 
 and conducted his first investigations on their plan. 
 These are not easily intelligible without a previous 
 knowledge of the researches of his great predecessors. 
 The Student, who, for initiation into the peculiar calculus 
 of Isoperimetrical problems, should resort to Euler's 
 first memoir, would, from the novelty of its terms, 
 principles, and methods, find it perplexed and abrupt. 
 But, he will fancy himself gliding along the same route, 
 if after Bernoulli, he takes Euler as the guide of his 
 enquiries*. 
 
 Euler's first memoir was inserted in vol. VI. of 
 the Ancient Commentaries of Petersburg for 1733. 
 He there distributes his problems into classes. In 
 
 * It is thus with other subjects. Investigation is easy when it is 
 made with the proper series of steps ; but difficult, when by our own 
 sagacity and labour, we must supply either steps that are wanting, 
 or approximate those which arc too widely separate. 
 
Euler's Principle of Solution, the same as Bernoulli's. S3 
 
 the first, are problems like that of the brachystochrone 
 and the curve of least resistance, with the property of 
 the minimum, but without the I soperi metrical condition 
 or any other. These are to be solved from the principle 
 of the property of a maximum, belonging; to the elements 
 of the curve, as well as to the curve itself; and from 
 the principle of the equality between two proximate 
 states of a quantity, when near its minimum or 
 maximum *; and they require for their solution the 
 variation of two elements only of the curve. Thus, in 
 the subjoined Figure which is constructed similarly to 
 Euler's f, let CLND be the element that possesses the 
 
 property of a proposed maximum or minimum: take 
 CMGD indefinitely near ; then Euler says, if we express 
 the property in CMGD and in CLND, the difference 
 of the two expressions ought to equal 0. 
 
 [a] Now CL is changed into CG = CM+MG=s 
 
 CL+—.LG, 
 
 * See p. 7. t Comm. Acad, Petrop, tarn. VI, p« 128, 
 F 
 
34 distances of Euler's Method. 
 
 \b] and LD is changed into DG = DL - LN= 
 
 or the variations of CL, LD are -ttt • LG, - jy=-. LG. 
 
 Hence, in an instance, in which fx n . ds, is to be 
 a minimum (x = abscissa, ds = differential of the curve,) 
 we have, 
 
 OA\ CL + 0B\ LD = OA\ CG + OR 1 . GD, 
 
 hence, by [a], [ft], 
 
 0+.™.LO=0B>.% i ,LQ, 
 
 or 0A».™ = OBr.£, 
 
 and consequently from the uniformity of this equation, 
 [p. 15, 20, &c] 
 
 T p 7 
 
 0A n .-^j-, or x n .-~^ — a the equation to the curve. 
 
 If W = — 4, , , ' , - = a, the equation to a cycloid, 
 
 and the expression for the minimum \$J — 7-, or fdt 
 (t the time,) or the time is a minimum down the curve, 
 
 whose equation is -. — '—— = a. This problem is that 
 as \/ x r 
 
 which was solved in pp. 4, 9. and on the same principles ; 
 for if we compare the solutions, the two latter will be 
 found to differ from the former, merely in the greater 
 compactness and regularity of their processes, and in 
 
His Classification of Isoperimetrical Problems-. 55 
 
 their furnishing something like a clue to the general 
 solution of similar problems*. 
 
 The preceding problem is easily generalised ; for if 
 P be a function of x, and J Pels is to be a minimum, 
 we should have, supposing P to become P' when x 
 oecomes x + dx, 
 
 P.CL + P'.LD = P.CG + P'MD 
 and by [a], [6], 
 
 an equation, which, according to Bernoulli, may be 
 called a specific equation. 
 
 Hence, P .— = «, the equation to the curve. 
 
 By these methods, hardly to be reckoned different 
 from those of the Bernoullis, Euler solves problems of 
 the first class : he then passes on to those of the second, 
 which besides the property \_B~] of the maximum, are 
 to have another property [A] -f ; of this class are those 
 
 * The first solution of James Bernoulli, as it stands [see p. 4.] 
 affords scarcely the least glimpse of a general method or formula of 
 solution. Of its peculiarity, the Editor of John Bernoulli's Works is 
 aware, since he remarks, " Quoniam autem synthesin meram parti- 
 cularem, nee ad similes casus facile applicandam, &c." 
 
 f The following is Euler's classification : 
 " I. Ex omnibus prorsus curvis earn determinare, quae proprietatem 
 A maximo vel minimo gradu contineat. II. Ex omnibus curvis pro- 
 prietate A sequaliter preeditis, earn determinare, quae proprietatem B 
 
 maximo 
 
36 
 
 Problems involving two Properties, 
 
 problems solved by James and John Bernoulli, in 
 which the property [A], is the Isoperimetrical one. 
 These require the variation of three elements ab, bc 3 ce ; 
 
 B 
 
 I 
 
 J I 
 
 y 
 
 1 
 
 \ 
 
 j 
 
 ; 
 
 
 a 
 
 X J 
 
 6 
 
 p 
 
 (■ 
 
 V 
 
 
 
 
 m\ 
 
 v« 
 
 
 
 
 
 
 
 y^ 
 
 
 
 
 
 
 
 
 \ 
 
 til 
 
 
 /, 
 
 
 
 
 
 and from such John Bernoulli procures, in fact, two 
 equations, involving bg, ci: one equation* called a 
 fundamental equation, from the Isoperimetrical property 
 [A] ; the other t from the property of the maximum 
 [Bl : and these equations may be thus represented : 
 |Y| . . . P-bg — Q.ci= from I sop 1 , condit. [A], 
 [d] . . . R.bg - S.ci = from max m . condit. [B] ; 
 whence results PS=QR, which, if the law of uni- 
 
 maximo vel miuimo gradu contineat. III. Ex omnibus curvis et 
 proprietate A et proprietate B aequaliter praeditis earn determinare, 
 quae proprietatem C maximo minimove gradu contineat. IV. Ex 
 omnibus curvis proprietatibus A et B et C singulis aequaliter praeditis, 
 earn determinare, quae proprietatem D maximo minimove gradu 
 tontineat.— — Siniili modo quinta classis curvas quatuor proprietatibus 
 praeditas contemplabitur et ita porro sequentes." Comm. Acad. Petrop. 
 torn. VI. p. 125. 
 
 p. 15. t p. 19. 
 
solved by two similar Equations. 37 
 
 formity prevailed in it, as in the two first instances *, 
 would be a specific equation : but, if the necessary 
 uniformity should be wanting, then it must be intro- 
 duced, by multiplying each side of the equation by 
 proper quantities, as was done in the solution of the 
 third problem +. Now Euler's plan and reasoning is 
 very similar to this : he deduces the equations [c] 
 [d~\ ; and then observes, that the quantities Q and S are 
 frequently so compounded, that Q~P-\-dP, and S = R 
 + dR ; or, if not so compounded, may, by multiplication, 
 be reduced to that form J . 
 
 Euler, in this part, gives no general proof of the 
 
 preceding assertion ; but, if it be admitted, then the 
 
 equation 
 
 QR = PS, becomes 
 
 R(P + dP) = P (R + dR), and consequently, 
 R.dP = P.dR, or -77- = -55-, and integrating 
 
 P -+- aR = O, the equation to the curve. 
 
 In point of practical convenience of solution, a great 
 
 step was made by Euler in the preceding process. For 
 
 the computist is directed to attend solely to the deriving, 
 
 from one of the proposed properties, an equation of the 
 
 form, 
 
 P.bg - (P+dP)ci; 
 
 * pp. 20, 24. f P- 26. 
 
 + " In quibus quantitates 2 et S plerumque ita sunt comparatae, ut 
 sit 2—P+dP, et S=R-\-dR. Si vero hujusmodi formam non 
 habuerint, poterunt semper multiplicando vel dividendo tequationes 
 ad talem reduci," Comm. Acad. Petrop. torn. VI. p. 134. 
 
38 Resulting Equation of Solution : 
 
 whence P will be known. He must then derive from 
 the other of the properties a similar equation, 
 
 R.bg - (R + dR).ci 
 
 whence R will be known : and the resulting equation 
 
 of solution will be 
 
 P + aR. 
 
 For instance, suppose one of the properties [-6], to 
 be such, that fy". dx shall be a maximum; then, since 
 dx is constant, 
 
 Na n + Pb n + Rc n = NaT + Pg n + Ri n 
 and .\ Pb' 1 - (Pb + bg)" = (Re - ci) n - Re", 
 and, Pb n ~ 1 bg= Rc n ~\ci, 
 or Pb n ~\bg - Rc n - l .ci - O, 
 which, since Re is the contiguous or consecutive ordi- 
 nate to Pb, and therefore (if Pb = y)=y + dy, is of 
 the proposed form, 
 
 P.bg- (P+dP)ci, 
 
 d y n 
 in which P=y n ~ 1 = — — -, or to render the equation 
 
 homogeneous, y n ~\dx; similarly, if T had been a func- 
 tion of y and the property J'T. dx ; P would have been 
 
 /IT fJT 
 
 = -j— or — . dx. Suppose the other property \_A~\ to 
 
 have been fx m . ds — a, a being constant and ds the 
 
 element of the curve, then 
 
 BN m .ab+BP m .bc + BR m .ce = BN m .ag + BP m .gi + 
 
 BR'.ie, 
 
 and substituting according to the forms [a], [b~\, p. 33. 
 there will result 
 
Application of it to Instances. 39 
 
 (BN m &- BP m ~).bg - (BP m ^-BR ml ^\.ci 
 
 \ ab be/ ° V be ce/ 
 
 = *, which is evidently under the proposed form, 
 if we make R = BN mb £- BP m jf, or symbolically = 
 
 (x m . di/\ 
 \ h Similarly, if instead ofyV* ds, the form had 
 
 beenyA'.^s, Ja function of x 3 then there would have 
 resulted, 
 
 V as / 
 
 The equation therefore to the curve with these two 
 properties, A and B, 
 
 , 7 7 /x ni .dy\ 
 
 or generally, -7- e?# + a.e? (A.-r~) • 
 
 Since the equation P.bg — (P + d P) ci, derived 
 from the property B, is precisely similar to the equation, 
 R.bg — (R + dR) ci, derived from the property A ; it 
 is plain, that the resulting equation, that which deter- 
 mines the curve, or the relation between y and x, will 
 be the same, if B and A were changed ; that is, if 
 
 P.bg- (JP+ dP) ci 
 be derived from A, and 
 
 R.bg - (R + dR) ci 
 
 * This is exactly under the form of John Bernoulli's specific equa- 
 tions, for the law of uniformity is manifest in it. 
 
40 Commutability of Properties. 
 
 from B. For instance, suppose the curve to be required 
 
 which, with a given length, should contain the greatest 
 
 area ; here the maximum property B is fy dx, and the 
 
 Isoperimetrical property A, is fy/{dx 1 + dy x ) = a. 
 
 x d x 
 The resulting equation is dy = a — an equation 
 
 \/ v ~ v ) 
 
 to a circle : which equation would equally result, if we 
 required the curve which, with a given area, should 
 contain the greatest arc. 
 
 This important remark of the commutability of the 
 properties was originally made by James Bernoulli *. 
 
 Euler, we have seen, reduced problems of the second 
 
 class to a dependance on two similar equations of the 
 
 form 
 
 P.bg-(P + dP) ci = 0, 
 
 the determination of P depending on the proposed pro- 
 perties : for instance, if the property A or B werefT.dx, 
 
 jrp j rp 
 
 T=f(y), P would equal -a — , or -7— . dx. If the pro- 
 
 " la quibus omnibus singularis quaedam observatur reciprocatio. 
 Quemadmodum enim, exempli gratia, inter omnes figuras ejusdem 
 Perimetri circulus maximam possidet aream, catenaria maximam 
 conversione sui gignit superficiern, solidumque maximum elastica ; 
 sic inter omnes vicissim figuras quae aut aequalibus gaudent areis, aut 
 aequales rotatione gignunt superficies, solidave aequalia ; circulus, 
 catenaria et elastica minimo clauduntur ambitu, quod pariter procedit in 
 omnibus aliis." Acta Erud. Lips. Mai, 1701, p. 213. or Opera, torn. II. 
 p. 919. or, John Bernoulli, torn. II. p. 234. — Euler makes the same 
 remark. Comm. Acad. Petrop. torn. VI. pp. 135, 150. and torn. VIII, 
 p. 175. and in his Methodus Jnveriiendi Curves, fyc. See also 
 Emerson's Fluxions, p. 170. 
 
Eider's Table of Formula;. 41 
 
 perty were JT.dy, T=f(;v), P would =-t- or -j-.dx. 
 
 If the property were fT.ds, T=f{x) } P would equal 
 
 d(T.-j-j ; and by observation on the resulting forms for 
 
 P, Euler generalised his conclusions, and arranged them 
 in a table, after the manner of the subjoined specimen*: 
 
 Proprietates 
 
 
 Valores Litters P 
 
 propositi. 
 
 
 respoiidentes. 
 
 i. JT.dx . . 
 
 . d,T=Mdy\ . 
 
 . P = M.dx 
 
 u.fT.dy . 
 
 . dT= Ndx . . 
 
 . P = N.dx 
 
 in. J'T.ds . . 
 
 . dT= Ndx . . 
 
 • *-*(*2) 
 
 iv. fr.ds. . 
 
 . dT=Mdy . . 
 
 •*=«•(*§>- 
 
 &c. 
 
 
 
 Mds, 
 
 and of these forms he gave fifteen, by reference to which, 
 any problem belonging to the second class might be 
 solved. 
 
 For instance, suppose the curve to be required, which, 
 amongst all others of the same length, should contain 
 the greatest area. Here, 
 
 * Cornm. Acad. Pet) op. torn. VI. p. 141. 
 
 f dT—Mdy, and M— — - or M is the differential coefficient of T, 
 dy ' 
 
 making in T, y to vary. Similarly, 2V r —- — is the differential 
 
 coefficient making in T, x to vary. If T should contain both x and y, 
 that is, if dT—Mdy -|- Ndx, then M and N would become partial 
 differential coefficients. See Princ. Anal. Calc. p. 79. 
 
42 Application of the Table of Formula, 
 
 the maximum property B, = fydx, 
 the Isoperimetrical A, =fds = a. 
 By Form i. T=y ; ;. M= 1, P = dx. 
 
 By Form in. T = 1 ; P = rf (^) or R = rf(^) 
 
 Hence, since the equation is P ± aR = Q [see p. 38.] 
 
 c?j?db a.d (-—) = 0, or a? -f « -r - c = O (c = cor- 
 \as/ as v 
 
 rection) ; and by reduction, 
 
 , (x-c).dx . . , 
 
 ay — — —-— — f -r-, an equation to a circle *. 
 
 J v/(a 2 ~[a7-cj 2 ) H 
 
 Again., suppose the curve to be required, which, 
 amongst all others of the same length, shall, by a rotation 
 round its axis, generate the greatest solid. Here, 
 B -fy\dx\ ;. by Form i. T = y% M=2y, P-2y.dx, 
 
 A z=fds; .\ by Form in. P or R =d(-£); 
 
 Hence, 2y.dx + a.d(-~) = 0, 
 
 7 ds.d % y — dy.d % s 
 
 or, 2;/ .ax + a. '^r~r^ 
 
 St 
 
 But since dx is constant, and ds* = dx 1 + dy% ds.d's = 
 dy.d 2 y; therefore, substituting, 
 
 2y.dx+a.- ^ =0, 
 
 •See Emerson's Fluxions, third edition, p. 1ST; also Simpson's 
 Fluxions, p. 485-. 
 
to the Solutio?i of Problems. 43 
 
 CI Cill d V 
 
 and, 2y -\ •'' '" 3 = 0; multiply by dy, and 
 
 (dx 2, -f- dy*Y 
 
 integrate, and we have 
 
 a .dx r -, 
 
 y* _ = c ^ c = correction J 
 
 (dx^+dy*)* 
 
 i t (V — c) dy 
 
 whence, dx = -rr-, — r , -,, v . an equation to the 
 
 \/(« - [y -c] *)' 
 
 elastic curve ; and which in a particular case, when 
 c = 0, becomes 
 
 dx - y'- d y 
 
 and the curve in this case is called the rectangular elastic 
 curve *. 
 
 As a third example, let the curve be required, which, 
 amongst all others of the same length, shall have its 
 center of gravity most remote from the axis. Here, 
 
 (calling x the distance from the axis) B —J — - — ; .*. by 
 Form in. [since * is a given quantity] P ~d(x-4-) 
 again A=fds-, :. by Form in. P, or R = d (-p) 
 
 Hence, «.rf(g)+rf(.40= ' 
 
 dti 
 .'. (a + x) -J--C, or cds — (a + x) . dy 
 
 an equation to the catenary. 
 
 * See Simpson's Fluxions, p. 486, where the solution is not 
 general. 
 
4,4 Problems of the third Class, 
 
 This example could not have been solved by Euler's 
 table, if the property had been any other than the 
 
 7 i 
 
 Isoperi metrical : for s, an integral, =fdxy/(\ +~j—. ,J ', 
 
 and Eider gives, in this memoir, no general method of 
 finding the resulting equation, such as P is in his table, 
 when the analytical expression of a property involves 
 integrals. See torn. VI, p. 144, 
 
 By means of this table, the practical solution of Iso- 
 perimetncal problems, was, as it has been already said, 
 very materially expedited. In a subsequent part of his 
 memoir*, Euler increases his table by nine new forms : 
 making the whole number twenty-four. And although 
 this table is now superseded, yet its examination is not 
 without interest, since we may discover in it the parcels 
 of that general formula, which the author afterwards ex- 
 hibited. 
 
 Having given rules of solution for all problems 
 belonging to the second class, Euler passes on to those 
 of the third. In this a curve is required, which, 
 amongst all others equally possessing the property A and 
 the property B, contains a property C of maximum or 
 minimum. For instance, if the curve be required, 
 which, amongst all curves of the same length and the 
 same area, is such, that the time down it is a minimum. 
 This class requires the variation of four elements of the 
 
 Ovum. Petrop. torn. VI, p. 146. 
 
solved by Three similar Equations. 45 
 
 curve ; and consequently the methods of the Bernoullis, 
 without extension, are insufficient. The course pursued 
 by Euler is like his former one in the second class. He 
 constructs a diagram similar to the one, p. 36, the sole 
 difference arising from the introduction of a fourth 
 element of the curve ; and he reduces the analytical 
 solution to a dependence on three similar equations 
 instead of two. 
 
 The general form of these equations is, 
 P.bg-Q.ci+ R.d* 
 d$* being a variation of the ordinate similar to the varia- 
 tions bg and ci ; and he makes an observation similar to 
 the one in p. 37 ; namely, that it frequently happens in 
 simple cases, that Q = P + dP, R = P + 2dP + d*Pi 
 but, if such should not be the form of Q and R, then 
 the skill of the analyst must be directed to reduce them 
 under that form. 
 
 When three equations, such as 
 
 P.bg - (P + dP) ci + (P + 2dP -f- #P)M = O 
 
 p .bg — (p + dp) ci -f (p + 2d 21 + d'p) .d$ = O 
 
 ir.bg - (ir + dir) ci + (w + 2 dir + d % Tr).d$ = O 
 
 are obtained, the resulting equation to the curve will bef 
 
 P + ap + b-rr = O [l] 
 for, taking the differential, dP +a.dp-\-b.dir= O [a] 
 and again, d'P + a.a 2 p+b.d 2 7r = [3] 
 And since, [l] x bg = ; ([l] + [2]) x ci = 0, and 
 ([l] + 2[2]+[3])x^=0; 
 
 * d$ has no connexion whatever with the separate symbols d, £ 
 f Euler, Comm, Acad. Pttrop. torn, VI, p. J 4!>. 
 
46 Instances of Problems of the third Class. 
 
 '. P.bg 4- ap.bg + bir.bg =0 
 
 - (P + dP).ci - a(p + rfp).ci - b(Tr-\-d>rr).ci = 
 JP + 2rlP + d*P).ctt + a(p + 2«fp + d'p)d$ + b(>rr + 
 2dir + d 2 ir) dS= 0; 
 
 which equations, adding the quantities that are placed 
 vertically, verify themselves ; and accordingly shew, 
 that the equation 
 
 P 4- up -f bir = 
 satisfies the three equations. 
 
 The solution of problems then would be similar to that 
 of those in the second class, and would be immediately 
 obtained, if the quantities P, p, ■*, were contained in his 
 table of forms *; and in many cases they are. For 
 instance, suppose the curve to be required, which 
 amongst all other curves, of the same length and same 
 area, generates, by a rotation round an axis parallel to 
 the ordinate ?/, the greatest solid. Here, 
 the max m . property C'lsfxdy ; .'. by n. p. 41. P = 2xdx, 
 the . . . property A isj'ydx ; .*. by i. p = dx, 
 
 the . . . property B is fds ; .*. by in. tr — d (-j~) ; 
 and consequently, the equation to the curve is 
 2xdx 4- a . dx 4- b.d (Jjty = O ; 
 
 and integrating, x z -\- ax 4- b -~ = c [c = correction] 
 
 . 7 (x* 4- ax - c) dx 
 
 whence dy = -—^ r - r - i : «— . 
 
 J x (b — [pf 4- ax - c] 3 ) 
 
 Comtn. Petrop. torn. VI, p- 151. 
 
Objects attained by Eider in his first Memoir. 47 
 
 Again, as a second instance, let us take a third case of 
 the brachystochrone, of which we have already had 
 two, pp. 4, 18 ; that is, besides the condition C of quick- 
 est descent, let the conditions A and B be those of a 
 given length, and a given area ; 
 
 here, C = / —r- ; .*. by in. P = d ( , '■ , - ^ 
 
 ^ \X x J \ds\/xj 
 
 A = fydx ; .*. by I. p = dx 
 
 B =fds; :. by in. * = ^(-r0 
 
 and consequently, the equation to the curve is 
 
 d(-r~~—\ +adx + b.d(~-) = O: 
 \ds\/ x/ \ds/ 
 
 , . . dii j dy 
 
 and integrating, -= — ^y -\- ax + b -~ = c. 
 8 & ' ds\/ x ds 
 
 Euler, as we have observed, commenced his researches 
 
 where the Bernoullis had terminated theirs ; but, at the 
 
 end of this his first investigation, he was considerably 
 
 removed from the original point of departure. Several 
 
 important objects had been attained by him ; the 
 
 solution of problems involving three or more properties ; 
 
 the reduction of such problems to a dependence on two 
 
 or more similar equations ; the solution of problems of 
 
 the first class, and of some of higher classes, by a more 
 
 general method *, and by reference to a table of 
 
 formulae. 
 
 These methods were held, by their author, to be so 
 
 * Tamen eas methodo paulisper diversa et multo latius patentt- 
 sum persecuturus, &c. Comm. Pet r op. torn. VI. p. 127. 
 
48 Imperfections of Euler's Method. 
 
 complete, that, on the ground of facility, nothing farther 
 was to be desired*. They are, however, not exempt 
 from several imperfections. Problems, involving the dif- 
 ferentials of x or y of an higher order than the second 
 cannot be solved by them: for instance, that which 
 
 requires, amongst all other curves, one, in which, /—. — ~ 
 
 is a maximum or minimum. Secondly, problems cannot 
 be generally solved by them which involve integrals, 
 such not being constant : for instance, that in which it 
 should be required to find a curve that amongst all other 
 curves, has its center of gravity lowest. Euler solves 
 this problem when another condition, that of the 
 Isoperimetrical property, is added : for then the arc s, the 
 
 integral of dx\/ ( 1 +~f~^) ? is in all curves supposed to 
 
 be the same -f-. 
 
 The researches of Maclaurin +, Emerson §, and 
 Simpson j|, on this subject, may here be noticed. With 
 regard to practical methods of solution they do not extend 
 so far as those of Euler, which we have been speaking of; 
 and in point of perspicuity, if we except Maclaurin, the 
 other two mathematicians are inferior to the learned 
 foreigner. 
 
 * Atque methodo tam facili solutum ut nihil amplius desiderari 
 posse videatur. Comm. Petrop. torn. VIII, p. 159. 
 
 + See p. 43. 
 
 X Fluxions, p. 478. § Fluxions, Third edition, p. 170. 
 
 \\ Fluxions, vol. II. p. 480. Third edition, 1750. 
 
Maclaurin's Formula. 49 
 
 The methods of Maclaurin and Simpson (for Emer- 
 son's is plainly taken from that of .the former) extend to 
 cases in which more than one property is involved : but 
 they are inapplicable to the three cases, and the connected 
 problems enumerated in p. 30. 
 
 Maclaurin's formula of solution is this : If X and Z 
 
 are functions of x, then if Xds - Zdy be a minimum or 
 
 maximum, Xdy — Zds. This result is included 
 
 amongst Euler's. [See Co mm. Acad. Petrop. torn. VI, 
 
 pp. 141, 142, 143.] For since Xds expresses one pro- 
 
 dX 
 perty, and dX= ~- dx, or since X is a function of x, we 
 
 have, by Form in. the quantity corresponding to P 
 
 [see p. 41.] —dfX.-^-j } and for Zdy expressing the 
 
 other property, by Form n. the quantity, corresponding 
 
 7 y 
 
 to P, = -j—.dx; consequently, the resulting equation is 
 
 d fX-~J =a.dZ and Xdy = a Zds, 
 
 the same result in fact as Maclaurin's. 
 
 Simpson's method is equally restricted with Mac- 
 laurin's : it rests too on the assumption of the principle, 
 that the property of minimum or maximum, true for the 
 whole curve, is true also for any portion of it. The want 
 of generality, therefore, in this principle, would vitiate 
 the method in its application to the excepted cases. 
 
 The methods just described solve not problems of 
 oreater depth and intricacy than those of the Bernoullis ; 
 although, it must be remarked, they are invested with 
 
 H 
 
50 Sbnpson's Method. 
 
 greater analytical neatness and compactness. They are 
 not, however, more perspicuous ; and, even if they did 
 possess greater extent and clearness, it would not suit 
 the purpose of the present Tract, longer to insist on them, 
 since they conduct us not towards that formula and 
 algorithm, with which the researches on this subject 
 have been closed. 
 
 The planof this Work now leads us to the consideration 
 of Euler's second memoir; in which, he very materially 
 improved the calculus of Isoperimetrical problems; 
 although he introduces his resumed researches with a 
 prefatory remark, that his former methods of solution, 
 on the footing of facility, left nothing farther to be 
 desired *. 
 
 * Atque methodo tarn facili solutum, ut nihil amplius desiderari 
 posse videatur. Comm. Petrop. torn. VIII, p. 159. 
 
CHAP. IV. 
 
 Eider's second Memoir — General Formulae of Solution — Characters 
 of distinction, which Problems admit of — Exceptions to the 
 general Formula?. 
 
 The formation of the table * in Euler's former 
 memoir depended on this principle : If fVdx is the ana- 
 lytical expression of the maximum property +, certain 
 functions of x 3 y, &c. are substituted for V, and thence 
 
 * P. 41. 
 
 f Since in the succeeding part of this Tract /Fe?.r will be frequently 
 used as the analytical expression of the maximum property, it may 
 not be improper to illustrate its meaning by one or two examples. 
 
 ds 
 
 V 
 
 rds r v{dx>+ df) r dx ^ ( 1+ d?) 
 J -7=—J -j= ==/ 77=- , which, 
 
 /ds 
 = a minimum j therefore 
 
 — - —J — — ==/ — =~ , which, compared with 
 
 rv . . r . V + dx r ' ■Jd+j?) dy^ 
 
 fVdx, gives V= t = — J ■ '- |p=~l. 
 
 v 1) ^ y L dx\ 
 
 Inthesolid of least resistance, the resistance ==/ T ' , „ =/ ■■ .dx ; 
 
 axr+dy* u 1 +p 
 
 therefore, comparing, V— \ - - 2 . 
 
 If, with a given area, the curve generating the solid of least surface 
 is required, then [see p. 37.] fVdx = f(2-7ry.ds + aydx) = 
 A2*y/[l+P*l-dx + aydx); ,\ V= ay+lvy-S^+tf). 
 
 And from these expressions, the value of d Fmay be found by the 
 common rules of the differential or fluxionary calculus. 
 
,j2 Expression for the Differential of the Maximum. 
 
 a resulting equation (according to Euler the equation 
 P = o), is obtained. In the present memoir, he represents 
 the differential of/ 7 ", or dV, as Taylor had done [p. 30.] ; 
 that is, he puts 
 
 dF= Mdx + Ndy + Pdp + &c. + Lds* 
 
 dy 
 
 \p = -f- , and .9 = arc.l 
 LX dx J 
 
 and from this more general formula of representation, he 
 
 deduces an equation comprehending almost all his former 
 
 particular equations which he had registered in a 
 
 table [p. 41 .]. His mode of proceeding is not materially 
 
 different from his former mode. If V contains no 
 
 integrals, but is merely a function of x } y, dy, &c, then 
 
 the property expressed by fVdx is common to the 
 curve, and to its elementary portions : hence, if we 
 suppose two portions only of the curve to vary, and if 
 /^^corresponds to CL, and V'.dxto LD, (V+V) 
 
 * P in this and the following pages is merely a coefficient, and 
 different from the P of the preceding Chapter. 
 
Eider's Method of estimating it. 53 
 
 dx is a minimum. Hence, that peculiar differential of 
 (F+V) dx, which arises from changing CL, LD, into 
 CG, GD, is equal to nothing : which differential is easily 
 obtained from the common differential, since each has this 
 in common, that of being the first term of the difference 
 of two successive values : now the common differential, 
 
 or, (dV+dl 7 ') dx = 
 (Mdx+Ndij+Pdp) .dx + (M'dx + N'di/' + P'dp^dx, 
 supposing the differential of V to consist of three 
 terms only, and M r , N', P', to be the values of M, N, 
 P, when V becomes V . Now, in order to compute the 
 changes in the several terms, we may observe, that since 
 AC—y, and since dx is supposed to be constant, the 
 terms Mdx, Ndy, are not affected by the translation of 
 
 t TP T 
 
 the point LtoG; but Pdp is ; for p ~-~- = -^7= : but when 
 
 (IX KsJli 
 
 T . . r EG EL + LG 
 
 L is transferred to G, p — -p-p, = ; conse~ 
 
 ' CE CE 
 
 quently the variation produced in p, or what dp 
 becomes, when it expresses the peculiar differential 
 
 LG* 
 
 which we are seeking, is — >,»■, • Hence the term Pdp 
 
 * Those who are possessed of the calculus of variations will here 
 perceive the very great advantage accruing from the mere in- 
 vention of a symbol 5 to denote an operation, bordering on, vet distin- 
 guished from, the operation which d denotes. Circumlocution and 
 ambiguity were both rescinded by that invention. It is wonderful that 
 Euler did not hit on it : for the rules belonging to such a symbol, he 
 plainly lays down in Co?nm. Petrop. torn. VIII, p. 163, and afterwards 
 
 more 
 
54 Formula: of Solution. 
 
 must be written P.— , putting v=LG, and dx=CE. 
 
 Again, i) = BL, and by the translation of L to G, = BG 
 = BL + v. Hence, dy' must be written v, and A 7 ', dy' , 
 
 N'v. again// = ^ — 3 and by the translation of L to G = 
 
 jL/ — //fr LI — v . — v 
 
 — -7 = — r- — . Hence, dp must be written -5 — , 
 
 p/ 
 
 and P' .dp', — j— . Collecting the several terms, we have 
 the whole variation of {V -f- V) dx equal to 
 
 ( N— -j — j v . dx, retaining quantities of the same order : 
 
 lastly, since the variation must equal nothing, we have for 
 cases comprehended within the expression dV '= Mdx 
 -\-Ndy + Pdp, this formula of solution, 
 
 iV- -7— = o. 
 dx 
 
 By a process similar, but longeron account of the 
 introduction of the quantity ds, Euler deduces from 
 
 dV = Mdx + Ndy + Pdp + Lds 
 this formula of solution, 
 
 _ T dP Ldy 
 N- -j—+ —j-^ = 0. 
 dx ds 
 
 more fully and distinctly in his tract on Isoperimetricals ; and in his 
 last memoir, Novi Comm. Petrop. torn. X. p. 54, he says, " Illud 
 problema Isoperimetricum latissimo sensu acceptum, prout id quidem 
 in libro singulari pertractavi ; quern qui attente legerit, non dubitabit, 
 quin hujus generis investigationes calculi speciem prorsus singularem 
 postulent, a consuetis aualyseos regulis non parum diversam." 
 
Problems of the second Class. 
 
 55 
 
 The two preceding formulae have been deduced for 
 the first class of problems that involve one propertv only. 
 But Euler's method, although more tedious, is not essen- 
 tially different for problems of the second class. These 
 require the variation of three elements of the curve ; 
 and accordingly, we must compute the variation in 
 
 s I 
 
 I 
 
 1 
 
 ■» 
 
 ] 
 
 tl 
 
 | 
 
 S 
 
 a 
 
 
 J 
 
 
 /' 
 
 
 7 
 
 
 
 9 
 
 b 
 
 
 A 
 
 
 
 
 \7l 
 
 
 
 
 
 
 & 
 
 
 f, 
 
 
 °£ 
 
 
 
 
 tV+V' + V) doc, when the three elements, instead of 
 ab, be, ce, become ag, gi, ie; that is, as before we 
 must note the variations in 
 
 M.dx + N.dy + P. dp + &c. 
 M'.dx+N'.dy'+ P .dp' + &c. 
 M".dcc + N".df+P". dp" + &c. 
 
 arising from the translation of the points b and c, to g 
 and i and from the introduction of two arbitrary quan- 
 tities or variations such as bg, ci, and thence will result an 
 equation of the form A.bg- B.ci = 0, which compared 
 with the equation, R.bg- (R-\-dR) .d = 0, which Euler 
 
5 6 Formula: for Problems of the second Class. 
 
 had previously obtained *, would give — ry = — , 
 
 whence R is known. 
 
 In the formula dV—Mdx + Ndy + Pdp (forming 
 the equation A.bg — B.ci, according to the precepts just 
 given,) 
 
 A = N'.dx - dP 
 
 B = N".dx - dP' 
 
 , dR d(X'dx-dP) , n _ T/ 7 irt 
 consequent! y , —jt = \t, , r-p— , and/t = JX dx - dP, 
 
 dP^ 
 
 or 
 
 '-*<*-£)■ 
 
 This is an equation derived from one property fVdx\ 
 deduce a similar equation from the pther f Wdx ; for 
 
 instance, such as ^ — dx ( v — -7- J , and the equation to 
 the curve will be 
 
 A' 
 
 r/P\ / d. 
 
 + at = 0,or (A^) + „( u _IT) = 0; 
 
 ., d(P + air) 
 or A + av i — r = ° 5 
 
 T dP . 
 
 and since this form is like that of A r — -5— , instead of two 
 
 operations for finding the values of R, R, {romfVdx^ 
 a maximum, wnAfWdx = a constant quantity, we may 
 substitute one, and deduce the resulting equation from 
 
 * P. 37. 
 
Similar to those for Problems of the First Class. 57 
 
 fVdx + afWdx, 
 or /( V + aW) dx, 
 and for this great simplicity introduced into the calculus, 
 we are indebted solely to Euler. 
 
 For the sake of stating and illustrating Euler's 
 method, we have taken a simple form to represent dV, 
 that is, Mdx + Ndy + Pdp: but Euler introduces a term 
 Lds, which, for problems of the second class, renders the 
 calculation longer, and the result more complicated : in 
 this case, however, 
 
 c]R _ - d*P + L'dx .de + dxd(L'e + N) r dy-i 
 R ~ -dP + dx(L'e+N) L £ ~ ds\ 
 
 L'd x. ds d\-dP + dx{L'*+Nj] 
 
 - -dP + dx{L't + N) + '-dP+dx(L't + N) ; 
 
 and, R = e^^^'+^x ( - dP+ dx [Lt+M]). 
 
 This memoir of Euler contains almost all the matter 
 
 which is to be found in his subsequent tract on Isoperi- 
 
 metricals : but, the matter is ill arranged: considerable 
 
 perplexity, and some error, is introduced by the term 
 
 Lds: the cases in which it enters, are mixed with those 
 
 in which L—-0. These latter, in the above mentioned 
 
 tract, the author separately considers ; and even here he 
 
 notices, that as from dV=Mdx + Ndy, dV—Mdx + 
 
 Ndy + Pdp: the resulting formulae of solution are, 
 
 dp 
 N— O, and N— -j- = O, so if we put 
 
 dV- Mdx + Ndy + Pdp + Qdq+ Rdr+kc, 
 
 then the resulting formula will be 
 
 i 
 
58 General Formula. 
 
 N- — + ^?- — &c =0 
 dx dx 2, dx 3 
 
 which, by the principle stated in page 52, will belong 
 
 to problems of all classes involving definite expressions. 
 
 This last formula, remarkable for the simplicity of its 
 
 law, supersedes Euler's table of the values of P*, given in 
 
 his former memoir. In fact, all problems, in which 
 
 integral expressions do not enter, are solved by it; 
 
 and although Euler himself, and afterwards Lagrange, 
 
 very materially simplified, and expedited its proof, yet, 
 
 as a formula of solution, it still remains as a final 
 
 result of all researches on this subject. 
 
 Problems are classed according to the number of 
 properties which they involve t ; and the number deter- 
 mines that of the ordinates that must be made to vary. 
 But, problems might be distinguished also from each other, 
 by the order of differentials which they involve ; and 
 then the order would determine the number of terms to 
 be used in the formula dV=Mdx + Ndy + Pdp + &c. 
 thus, if the curve were required, in which f{ax—y z )ydx 
 is a maximum, we have V—iax—y^y, dV = 
 ay.dx-]-(ax — 3y") dy : consequently, two terms are suf- 
 ficient, namely, Mdx, Ndy, and the formula of solution is 
 N = o, or ax - 3y 2 = o. 
 
 This problem is, according to Euler's classification, of 
 the first class, and might be said also to be of the first 
 order. 
 
 • See p. 41. t Seep. 35. 
 
Distribution of Problems into Classes and Orders. 59 
 
 If the br achy stochr one, with no limitation, were 
 required, since the time 
 
 . r ds _/* ds ( V(d& + dy 2 ) _ fi„ \/ (1 +P X ) 
 - J vel'. ~ J -V~y- J y/y ~ J dX —VT~ ' 
 
 by comparing it vt\t\\f Vdx, we have V— — v — f *, 
 
 s/y 
 
 and thence dV = '—.dy-i- — 7—7- -dp; so 
 
 2.y- ^ \/3/(l+F) ^ 
 
 that, the formula Mdx + Ndy + Pdp, or three terms 
 
 are sufficient, one of which Mdx = 0. 
 
 This problem of the first class, might be said to be 
 of the second order. 
 
 Again, the solid of least resistance is of the same 
 class and same order with the preceding problem ; for 
 
 since J -rj ■> a = a minimum —J^Vdx, by comparison, 
 we have, V= , "; a , and </F= — - — - . dy + 7 * , / z . - _ . y dp. 
 
 But, if a curve were required, in which^-^- should 
 
 d'ti d^u dx 
 
 be a minimum, then since Vdx — -f-—~r^' — , ,, . — 7? 
 
 as ax- v ( 1 +p) 
 
 V-—r^- - , and dF= - M X3 .dp^—yj^ ^ . dq 
 
 so that this problem would require four terms, or this 
 
 formula 
 
 dF=Mdx + Ndy + Pdp+Qdq; 
 
 (in which M and iV= 0), and therefore might be said to 
 
 * See p. 5 1 . 
 
60 The Class determines the Number of Ordinates that jnust vary. 
 
 be of the third order, and, according to Euler's classifi- 
 cation, would be of the first class, if it depended on the 
 minimum property alone. 
 
 The class of problems then determines the number 
 of ordinates that must be made to vary ; but, on what 
 might be called their order, would depend the 
 number of ordinates to be taken account of, in the 
 computation of the variations. For, as more terms of 
 the series Mdx-\-Ndy+h.c. enter, more ordinates are 
 required to estimate the variations of the terms : for 
 instance, let dV— Mdx+Ndy-\- Pdp : then if the pro- 
 blem is of the first class, the variation of one ordinate is 
 sufficient. Let * BL =y be the ordinate that varies, and let 
 the two adjacent ordinates be y, y", then it is plain, from 
 the subjoined table, that three ordinates y, y', y" are suffi- 
 cient for the computation of the variations in 
 Mdx+Ndy + Pdp, and in M'dx + N'dy' + P ,dp\ 
 
 dg EjL 
 
 dx dx 
 
 d£ LI 
 
 dx ' dx 
 
 y =,AC 
 
 ;</ = BL 
 
 V 
 
 = KD 
 
 p = 
 
 P = 
 
 P'= ±r 
 
 dx 
 
 Variation in p = 
 
 V 
 
 dx' 
 
 variation in p' = — 
 
 dx' 
 
 The introduction of other ordinates, such as y"', v/' v , 
 &c. or y x y^ &c. on cither side, is plainly unnecessary. 
 
 If, however, dq—~~- — -~ enters into the form, 
 1 ax* dx° 
 
 dF= Mdx + Ndy + Pdp + Qdq, 
 
 or if 
 
 See Fig. p. 52. 
 
The Order, the Number that must be brought into Computation. 61 
 
 then three ordinates are not sufficient: an additional 
 one y^ must be taken account of: for the element 
 Vdx of the area analogous to Vdx y and immediately 
 preceding it, is M x dx + N x dy^ + P x dp x + Q x .dq x ; 
 now, the last term Q x dq^ of this formula, is af- 
 fected by the increment v of the ordinate if; for, 
 
 dp v'-p P-?\ i r 
 
 since q = ~r=- 7 l ,q= f ' x ; .; <7 is varied, if p is 
 * dx dx lK dx JN 
 
 varied; and p is varied, if if is. Hence we must intro- 
 duce into the computation an additional ordinate y x ; but 
 that is sufficient; for none of the terms representing 
 V".dx> V'.dx, &c. will be affected by the variation of 
 the ordinate y' . 
 
 What has been said relates to those cases in which V 
 involves no indefinite expressions ; for if a term, such as 
 Lds be introduced, a fourth ordinate must be taken 
 account of. If the problem is of the second class, then 
 two ordinates must be made to vary, and the number of 
 ordinates that must be taken account of is easily deter- 
 mined from the preceding reasonings. We shall have 
 occasion, however, to notice this case, when we speak of 
 Euler's treatise on Isoperimetrical problems. 
 
 In the conclusion of his memoir, Euler notices, that 
 his methods fail, or require peculiar artifices, if the 
 quantity V in fVdx, contains the arc s, or other inte- 
 gral quantities. For, in that case, the principle of the 
 whole curve possessing a certain property of maximum 
 or minimum, if a portion of it does, is not true. For 
 if V contains an integral expression, then the curve 
 OCD may contain j'Vdx a maximum or minimum, 
 
62 Exceptions to Euler's Method, caused by Integral Expressions. 
 
 although no separate portion of it has that property. If, 
 
 indeed, 
 
 dV = Mdx + Ndy + Pdp + &c. 
 
 dn 
 then, since the quantities x, y, -j-, or p depend on the 
 
 O H 
 
 \ 
 
 
 
 \ 
 
 Vc, 
 
 
 c 
 
 I 
 
 E 
 
 L #^^ 
 
 F 
 
 
 D 
 
 point C, the part CLD may be changed at pleasure, 
 without any change being produced in OC; but if V 
 contains an integral, then if CLD is changed, OC will 
 take another value from f Vdx. In these cases then, 
 the preceding methods of Euler, deduced from assuming 
 a portion of the curve possessing the requisite property, 
 fail : so likewise do the methods of Bernoulli and 
 Taylor, which are founded on the same principle # . 
 
 There are cases, however, Euler observes, in which 
 
 * Bernoulli's has been already mentioned ; and Taylor, in his 
 Methodus Incrementoruiri, p. G7. prefixes to his solution, a similar one. 
 Its truth depends, as it has been said, on the two portions adjacent to 
 CLD not varying whilst CLD varies, which is to be granted, when 
 / Vdx contains no integral. 
 
Objects attained by Eider. Q$ 
 
 the ordinary solutions may be applied; although the 
 quantity fVdx contains an integral : and these happen 
 when the integral is by the conditions common, or the 
 same to all curves, amongst which a required curve is to 
 be found. Included in these cases, are the Isoperimetrical 
 problems of the Bernoullis ; for there s, the integral of 
 
 dx y/ ( 1 + ~T~i) ls given: had it not been given, or if, 
 
 instead of the Isoperimetrical property, another had been 
 substituted, the solution of the second case proposed in 
 James Bernoulli's programma, would have been faulty. 
 
 Very important objects were obtained by Euler in 
 this memoir. The solution of problems involving differ- 
 entials of any order ; the invention of a formula includ- 
 ing his former formulae, which, to the number of twenty- 
 four, he had inserted in a table ; the partial solution of 
 problems involving integral expressions ; the establish- 
 ment of his theorems and formulae by easier pro- 
 cesses*. 
 
 An author is usually, more than justly, fond of his last 
 inventions : and Euler, by this memoir, thought he had 
 nearly perfected the method of solving Isoperimetrical 
 
 * " Non solum faciliorem quam ante detexi viam ad solutiones 
 hujusmodi problematum perveniendi, sed etiam omnes 24 formulas, 
 quas ante tractaveram, in unicam sum complexus, &c." Comm. 
 Petrop. torn. VIII. p. 159. 
 
()4 Imperfections of his Methods. 
 
 problems * : yet his methods were not without their 
 imperfections. They afforded no general solutions of 
 problems involving integral expressions; and, erroneous 
 solutions, when the differential function depended on 
 a quantity given solely by a differential equation not 
 generally integrable : and the cause of these imperfec- 
 tions was the assumption of the principle, that the 
 whole curve will be endowed with the property of max- 
 imum or minimum, if any portion whatever of it possess 
 the same [see p. 52, 62.]. 
 
 We are now led, both according to the order of his- 
 torical and scientific succession, to the consideration of 
 Euler's tract on Isoperimetrical problems. 
 
 * " Triplici rat ion e illam priorera methodum ad majorem perfec- 
 tionis gradum sum evecturus." Comm. Petrop. torn, VIII. p. 175. 
 
CHAP. V. 
 
 Euler's Tract, entitled "Method us inveniendi LineasCurvasProprietate 
 Maxinii Minimive gaudentes" — Distribution of Cases into absolute 
 and relative Maxima and Minima — Rules for finding the Increment 
 of Quantities dependent on their varied State — Formulae of 
 Solution. 
 
 This Work appeared in the year 1744, about three 
 years after the publication of his last memoir *, on the 
 same subject. It was inr nded, and, with a few excep- 
 tions, it must be conceded, to be a complete treatise ; 
 containing essentially all the requisite methods of solu- 
 tion, with great abundance and variety of examples and 
 illustrations. There is wanting, however, to make it a 
 perfect work, and on the subject, the best extant, a new 
 algorithm ; a more compendious process of establishing 
 the theorems; and certain supplemental formulae, that 
 determine, not the nature of the curve, if a curve be the 
 object of enquiry, but the conditions according to which 
 it must be drawn. These desiderata were afterwards 
 supplied by the fertile genius of Lagrange. 
 
 The former memoir contained, as it has been already 
 stated, abundance of valuable matter, but ill arranged. The 
 
 * The volume of the Commentaries of Petersburg, in which this 
 memoir is inserted, is said to be for the year 173o' ; v\liich determines 
 nothing concerning the dates in which the memoirs contained in it 
 were written. Several of them, however, were undoubtedly written 
 after 1736; since they contain accounts of observations made in 1740. 
 
66 Pl«- n of Euler's Work. 
 
 distribution and arrangement, however, of the present 
 work is extremely luminous and regular. Absolute 
 maxima and minima are first treated of, which concern 
 curves that are to be determined solely by the property 
 of maximum or minimum ; such a curve is the brachysto- 
 chrone*, which has the property of the least time, out of 
 all curves whatever that can be drawn between two given 
 points. The curve generating by its rotation round its 
 axis the solid of least resistance, is another. 
 
 If/' Vdx be the analytical expression of the maximum 
 or minimum, V may contain either determinate or inde- 
 terminate quantities, such as integrals. Euler first con- 
 siders the former cases, that is, when V contains only 
 
 quantities, such as x, y, -~, -r^, &c. which are plainly 
 
 determinate quantities, that is, of assignable value, when 
 x or y is given. 
 
 After absolute, relative maxima and minima are 
 treated of: these relate to curves that are to be deter- 
 mined not solely by the maximum property, but con- 
 jointly by that and other properties. Such a curve is 
 the brachystochrone f, when the property of equal 
 length becomes an additional condition ; that is, when 
 the curve of quickestdescent is required, not amongst all 
 curves whatever, that can be drawn between two given 
 points, but only amongst those that are of a given length : 
 such also is the brachystochrone, when a third condition, 
 that of equal area, is added +. 
 
 * See pp. -J-, 9, 34-. t See p. 18. + See p. 47. 
 
Method of solving Cases of absolute Maxima. 6*7 
 
 In these cases of relative maxima and minima, the 
 quantity V t when JVdx represents a property, may, or 
 may not, include integral expressions : and since, by an, 
 artifice like that which we have stated*, Euler re- 
 duces all questions in which are involved two or more 
 properties, analytically expressed by J^Vdx, fldx, 
 fXdx, to this form, 
 
 JVdx + afYdx + bfXdx + &c. 
 the determination of all cases is reduced, ultimately, to 
 that of an absolute maximum or minimum. 
 
 The method employed by Euler in treating the first 
 and simplest cases of absolute maxima and minima, is 
 similar to that used by him in the eighth volume of the 
 Commentaries of Petersburg : thus, suppose the expression 
 of the maximum property to befFdx, and V to be a 
 
 determinate function f of x, ?/, ~r , &c. ; then, as it has 
 
 9 J dx 
 
 L M N 
 
 appeared, the property belongs equally to the curve and 
 to its element. Let PQR be a curve, and let its 
 
 P. 56. 
 
 t P. 62. 
 
(jS Mode of Computing the Variations, 
 
 ordinates LP, MQ, NR, &e. be ?/, ;/',/', &c. and 
 Qq = v; now j % Vdx is to be a minimum or maximum; 
 Suppose jV^.dx to be its value up to the ordinate y, the 
 JDne immediately preceding LP ; then, beyond that or- 
 dinate, to the right of y, the value will be increased by 
 Vdx+V .dx + V".dx + hc. ; so that, the whole value 
 of fVdx will be 
 
 JVAx + (*"+ V + V" + &c.) dx 
 l maximum or minimum, and in which, for the reasons 
 just assigned, 
 
 {V + V + V" + &c.) dx must be a maximum 
 or minimum, that is, {dV+dV + dV" + &c.) dx must 
 be put equal to nothing # , when the changes arising in 
 dV, dV ', &c. from the translation of the point Q to q, 
 or from the increment or variation v are properly ex- 
 pressed ; or, if d V— M dx + Ndy + Pdp + Q dq, when 
 the changes or variations of . 
 
 dxQIdx + Ndy + Pdp + Qdq) [l] 
 
 dx {M'dx + N.dy' + P'.dp' + Q. dq) [2] 
 dx {M"dx+N".dy"+P"df +Q".dq") |_ 3 ] 
 ire. expressed. 
 
 Now in the Figure p. 67. the ordinate y" is changed 
 by the quantity v, and since 
 
 p 
 
 y' - y 
 
 dx " 
 
 p' - p 
 
 1 = dx 
 
 y" - 2 ?/ + y 
 dx z 
 
 pi 
 
 y" -y 
 
 dx 
 
 1 dx 
 
 y'"-<2y"+y' 
 dx 2 
 
 f 
 
 in a 
 
 y -y 
 
 dx 
 
 t - "V 
 
 y v -2!/" + i/' 
 dx 2 
 
 M 
 
 * See p. 53. 
 
from the Differential Expressions for V, QQ 
 
 it is easy to see the change, or increment, or variation pro- 
 duced \np,p f , &c. q, q', &c. from thechange v my", which is 
 the sole ordinate that admits of any : thus, the variation* 
 
 in p is = ; in »' is -7— ; in p" is - T — : in a is -, — ; in q' 
 dx c ax 2 ax 2 
 
 . - 2» „ . v 
 
 ls ^r iln ? ,s zf- 
 
 Hence, substituting in the three expressions [l], [2], 
 [3], 
 
 the variation in Til is -7-^- dx; 
 u J ax* 
 
 ■' m ^ is ( N "- p "-7i+^-77)^ 
 
 Hence, the whole variation is 
 
 /„„ P'-P' , a'-gq + ax . 
 
 P'—P' dP' 
 
 but, — -7- — = --7-y (the increment of P' being diminished 
 
 in infinitum,) 
 
 . Q"-2Q' + Q 
 
 also — -j-r = 
 
 ax* 
 
 (Q + 2dQ + d*Q)-2(Q-rdQ) + Q d*Q 
 dx* ~ ~ dx* ' 
 
 * Variation, now used technically, is not in this Tract used so by 
 Euler, 
 
70 Resulting Formula of Solution. 
 
 Hence, retaining, for the sake of homogeneity, quan- 
 tities of the same order, 
 
 %r dP dQ 
 
 which formula will solve all questions of absolute maxima 
 that do not involve integral expressions, or differential 
 
 expressions, of a higher order than dq, or -~; and by- 
 means of the principle and formula stated in p. 56, will 
 solve all questions of relative maxima that neither involve 
 integral, nor differential expressions of a higher order 
 . d z y 
 
 than ■&■ 
 
 There are two points, in the preceding process, that 
 require explanation. The increment or variation v is 
 attributed not to the ordinate y ory', but to y" ; and three 
 values of dV, such as Mdx+Ndy -\-Pdp-\-Qdq, only 
 were taken. With regard to the first point, the value of 
 CVdX) that is, JV^dx, from the left hand up to the 
 ordinate y, is supposed to be determined, and not to be 
 affected by any change in that portion of the curve that 
 lies to the right of y ; now if y" be changed, this happens, 
 or /' V dx is not affected : for, an element of it to the 
 left of y and immediately preceding y, would be, analo- 
 gously to the mode of expression used, 
 
 M % dx + N K .dy x +P K .dp K + Q x .dq s 
 and according to the table [a] p. 68, we should have 
 
 P *~ dx ' q *~ dx' 
 neither of which quantities depend on y", the ordinate that 
 
Explanation of Eider's Method. 71 
 
 is changed ; and consequently, these quantities would 
 suffer no change or variation, and therefore y V s dx will 
 not be affected. But, if to 1/ instead of y" the incre- 
 ment > had been attributed, then q s would have been 
 
 v 
 changed, and its variation would have been -7-5 . 
 
 This explanation includes, in fact, that of the second 
 point : if a fourth horizontal series 
 
 M". dx + N'\ di/" + P". dp'" + Q'" . d c(" 
 
 had been taken, then in each term, there would have 
 been no variation from the variation v of the ordinate 
 y". For, according to the table [a] 
 
 p'» — -' >' 9 j w = £ -' ^ •' 9 which quantities are 
 
 independent of y". 
 
 If the quantity V involve differentials of a higher 
 order than d ! y, the variation v must be attributed to an 
 ordinate more remote from y than y" \ and more hori- 
 zontal rows like [l], [2], [3], p. 68, must be taken 
 into the computation. We may easily learn from the 
 preceding explanation, the exact process that must be 
 instituted ; for instance, suppose 
 
 \b~\ dF=Mdx + Ndy+Pdp + Qdq + Rdr 
 
 L dx dx" dx 3 } 
 then, forming r', r", &c. as jf 9 p", q\ q", &c. were 
 formed in table [«] p. 68, we have 
 
>3 
 
 72 Formula of Solution xvhcn ~ enters into the Expression for dV. 
 
 dx 3 r 
 
 r = (/f ~ r/ _ n'" -*v" + ?>y' -v 
 
 dx dx 3 * 
 
 dx dx 3 ' 
 
 T ,>_ <i"-<l" - y v -?>}r± ?>y'"-y" 
 dx dx 3 * 
 
 r '» = ? ~ r / _ y ~ 3 y ^~ 3 y ~y . 
 
 dx dx 3 
 
 now, if v be attributed to ?/', then the preceding value of 
 
 r. or r = - * ' j ' •' — ^ would be affected bv the 
 
 x ax 3 
 
 change in r ; therefore, we must, instead of?/", make the 
 
 next ordinate, or?/", to vary by the quantity v, and then 
 
 r, and consequently, j'Vjlx will not be affected by any 
 
 change in the curve that lies to the right of?/: again, 
 
 four rows like to \b~] p. 71? arc sufficient; for, if we 
 
 introduce a fifth, since ?/ IV , /> IV , q lv , r ly do not involve ?/", 
 
 no change would be introduced in it, or its variation, 
 
 arising from v, would be nothing. 
 
 If the several variations in the four values of dVhe 
 collected and reduced, after the manner given in p. 6q, 
 the result will be 
 
 s dP d'Q d*R\ , 
 
 V dx dx~ dx 3 / ' 
 
 which, by the nature of the question, that is, by the pro- 
 perty of maximum, must equal nothing. 
 
 The above mode of computing that peculiar incre- 
 ment of J' I'd x, which arises from a change in the curve 
 itself, and which Lagrange technically, and for distinc- 
 
Rule for finding the Variation. 73 
 
 tion, called the variation, is sufficiently plain and direct. 
 The rule too that Euler gives *, is only in different terms, 
 the very rule now in use; for instance, to find the varia- 
 tion of ?/V(l + jo 2 ) 
 
 take, says he, the differential of this quantity as in the 
 ordinary calculus ; and then, instead of the differentials 
 of ?/", p, substitute those peculiar differentials, that is, 
 variations, that arise from a change in one of the or- 
 dinates (y') ; accordingly, the common differential being 
 
 (dif) x/(i + p.)+_Jj£__(rfp) 
 
 the peculiar differential, or variation, if {dy') = v and 
 ( d P) =-^,> willbe 
 
 In the preceding formula, V contains definite quan- 
 tities, such as x, y, dy , &c. but it may contain an inte- 
 gral expression, or be of this form TfZdx : in such case, 
 Euler employs a method similar to his preceding one ; 
 but, the method becomes very complicated: for then 
 corresponding to V would be TfZdx, and since the 
 consecutive value of T would be T, and of fZdx, 
 fZdx+Z.dx, the quantity corresponding 
 to V would be TfZdx+T.Zdx, 
 
 toT T'fZdx + T'Zdx + T'.Z'dx, 
 
 and then from these expressions, dF, dV, &c. must be 
 expressed, and their values put down, such as arise from 
 the variation i» of any ordinate as y ( " } . 
 
 * Mcthodns tnveniendi, p. 33, &c. 
 L 
 
^4 Formula, ivhen Integral Expressions are involved. 
 
 It will be easily perceived from what is already said, 
 that this process must be very tedious ; and it is not 
 necessary more fully to explain it, since the method, 
 with regard to its principle, contains nothing either novel 
 or abstruse, and as a method of computation has been 
 superseded by another more regular and concise. 
 
 Since s=fy/(dx> + df) =fdx^/^l + (^)*1 
 
 or since s is an integral, if 
 
 dV ' — Mdx + Ndy + Lds, 
 the formula for the peculiar differential or variation of 
 j'Vdx would be given by the preceding method ; which 
 formula is to be adopted to the exclusion of those other 
 given {dV having the same expression) by Euler in 
 the eighth volume of the Petersburg Commentaries. 
 
 V has been supposed to contain an integral J Zdx, 
 and we may go on farther, and suppose Z also to contain 
 an integral such z&CXdx, or /^to contain a double inte- 
 gral, and so on : the methods to be pursued for such cases 
 will be similar to the preceding : they will, however, 
 from their complication shew how inconvenient Euler's 
 process is ; although, if we attend to its management by 
 the author himself, we cannot help admiring the sin- 
 gular art and dexterity with which, from a perplexed 
 mass of symbols, he at length extricates compact and 
 simple formulae of solution. 
 
 Euler in the sixth volume of the Commentaries of 
 Petersburg reduced problems of relative maxima and 
 
Relative Maxima reduced to Absolute. 7<5 
 
 minima to a dependance on as many similar equations, 
 as the properties proposed. For instance, if the curve 
 required was to possess two properties, the equations 
 would be of the form R.bg— S.ci*, in which S=R-\- 
 dR. But, Euler gives there no satisfactory nor general 
 proof, that S = R+dR- y but contents himself with 
 saying, that from the analytical process, such will 
 appear to be the value of S; and if S should not be 
 under the proper form, he then directs the computist so 
 to reduce the equation R.bg - S.ci, that the coefficient 
 analogous to S, shall equal R-\-d R. 
 
 In the present Work, however, he gives a much more 
 satisfactory proof of this important principle : the nature 
 of which will be easily understood from one or two 
 simple examples. Let 
 
 dV= 31 dx + Ndy + Pdp, 
 
 and let v be the variation of the ordinate y" and u that of 
 y' '; y and u representing quantities such as hg and ci are 
 in Bernoulli's and Eider's former methodf : then since 
 [see p. 6 8.] 
 
 1 - dx ' P dx ' P "" dx 
 
 the variation of p = ^ of ;/ = ^ , of/ = - ^ 
 
 Pa 
 
 consequently, the variation of dV .dx ~ -j- t dx 
 
 * P. 37. t Pp. 15, 36. 
 
76 Proof of that Reduction, 
 
 the variation ofii&V.dx— (N'u -f P. " , " \dx 
 
 oMr\dx=(N'\v-P\-^)dx. 
 
 Hence, the whole variation is [see pp. 55, 68.] 
 
 (P'_ P\ • P" — P 7 
 
 N'-Z-j~?-).dx.*+ (N»- 1 —JL. ).dx.v. 
 ax / \ ax / 
 
 Now, A T " is the consecutive value of N', or = N' + dN' 
 P- P' is the consecutive value of P-P, or = (P' - P) 
 
 P-P 
 
 + d(P' — P) ; consequently, if we make N' -* — = 
 
 P" - P 
 
 R, the coefficient of the second term, or A" ■? s 
 
 7 dx 
 
 will be R + dR. Hence, the form of the resulting 
 equation, will be 
 
 R» + (R + dR)v. 
 Again, suppose 
 
 dF= Mdx + Ndy + Pdp + Qdg ; 
 
 here, if the increments v and w are attributed to y" and 
 y', we must introduce an ordinate ^preceding y\ since 
 
 q^ = — — -~ — —>. Hence, referring to the values o£p, q, 
 given, p. 68, and operating as before; 
 
 variation of »= 0, of» = -j--, of // = — ; — , of »"=-^- 
 rs dx * dx J dx 
 
 c « r v—2w „ , — 2v+w c ,, v 
 
by Examples, 77 
 
 Hence the variat n . of dV .dx — Q -r— • dx. 
 
 x s dx % 
 
 €£dr.dx = (p£-+Qi=Bi).dx 
 
 \ ax ax / 
 o£dF\dx=(N f a +P^+Q.^^dx 9 
 
 of dF".dx= (lX''. v - P''~ + Q".~j^).dx 
 
 \ ax ax*/ 
 
 or, collecting the terms affected with «, and v, the 
 
 variation is 
 
 C A — s— + — I* — )"**+ C A — ir + 
 
 j— jv.dx ; where it is plain, the coefficient of 
 
 the term involving vdx, is consecutive to the coefficient 
 of the term involving u.dx; or is R + dR, if the co- 
 efficient of the first term is R; hence, as before, the 
 resulting formula is 
 
 R.<* + (R + dR).*. 
 
 The truth of this result may be also easily perceived 
 from the form which the variations of j), j/, p", and of 
 q, q', q", &c. necessarily assume from their expressions 
 in terms of the consecutive ordinates, y, y', y'\ &c. 
 
 We may here again notice the important result, 
 which Euler drew from the above formula. Since Ra+ 
 (R + dR)v is derived from a property, of maximum or 
 other, represented hy fVdx ; if another property fVdx 
 be added, a similar equation, Qu + (Cl + dQ) v may be 
 derived : and exterminating v, the resulting equation is 
 R + aQ=;0 ; that is 3 
 
78 Practical Convenience of Eiders Formula: of Solution. 
 
 if V = Mdx + Ndy + Pdp + &c. 
 and Y = p.dx + v.dy + ir.dp + &c. 
 
 we shall have R + aQ= (N- ^ f &c.) +a(v - ^+&c.) 
 
 = (W+av) ^ ■'■ + &<:.; 
 
 that is, the same formula that would result, if the pro- 
 perty proposed had been 
 
 [Vdx + afY.dx. 
 By this, all questions of relative are reduced to those of 
 absolute maxima and minima : for, similar reasonings 
 and properties hold, when the curve sought instead of 
 two, has three, four, &c. properties : and if such pro- 
 perties be expressed by 
 
 J Vdx, [Ydx^JWdx, fUdx, &c. 
 then we must solve the question as one of absolute 
 maximum and minimum ; and inquire, what the 
 curve is that has the property expressed by 
 
 [Vdx + aj'Ydx + bfWdx + c/Udx. 
 
 Euler, besides the cases already mentioned, solves also 
 those, in which V contains quantities, neither determi- 
 nate such as x, y, p, &c. nor integrals ; but expressed 
 solely under the forms of differential equations. What we 
 have given however, is sufficient to explain and illustrate 
 Euler's method. The results of that method are, for the 
 practical solution of problems, under a most convenient 
 form. On that head there is nothing to desire. Neither 
 is there any want of perspicuity in the principle or in the 
 conduct of his method. It is the length of the opera- 
 
Tediousness of Processes establishing them. 79 
 
 tions attendant on his method, the want of mechanism in 
 his calculus, that are objectionable *. These inconveni- 
 ences Lagrange removed : but, as in such cases it not 
 unfrequently happens, whilst he rendered the process of 
 calculation more expeditious, he deprived its principles of 
 a considerable portion of their plainness and perspicuity. 
 
 * Euler, we learn from the historical account prefixed to vol. X. 
 of the Novi Comin. Petrop., was sensible of the inconveniences of his 
 method : " Interim tamen ipsa methodus, etiamsi totum negotium satis 
 expedite conficiat, tamen ipsi non satis naturalis est visa, propterea 
 quod vis solutionis tota in consideratione elementorum curvae investi- 
 gandae erat posita, ista vero quaestio facile ita adornari possit, ut ex 
 geometria penitus ad solam analysin puram revocetur, &c . . . Tametsi 
 autem auctor de hoc diu multumque esset meditatus, atque amicis'hoc 
 desiderium aperuisset tamen gloria primae inventionis acutissimo 
 Geometraj Taurinensi Lagrange erat reservata, qui sola analysi usus 
 eandem plane solutionem est adeptus, quam auctor ex considerate 
 onibus geometricis elicuerat." Novi Comm. Petrop. torn. X. p. 12. 
 
CHAP. VI. 
 
 Lagrange's Memoir — Use of an appropriate Symbol to denote the 
 Variation of a Quantity — Rules for finding the Variation — New 
 Process of deducing Euler's Formulae — Invention of new Formula?. 
 
 In the preceding account of Euler's memoir, it can- 
 not have escaped notice, that some ambiguity and much 
 circumlocution took place in describing the process of 
 rinding that peculiar increment which depends on the 
 quantity v, by which the ordinate y or y' is increased. 
 
 These inconveniences Lagrange obviated by a simple 
 invention ; that of a symbol such as $ # ; which, analo- 
 gously to the symbol d of the differential calculus, was 
 to be the means of representing either a quantity or an 
 operation. Thus, the quantities v and w, by which y 
 and y' are increased, are symbolically denoted by Sy, $y' : 
 and as d in d (ay n + by' m ) signifies, that the operation 
 of taking the differential of ay n -\-by' m is to be made; so, 
 $ in $ (ay n + by'" 1 ), signifies that the peculiar differential 
 of ay n + by'" 1 is to be taken, when instead of dy, dy' ; v 
 and w, or Sy, iy, are to be used. 
 
 * " Mais avant tout je dois avertir que comme cette methode exige 
 que les m&mes quantites varient de deux manieres differentes, pour ne 
 pas confondre ces variations j'ai introduit dans mes calculs unenouvelle 
 characteristique J." Miscell. Taurhu torn. II* p. 174. 
 
Similarity between the Differential Calculus and that of Variations. 8 1 
 
 With this change in the system of symbols, and with 
 certain changes in the processes of establishing the fun- 
 damental formula?, Lagrange called his method, a new- 
 method of determining the maxima and minima of inde- 
 finite integrals. But Euler, resuming the subject in the 
 10th volume of the New Commentaries of Petersburg, 
 p. 54, called it by a name which still adheres to it, 
 The Calculus of Variations. 
 
 The variation of ay 11 + by' m is nay n ~ l Jy -\- m by" n ~\$y'. 
 For if we recur to the problems solved by Bernoulli and 
 Euler *, we shall find the difference of two contiguous 
 states of a quantity at its maximum or minimum put 
 equal to nothing ; not indeed the whole difference, but 
 the first term of the difference ; for the terms involving 
 (b?i)'\ (coy, &c. f are neglected. Now, the difference in 
 
 that example, is -— // p/ , , , ; Pb or y being 
 
 1 v Fu V (Pb + bri) 
 
 increased by bn : if y had been increased by dy, the first 
 
 term of —rm ~ , , t>, T~v would have been the 
 
 \/Pb \/(l J b-\-dy) 
 
 common differential ; consequently the peculiar differ- 
 ential, that which is now to be called the variation, differs 
 from the common differential, only inasmuch as bn or v 
 or Si/ takes the place of dy. The process then for 
 finding the coefficient of the term involving b n or $y, is 
 
 *Pp. 7, 33, &c. fP. 19. 
 
 M 
 
8^ Rules for finding the Variations of Quantities. 
 
 precisely the same as that for finding the coefficient of 
 the term involving dy ; that is, for finding, what tech- 
 nically* is called the differential coefficient. The rules 
 must consequently be the same: in other words, the 
 rules of the differential calculus, become, with the 
 alteration pointed out, those of the calculus of variations. 
 
 Hence $(y n ) — ny n ~\$y, which is the first term of 
 [(# + fy) n - /'J expanded. 
 Again, $ (ay z + by*)* = J (ay 3 + hy*)~*. (2 ay + 4by 3 ) $y. 
 
 If §x be the variation of x; then since the 
 differential of xy, or d(xy) \s=xdy -\-ydix, 
 the variation of xy, or$(xy) is = x$y + ySx. 
 
 If we take the example, p. 73, 
 
 since d\y.y/{\ +jp«)] = dy </(!+?*)+ v ?f^r ) 
 
 dp 
 
 ,. tptr i^*i- arm 'Jx, but ^=p. 
 
 r ax dx ax ax 
 
 Sdu dSy 
 
 Hence, Sp=~ or = -£. 
 
 If V - ax z s/{\ -f y x ) hyp. log. p x eT" 
 
 [e — number whose hyp. log. = 1,] 
 
 * Princ. Anal. Culc. p. 7i. 
 
Examples to preceding Rules. S3 
 
 then since dF=3ax'\/(l^t z )hA.p.e m ^.dx + ^^^-d?/ 
 
 v J ' _ v (1 +y) J 
 
 + a# 3 N /(l +#*) ~e mq Jp-^ax"x/(l+y)h.\.p.e m ^?n.dq 
 
 therefore putting M, N, P, Q, for the coefficients of the 
 terms involving dx, dt/ y dp, dq, we have 
 $ F= M. tx + N.ty+P.tp + Q. Sq, or expressing M, N, 
 P, Q, as partial differential coefficients *, 
 
 .- dV % dV^ dp\ dV . 
 
 Hence, generally, whatever be the function V> 
 
 if dV=Mdx+Ndy + Pdp + Qdq + hc. 
 
 then $V=MSx + 2Vfy + Pty + Q<ty + &c. 
 
 Since the processes for finding the differential and 
 variation differ only in the symbols dy, $y, which are 
 arbitrary ; it is plain, if both operations are to be per- 
 formed on an analytical expression, that it is matter of 
 
 » Princ. Anal. Calc. p. 79. 
 
 \ This rule is not only in the first solutions of Isoperimetrical pro- 
 blems, virtually acted upon, but expressed. " II faut bien remarquer 
 que la difference des fonctions de deux lignes comme RO, RT 
 (j/> y~\-fy) °t u i s e surpassent d'une quantite TO infinitement petite 
 du second genre, se trouve en differential simplement la fonction de 
 RO, et en multipliant par TO (Sy) ce qui en vient, ayant omis les 
 difrerentielles. Par exemple, si RL (F) fonction de RO (y) etoit 
 seulement la puissance n de la meme.RO (y), en quoi consiste le cas 
 de mon frere, e'est a dire, que si la courbe BH etoit une parabole du 
 degre n, alors LM($Y) ou RO n — RT n fXy + ty) — j/ n ] seroit=«. 
 RO"- 1 x TO [ny n - l Jy): y John Bernoulli. Acad, des Sciences, 
 1706, p. 236'; also Opera, torn. I. p. 424; and Euler, Methodus 
 Inveniendi, p. 33, &c. gives'the very rule and method for finding the 
 variation, which Lagrange invested with appropriate symbols, 
 [see p. 73.] 
 
84 Order of the Symbols d, $, changeable at Pleasure : 
 
 indifference, which operation is performed first : or, if the 
 symbols d, 8, meet together denoting operations, we may, 
 at our pleasure, change their order: for instance dSy 
 and Sdy are alike significant ; for dty means the first 
 term of two successive values of $y, or = <? {y+dy)~ty 
 — Sy + <? dy — ty = idy : again, if V for instance be a 
 function of y ; then 
 
 dV d'V 
 
 dV=-^dy, and tdF=-^ 2 .Sy.dy 
 
 dV d 2 F 
 
 ir^iy, znd d*F=^.dy.ty; 
 
 :.$dr= dSV, 
 or, in a particular instance, when V=.y", 
 d (if) = 1 st term of [{y+dy) n -y n ~\ = ny n ~\dy 
 $d{y n ) = 1 st termof ndyx [(y + fy)"" 1 -^"" 1 ] = 
 
 ?i()i-l)y"-\dy.3y, 
 $(y») = 1 st term of [(y -f Sy) n - ?/"] = ny*~\9y 
 di{f) = 1 st term of >%x [_(y+dy) n - x -y n - l ~] = 
 n(n — l)y n ~ 2 -ty-di/ ; 
 
 /. Sd {if) = dS Q/ n ). 
 
 And, by similar processes, d z $V= $d z F= SddV = dSdV 
 
 $$V= td?V = dU % V= drt&V. 
 
 This rule is, in Lagrange's method, of the greatest 
 importance ; it is an essential part of it. Amongst other 
 uses, it enables us when an integral is concerned, to 
 introduce the symbol $ within the symbol (/') of the 
 integral: thus, since the symbols d andy indicate re- 
 verse operations. 
 
also of the Symbols $, f. 
 
 8,5 
 
 V=dfV\ :. *V= SdfF= [by Rule, p. 84.] difF. 
 Hence, taking the integrals on each side 
 
 This result may be easily extended to double and 
 treble integrals : for if V=flV, then SV—ifW=f$W 
 by [a] ; 
 /. JW=ffnV-. hxAf*V=lfV=iffW 9 consequently 
 
 SffW=ffSPT. 
 
 Before we proceed to explain the last improvement 
 made by Lagrange, we will give, under the symbols of 
 his new Algorithm, another solution of the brachysto- 
 chrotie, [see pp. 4, 9.] 
 
 Let AC=y, BL=y', CL = ds, LD = ds\ OA=x, 
 OB — x', g = 32% feet; then the velocity = \/ 2gx, 
 
 
 
 A 
 B 
 K 
 
 
 
 C 
 
 VSJS- I 
 
 E 
 
 
 F 
 
 D 
 
 hence, 
 
 ds 
 
 ds' 
 
 \/(2gx) x/(2gaf) 
 
 ds ds' \ dts 
 
 = a minimum ; 
 di* 
 
 or -7- + -7—, 
 
 \/ x \/ x 
 
 but 
 
 . / as , as \ 
 ds=y/(dx> + df); :. dts= dy -f 3y and did = 
 
86 Brachystochrone solved in the Symbols of the Calc. of Variat. 
 
 " /U - 5 hence, ~—f- + , ' ', = O [<*]. But 
 
 .*. since A'Z) and ^C are not changed in the trans- 
 lation of L to G, dy + dy' is constant ; /. dhj = — dhf, 
 
 and substituting in [a~\, 1. 1 . - , ?/ . - = y^ , - , = a constant 
 L J ds\/ x asy/x 
 
 quantity, the property of the cycloid. 
 
 This solution is on the same principle, as the first 
 and second, but is less peculiar and geometrical than 
 either ; instead of the similar triangles in the second, 
 one of the proeesses of the new method above described, 
 has been used. 
 
 The rules in pp. S4, 85, relate to the second improve- 
 ment made by Lagrange in thecalculus of variations which 
 we shall now describe. If we refer to the method which 
 
 dP 
 Euler used for deducing the formula, iV— -3 — |-&.c., we 
 
 shall perceive that its length and complication arises 
 from the integral fVdx being broken down into parcels 
 and elements Vdx, V'dx> &c. and from the calculation of 
 the variations in the differential expressions that represent 
 dF, dV\ dV", &c. [pp. 68. 69.] 
 
 Lagrange precluded the necessity of this resolution 
 of j'Vdx into its elements, by combining with the 
 preceding variation processes, an integral process. This 
 will be understood from the solution of the following- 
 problem, in which it is required to find an expression for 
 SfVdxy V being a function of x, y, p, (j, r 3 &c. the law of 
 
Formula for the Variation of/Vdx. 87 
 
 ii- r o i • dy dp dq 
 
 the formation or p, q, r, &c. being p=z-j-, q = — , r =~ t 
 
 &c. 
 
 lfV4x=f$(Vdx) [p. 85. 1. 3.] =f$F.dx+fFdtix. 
 
 Now on the principle of this formula, J'xdy — xy —jydx 
 
 fVdtix = F.to -fdVJx. 
 
 Hence, J/Ttf a; = F. to +f{*V. dx - dr. to) 
 Now since £F= 31 tix + A T <ty + Ptip + Q2q+ kc. 
 
 if tfF= Mdx + A% + Pd/? + Qdq + &c. 
 ^r. da - rfr. to = iV(fy • <*# - ^ .to) + P(ty . rfa; - dp . to) 
 + Q (tiq dx - d q.ti x) + &c. 
 
 = Ndx{tiy - p. tix) + Pdx(tip-qtix) 
 
 + Qd*(ty-..r#*) + &c. (since ;? = ^|, 9 = ^, &c). 
 
 tv- '^/ . of to dridtix 1 ,,„ 7fc N 
 
 .-. tip—qtix = r-^(dtiy—pdtix — qtix .dx) — 
 
 <^{d$y- pd$x-dp.tix)=j^.d(Sy-ptix) 
 
 Now since the quantities jo, ^, r, &c. are formed after the 
 same law, the same relation that subsists between tip — qtix 
 and tiy — p tix, must subsist between $ q — rtix and tip — qtix, 
 and so on; but the relation between the two former 
 quantities is thus expressed : 
 
 *p-q2x=~d(tiy-ptix) 
 -'-ty-rtx = ±d(tip-qtix)=;*±;.d>(tiy-ptix) 
 
 * dx bein£ constant, otherwise =— d— .d(o>/—p$x). 
 * dx ax ^ r 
 
88 Formula for the Variation of fV d x. 
 
 and fr-slx^apq-rte) ='^.#(ty-j>#*). 
 
 Hence, for the purpose of abridgment, putting 
 &y-p$x = Su, we have, 
 
 ifvdx=r.t x +jdx & j. + 12a. + «£* +&c .). 
 
 ax ax 
 
 But by Formula, p. 87. 1.3. 
 fPdfc=PJm--fdP.im s 
 fQ&fa = QdJ w -fdQd$<o = Qd$o-dQJo+fd 2 QJ» 
 
 flld j u = Rd fu, -fd Rd 2u 
 
 = R#fo-dRd*»+fd*RM*, 
 
 = R.&tu - dR.dfa + d*Rt» -fd?R . Su. 
 
 Hence, collecting quantities involving like symbols, 
 
 tfVdx = VSx + fdx.t*(N- ~ + ~- ^-?+ &c) * 
 J ,J \ ax dx" ax' 
 
 L J x dx dx- / 
 
 +&C. +A'. 
 
 The quantity A' represents the sum of the corrections 
 introduced by the integrations. 
 
 * -3—, -3-3, &c. are not partial differential coefficients, but the first, 
 
 second, &c. entire differentials of P, 2, &c. divided by dx, dx 9 , fyc. 
 
 respectively. They ought in strictness, to be written -r-.dP* — .cf£, &c. 
 
 dx dx 
 
Formula when the Variation is taken betiveen tivo Limits. gO, 
 
 If x be supposed to have no variation, or if $x=0, 
 then Su = $y — p$x = $y ; and consequently we shall have 
 the variation of SfVdx by omitting in the preceding form 
 Vix, and writing $y, d$y, &c. instead of <5w, dSu, &c. 
 
 The preceding value of the variation may be sup- 
 posed to be taken between two limits corresponding to 
 values a, and b, of x. Let y , P Q , Q Q , &c. be the 
 values of?/, P, Q, &c. at the first limit when x = a ; and 
 and y t , P l9 Q l , &c. when x = b; 
 
 then, $fFdx= F l Jx 1 - ^ Q .(te Q 
 
 + &c. 
 
 If at the two limits # = «, x = b, the values of y Q ,y t 
 are given, then <^w x , <?w are both equal to nothing, 
 (since $<a = $ij- p$x) ; and in this case the variation of 
 fVdx is reduced to the quantity under the integral 
 sign (/'), which, in the case of fp r dx = s. maximum, 
 since $fVdx = 0, must equal nothing; or 
 
 y _dP d'Q iPR &c _ Q 
 dx dx 9 dx 6 
 
 This last formula, is that which Etxier, by the method 
 
 N 
 
90 Definite and Indefinite Parts of the Formula. 
 
 described in the last Chapter, arrived at # ; and it is here 
 deduced on his hypothesis of the evanescence of £y Q . 
 
 dP d 1 Q 
 The formula A T — 7— + —.—-— &c. is not, however- 
 ax dx % 
 
 equal 0, solely in the case when $jc oi $x t , Sy Qi jy t , are, 
 equal 0, but also, when these are, from certain equations, 
 assignable quantities. For, the formula [Ad] p. 89, is 
 composed of two parts : one, affected by the integral sign, 
 expresses the sum of all the separate variations throughout 
 the whole extent of the curve or integrated quantity; 
 the other part, independant of the integral sign, is 
 affected only by the variations at the extreme points, and 
 therefore cannot by any combination with the other, 
 (which by changing $x and &y may be varied at will) 
 forma sum equal to nothing. Hence, since tij'Vdx 
 must = ; each part separately, the one under the inte- 
 gral sign^j the other not affected by it, must = 0. 
 
 M. Lagrange is the inventor of that part of the 
 
 * Methodus Inveniendi, &c. Prob. V. Lagrange, Miscell. Taurin. 
 torn. II., says, " Mais les formules de cet auteur (Euler) sont moins ge- 
 nerates que les notres 1° &c. — 2°, parcequ'il suppose que le premier et Je 
 dernier point de la courbe sont fixes, Sec.''' Eider also, in his memoir 
 subsequent to Lagrange's [Novi. Comm. torn X. p. 110.] acknowledges 
 that his first formula did not contain the absolute or definite parts in- 
 volving §u!, d$u, &c. " Neque tamen ha? partes absolutas frustra sunt 
 invent*, sed singularem pnsebent usum, ad quern methodus mca prior,, 
 
 ., dP 
 quae tantum aequationem Z\ — - — |-5cc. = .suppeditavit, minus est aa- 
 
 commodita; quajn ob cauaam haec methodus illi longe est anteferenda/' 
 
Lagrange's second Problem. t)l 
 
 general formula which is not affected by the integral 
 sign. Its use, as it will be shewn hereafter, is great and 
 extensive: without it, the solution of problems would be 
 incomplete ; for, Euler s formula, that under the integral 
 sign, merely determines in general terms the relation of 
 x and y ; undoubtedly the chief, but not the sole object 
 of search. 
 
 We will now proceed to that, which, in fact, is 
 Lagrange's second problem*. 
 
 It is required to find an expression for fJ'dx. in 
 
 which dV— Tdt, dt not being similar to dy, dp, dq, &c... 
 
 but determined by this equation t—jZdx. 
 
 By p. 87, tfVdx = V9x +f(9V.dx - dF.ix) [l] 
 
 hwt 9F=T.9t\ .'. 9Fdx—dVJx=Tdx.9t- T9x,dt. 
 
 again, by [l] 9t=9f Zdx = Z .9x+f '(tZ.dx-dZ.9x) ; 
 
 .-. Tdx.9t-T9x.dt = TZ.dx.9x-TZ.dx.9x + 
 
 Tdxf(9Zdx-dZ9x), 
 
 = Tdxj\9Zdx - dZJx) 
 
 and .-. 9jFdx=V9x+fTdxj\9Zdx-dZ9x). 
 
 Now by virtue of this formula. f(vdu) = vu-J (udv), 
 
 if h be put=/Td#, the latter part of the value of 9fVdx 
 
 is, hf(9Zdx - dZ9x) - fh (9Zdx - dZ9x) 
 
 Also by Form [A], p. 88. ifdZ = Mdx+Ndy + Pdp+ke. 
 
 /»r dP dQ 9 x 
 hf(?Zdx-dZix)=hfdx.9<* (A - { 7J + ^r- &c - y )- f 
 
 h.L ... . [2] 
 
 * Miscellanea Taurimnsia, lorn. II, p. 183, 
 
93 Lagrange's second Problem. 
 
 and fh {SZdx — dZtx) = 
 
 (L, U being put for the definite parts,) 
 
 Subtracting this from [2] and adding VSx, we shall 
 have the value of Sj'Vdx. 
 
 In general, it is required to find the whole variation of 
 fVdx from x = 0, for instance, to x = a. In such case, let 
 the whole integral of Tdx = H; then the part [2], of the 
 
 ™riation= Hfdx§^N~~^~~ kc.^ + HL 
 
 =fdx^(HN-^P + £Jgfi - &c.) + tf£ 
 
 For since //is constant, HJN=fHX, and ZT.dP = rf(i/P) ; 
 subtracting therefore, as before, [3] from [2], we have 
 
 ifFdx=zV$x 
 +/*.* ((JET- k)N^ H ^ P ^ H 2J )a - &c.) 
 
 [*] +((ir.*)P.fi^fi? + te.)*i 
 
 + &C. 
 
 If we put//—//, that is, H—J' r J'dx = k, and suppose 
 £r, £w, the variations at the limits, to equal nothing, we 
 shall have 
 
application of his Formula to particular Cases. 93 
 
 which is Euler's formula, Method. Inveniendi, &c. p. Ql, 
 and the subject of his third Chapter. 
 
 If dy^n.dx + v.dy+Tr.dp + kc. + Tdt, 
 which is Lagrange's case. Misc. Taur. torn. II. p. 183. 
 
 *fFdx=jnx+fdxJ»Q - ^-+&c.) . . . [C] 
 +fdx3o> (/nV-^^ + &c.) 
 
 S being put for the sum of the definite terms. 
 
 If dt = ds~\/(dx'-\-dif) and dF=pdx+vdy+Tds 
 
 d(lH-fTdx] /( * \ 
 *fFdx = fdxl»\*- - rf , ]•»[£*] 
 
 not taking account of the definite parts; 
 
 For,dt = ds=zdxy/(l+p 1 )=Zdx; .-. Z = x /(l+^ 2 ) and 
 
 dZ = — 777-- — w .«» ; which, compared with the value of 
 dZ [p. 91 J gives ,V=0, P = _£_,Q =0 , &c. 
 
 Hence, in the case of maximum or minimum, since 
 ffVdx — Q, there results this equation of solution, 
 
 v.&-rf([/f-yT&]^— ) )=o ; 
 
 whereas Euler, Novi Comm, torn. VI. p. 141, gives this 
 
 , + T ,, V ., =0, 
 
 v/(l+/r) 
 
94 Lagrange's third Problem. 
 
 v answering to M, T to L, and - being = q 
 
 ~ ds ' 
 
 This is the case before alluded to [p. 6l.] ; when it 
 was remarked that Euler's formula? contained in the 
 sixth and eighth volume of the Petersburg Commentaries, 
 were erroneous, when V contained an integral ; which 
 integral in the above instance is s. 
 
 In the third problem of Lagrange [p. 185.], it is re- 
 quired to find the variation of fV when V is given 
 simply by a differential equation involving no differentials 
 of ^higher than the first. 
 
 Let dV — X. dx + Udx = be the differential 
 equation, X being a function of x, y, p, q, &c. and U 
 a function of x, y, p, q, &c. and of V\ then taking the 
 differential, and supposing dx to be constant, 
 
 d%v ~hfc dx +ihj d y + ~dt dp + kc ') dx 
 
 (dU , dU 7 dU j \ j 
 
 + -rp->dF.dx = 0, 
 
 or d % V - dfp.dx -f -jp.dV.dx = o, . . . . [a] 
 
 Substituting dtp for the collection of terms involving dx, 
 dy, dp, &c. Hence, if instead of deducing the diffe- 
 rential in the last operation, we suppose the variation to 
 be deduced, 
 
 dtr - ty.dx + TM'.dr = o [ r =^] 
 
Lagrange's third Problem, 95 
 
 multiply by A, and A d$F— {\$p) dx + (a TJF) . dx - 0, 
 or, d(\SF)-d\.SF-(\$p)dx + (\TJV).dx = o. .. [b] 
 
 Assume the sum of the second and fourth terms to =0 - y 
 
 .'. dx - xT.dx = ; and — = Tdx, and integrating A = € rrdx 
 
 \_e = number whose hyp. log. = 1 .] 
 Hence substituting this value of A in the sum of the 
 first and third terms, 
 
 e fTdx JF = fe fT ^J<p.dx; 
 .'. f$V or tfV = fe- fTdx .fe' Tdx J(?.dx. 
 
 Euler, by a different process, solves this problem in 
 the third Chapter of his treatise on Isoperimetricals. 
 
 Euler, however, stopped at this problem ; and did 
 not, as Lagrange has done, proceed on to those which 
 besides c?/ 7 , involve d*V, d i V i &c. Suppose ?7to be a 
 
 IT/' 
 
 function of x, y, p, &c, V and of j— ; then in taking the 
 
 differential [a] p. 94. of the equation dV— Xdx+Udx = 0, 
 
 dU v dl\ 
 
 an additional term -7- . d 1 1 ( v = — - ) will be introduced, 
 
 dv V a x' 
 
 ITT 
 
 and consequently in the variation, this term -j- ,d<iV ; 
 
 therefore when the equation is multiplied by A (for the 
 same process must be used,) there will be an additional 
 
 term xT'MV (T = ~) =d(xT'JF) - d{xT')x^ 
 
 ,-. equation [b] 1. 2, would be 
 
96 Third Problem made more general. 
 
 d (k$F) - r/x . SV- \$<p.dx+ (* TJF) dx 4- d (\T.3F) - 
 d{\T)!V. 
 
 Hence, making the sum of the 2 d , 4 th , and 6 th terms = 0, 
 - d\ + x. Tdx - d(xT) = . . . [c] 
 
 and d[(\ + \T)3V~\ - x$<pdx = [d] 
 
 in which two equations, A must be deduced from the first 
 and substituted in the second. 
 
 The process is the same if U besides a function of 
 the former quantities, is also a function of d?V \ for then in 
 the equation of variation there would be introduced a 
 
 term T".d^V (T" = ~, «=jr) 5 and consequently 
 
 in the equation multiplied by x, a term = \T".d x iV— 
 d {\T"d$V) - d{\T") J§V=d{\T".dSV)-d\\T"JV) 
 + d 2 xT'xtr. 
 
 Hence the equation [c] 1. 4. will become 
 - dx + x.Tdx - d{xT) + &.(\T U ) = 0, 
 and the equation [d~\ 1. 5. 
 
 d [(x + xr- d{xT")) W]-.d(\r'.dSF)-\tydx=0: 
 and similarly, if differentials of a still higher order are 
 introduced. 
 
 In the preceding cases, the maxima and minima 
 depend on a function of one variable quantity alone : but 
 a variable quantity z may be introduced, a function of 
 two others x and y. For instance, suppose it were requir- 
 ed to find amongst equal solids, that which is bounded 
 by the least surface, (which is the problem given by 
 
Problem of the Solid of least Surface amongst equal Solids, QJ 
 
 Lagrange in his first Appendix, p. 188.): then, if the 
 points of the surface be referred to three rectangular 
 co-ordinates x 9 y, %, we have 
 
 ( -r- V (-7-) 3 being the partial differential coefficients *, 
 vve have also the solidity =ffz.dx.dy, and the surfaces 
 
 the integrals being taken, first relativelytojr, then relatively 
 to y. The problem therefore analytically expressed is 
 
 \_a] ff$(z.dx.dy) 
 
 sinnlarly /Q>(g) = Q ± -f 
 
 dy 
 
 Princ. Anal. Calc. p. 79, 
 Q 
 
98 Problem of the Solid of least Surface, fyc. 
 
 Hence the equation [a] p. 97. 1. 10. becomes 
 j'fdx . dy . $z -+- a/P dy Jz + af Q dx . iz 
 
 - affdx.dy -j — H — affdx.dy -7- ^ =0. 
 
 Hence, making the sum of the second and third 
 
 terms, which are affected with one sign of integration, 
 
 and belong to the extreme points of the curve surface, = 
 
 Pdy-\-Qdx~0, and consequently the remaining 
 
 dP dQ r -, 
 
 terms = 0, or, 1 — a.-, a.-r- — O, \p\ 
 
 dx dy ' Ujrj 
 
 now the condition under which 
 
 Mdx + Ndy 
 
 is a complete differential, is this * 
 
 dM _dX 
 
 dy dx 
 
 Hence, putting N=x — aP, 31= aQ, we have 
 
 dQ dP dP dQ 
 
 a— 7-— 1 - a. -r— . or, 1 — u-, 0-7— =0, 
 
 dy dx dx dy 
 
 which is the equation [/?] 1.8.; hence, (x~aP)dy-\- 
 
 aQdx is a complete ditferen'. : but that, dz—-r-dx + -r—du 
 r dx dy * 
 
 may be real, ~dx -\-~ dy must be also a complete 
 iix (i 11 
 
 d x z d z z 
 
 differential; that is, -, — j- must= -j — ~ . These two 
 dx.dy ay. ax 
 
 * Woodhouse, Princ. Anal. Calc, p ;r, Calc. Int. p. 315 
 
Portion of a Sphere satisfies the Conditions of the Problem. 90, 
 
 conditions therefore must be fulfilled, in order that the 
 problem may be solved. 
 
 If r be the radius of a sphere, and oc, |3, y, the three 
 co-ordinates of its center, 
 
 r= s/ [(* - «)' + (y - j3)* + (* - ?)'] ; 
 
 .. -7-= ; r (#-<*), and -t~= ; rCv — /3) ; 
 
 ax (%-y) dy (z-y) ^ J 
 
 d z z d" v 
 
 .-. (putting for % — y its value) , j '~ j j ~ one con- 
 dition therefore is satisfied : 
 again P=~ , Q=^; .\ (a - aP) <ty ^aO***- 
 
 ( 1 - - ) xdy + -ydx + -^dy-—.dx: which is a com- 
 
 plete differential, making 1 = - ; .-. the portion of 
 
 the sphere corresponding to x, y, z, satisfies the problem. 
 
 The formulas of solution, on which the nature of the 
 curve, or the relation of x and y, depends, were all, ex- 
 cepting the two last, invented by Euler. To Lagrange 
 belongs the merit of having deduced them by neater 
 processes. The latter author, however, is the sole, 
 inventor of those definite and absolute formulae [see p. 88. 
 1. 13, 14.] which are requisite for the complete solution 
 of I soperi metrical problems. 
 
 In his first memoir, Lagrange seems not exactly to 
 have comprehended the nature of these definite formulae. 
 
ioo 
 
 He drew some conclusions, which the Chevalier Borda * 
 ]) roved to be not general. These defects, however, 
 M. Lagrange remedied in a subsequent memoir in the 
 fourth volume of the Miscellanea Taurinensia : and 
 without acknowledging the detection of the defects, or 
 rather, with a faint endeavour of denying it, extended 
 his formulae, more accurately applied them, but con- 
 firmed the truth of M. Borda's results. 
 
 In the same memoir Lagrange considers the subject 
 under a new point of view, and gives a method of solu- 
 tion including all his former ones. 
 
 The substance of Lagrange's first researches on this 
 subject have been given, with some deviation, not essen- 
 tial, from their mode. Euler's last manner t of treating 
 the subject has been followed. That is commended and 
 adopted by Lagrange in the latest of his publications J. 
 In the same Tract he has resumed the consideration ot 
 the general problem, and deduced formulae applicable 
 to all cases. 
 
 This last method of Lagrange's is distinguished 
 rather by its mode of treating the question^ than by any 
 thing novel in its principles; and therefore, it will be 
 separately considered in the following Chapter, which 
 the Student who hastens towards the end of the inquiry, 
 may, without inconvenience, pass over. 
 
 * Acad, des Sciences, 1707. -f- Novi Comm. Pdrop. torn. X, p. 51. 
 t Lemons sur le Culcul des Fohctions. 
 
CHAP. V|l. 
 
 Lagrange's general Method of treating Isoperimetrical Problems- 
 Equation of Liniits- i -Cases of relative Maxima and Minima re- 
 duced to those of Absolute. 
 
 Let V be a function of x, y, p, q, z, p', q' y &,c 
 
 dif dp , _ dz , dp' 
 
 1)= "i~v> qz= Jbc ,P = di> q = dx' 
 
 and let dV = Mdx + Ndy + Pdp + Qdq + &c. 
 
 + v.dz + it. dp + <r.d(/ -j- &c. 
 
 then, if J = JV - — + -r— - &c, 
 ax dx 
 
 dx 
 I'= Q -&c. 
 
 &.£. = &c. 
 
 £ = , _ ± +*1 _ &c. 
 
 r/.c dx z 
 
 d<r 
 
 Z' — 7T — + &e. 
 
 dx 
 
 Z"= <r - fee. 
 
 &c. == &c. 
 
 we shall have, by processes already described, [pp. 8/, 88.] 
 
 S/Fdx = Fix +fdxJo.r + /'dxJu,'Z 
 
 -\- T ' .£w + Z '. <JV 
 
 + Y".d$<*+ Z".dte 
 + &c 
 
102 Lagrange's general Method. 
 
 Hence, in the case of a maximum, when $fVdx = 0, 
 
 YJa + ZJa = 0, 
 or Yty ■+ Ziz - 0, if $x = O, since [p. 88.] 
 Sv> = §y — p$x, and 8u = Sz — p'Jx 
 
 If y, z, are quantities independent of each other, 
 then, Y = O, and Z = O. 
 
 But, if the quantities y and z are connected together 
 by an equation, such as f (x, ?/, z) = 0, or W — 0. 
 then, supposing x invariable, we have 
 
 div . div 
 
 which combined with the former equation [1.3.] gives 
 
 dW dW 
 
 dz dy 
 
 The same result will be obtained, if x be supposed 
 
 to vary; for then 
 
 dW dW ' dW\ 
 
 d# ^ dy * dz 
 
 which combined with the common differential equation 
 
 dW , dW dW , .. 
 
 r/x dy J dz 
 
 gives, ^. (ty - ^ >*) + -ft (>* " E '*) = o , 
 
 or, -7 — .ou -\ — -j— .dw=0, 
 dy dz 
 
 which combined with J r <Tw -f- Z$u>', gives 
 
Lagrange's general Method. \03 
 
 v dW ^dW 
 1 .—j— — Z.—j— = O, as before. 
 dz dy 
 
 Hence, this conclusion follows ; that the variation of 
 x in no wise affects the general equation of maximum or 
 minimum, but solely the equation at the limits. 
 
 The equation at the limits depends on the parts of 
 the formula for SfVdx, which are definite and freed 
 from the integral sign, and is 
 
 + Y\J^ + Z\M 1 + &c. 
 
 - V' .im - Z' .t»' +&c. 
 V li V oi &x l3 $x oi Sec. representing the values of V, $x, 
 &c. at the end and beginning of the integral. 
 
 Instead of eliminating (Jw, §u, by the combination of 
 the two equations [p. 102. I. 19, 20.] we may multiply 
 the latter by an indeterminate quantity A, add it to the 
 jirst equation, and then determine the resulting equa- 
 tion, by the elimination of A; thus 
 
 r , „ % , dW . dW % . 
 
 }Ju+ ZJa -f A-t-.^cj + A -7— $w = O. 
 ay dz 
 
 Eliminate A from the two equations 
 
 „, dW „ dW 
 
 1 + A— 7— = O, and Z + A -7— = 0. 
 dy dz 
 
 and there results, as before, L 
 
 „dW „ dW 
 cu ay 
 
 which combined with fV=zO, will give the values of y 
 and z in terms of x. 
 
104 Lagrange l s general Method. 
 
 -i his last method extended, will afford a general for- 
 mula of 'solution ; for, suppose JVto be a function, not of 
 x ) y- 3 %, only, but also of the differentials of these quan- 
 tities ; that is, suppose 
 
 dJV^M'.dx +N'.dy + P'.dp +Q'.dq +&c. 
 + v.dz + Tr'.dp' + a.dq'+kc. 
 
 (J'') = xP'-^Q+kc. 
 
 (Z) ^ X j- *$£$+.&, 
 
 v ' dx 
 
 <z) = *„' _ i^il + &o. 
 
 v ' dx 
 
 &c. 
 
 then we shall have 
 J/F<fa -f $f(\W).dx = (F+ xJT).^ 
 
 4-/[J'-f(K).^J w ]+/[Z+(Z)^<J w / ] 
 
 + &c. 
 
 In this formula since W—O, xW.§x = 0, and since J«, 
 <$V are independent quantities, we have, on principles such 
 as have been already stated, 
 
 r+(F)=o, Z+(Z)=o, 
 whence A is to be eliminated ; and then by the aid of the 
 equation of condition IV— O, y and z are to be deter- 
 mined in terms of x. 
 
 This method comprehends all the former ones which 
 
Lagrange's general Method, 105 
 
 were separately instituted [pp. 88, 92.] to determine the 
 variation of f Vdx. 
 
 1st. When dV contains merely x, y, /;, q, &c. and 
 no relation is assigned between x, t/, z } then the equation 
 W ' — O is not introduced, and the formula becomes 
 Y=Q, Z = 0; 
 
 ,, dP d*Q 
 or N- _ + -— - &c. = Q, 
 dx ax 
 
 , drr d 2 <r Q 
 
 and v 5 1- 77 — ccc. = 0. 
 
 ax ax 
 
 2dly, If V contains an integral expression; that is, if 
 
 dV = Mdx + Ndy + Pdp + &c. + Tdt, 
 in which t —fSdx, then the equation of condition W=. 
 
 becomes dt- S.dx = 0, and consequently^/ f-j Sjdx 
 
 = a. Hence, substituting t for z, we have 
 
 r= a - -5- + -T-, - &c. 
 
 Z = T [since » = T, o- = 0, &c.j 
 
 [if dS=M'Jx + iV'.<fy + &c] 
 (Z) = -Jp [since v' = 0, ir'=l, <r' = s &c,] 
 
 hence we have from the equation Z + (Z) = O. 
 
 T - — = O, whence, A =JTdx=JTdx- H, 
 dx 
 
 UfTdx = H, when A = 0: substituting this value of A in 
 
 the equation Y + (Y) =0, there results 
 
106 Lagrange's general Method. 
 
 + &c. as before, p. 92. 
 
 3dl y, If dV— (p.dxi- U. dx = 0, and Uis a function of 
 
 dV 
 V) this answers to the equation fY=o, and IV— —, p 
 
 dV 
 + V } and dW— T. dP +d.-j—- d(p ; instead of z, consider 
 
 /'"to be the variable quantity, then in 
 
 dW=d(p 4- v'.dz + v. dp' + &c. v = T, v = 1, 
 the other coefficients being O ; 
 
 Hence, Y = 0, (Y) = x\' - *&^ + &c. 
 
 x ax 
 
 .-. Z+ (Z) = 0, or a T- C ~j- = 0, whence a = e rrdx as before, 
 v ' dx 
 
 p. 93. 
 
 The same method applies to equations in which 
 higher differentials than the first are involved ; for in- 
 stance, to equations such as 
 
 y, #V , „ d*V L dV A 
 
 * dxT +t -dJ* +t 'dx- + Q = °' 
 
 Jn the variation of fVdx, the parts under the in- 
 tegral sign, that is, Y+{Y), and Z+(Z) equal nothing ; 
 there remains therefore, supposing $x ~ 0, 
 
 SfFdx = \T + {Y>)-\$y + \Z' + (Z)]ta 
 4- [Y»+ (Y")-]d*y + [Z" + {Z")]d$z 
 + &c. 
 which expression must be substituted in the equation of 
 limits, that is, in 
 
Lagrange's general Method. 107 
 
 iU x - iU =* o, 
 putting U—JVdx. 
 
 At the limits, there may be particular relations 
 
 between x and y ; for instance, it may be required to 
 
 draw the brachystochrone between certain curves. Let 
 
 L = Obe the equation to the curve at the first point, and 
 
 il/=0 at the last point, that is, let L be a function of .r , 
 
 y Q , and M a function of x t , y , ; then 
 
 dL % dL m , dM % d5'L 
 
 ^*o+^o = 0, and s ^+ ^, = 0; 
 
 and since, by taking the differential equations, we have 
 
 dL , dL , , f/M , dM , 
 
 _^ o+ _.^ o = 0; and — dx l+ -^dy l = O, 
 
 by elimination, there results 
 
 ^•^o-tyo = °, and Jk.fay-fy, = 0, 
 
 from these equations, and from the equation of limits 
 $U l — 8U Q =iO i which must contain <J> r , £r , Jy i9 fy oJ 
 the values of $x Q , ty Q must be eliminated, and the result- 
 ing equation will determine the conditions of the problem 
 that must be fulfilled at the limits. 
 
 The same conclusion will result, if, instead of an 
 elimination, we multiply the equations SL — O, $M=o 
 by two indeterminate quantities A, /a, and then add them 
 to the equation of limits, which will become 
 
 3U, - 3U + \ZL + p$M= 0. 
 
 This equation will contain terms affected with $x , 
 ix l9 Sy , $y x , the coefficients of which are separately to 
 be made equal nothing. 
 
108 Lagrange s general Method. 
 
 The origin of the abscissas and ordinates (when the 
 discussion is concerning curves) is supposed to remain 
 constant; but, it may be supposed to vary, which will 
 be equivalent to the hypothesis of making quantities, 
 such as a, b, contained in V, to vary. In that case, if 
 IV = Mix + Nfrj + P3p + hc.+Ahi + Btb + &c. 
 
 1 1XJ dV I 717 dJ/ \ A dV J 
 
 we have, as M= -r- , and i\ = —7- &c. A =~r~ , and 
 dx dy da 
 
 dV 
 B = -jT-; and consequently the variation HfV dx will=s 
 
 rJ i / -y~ dP dQ „ \ 
 
 +f $a dx .A + J'Sbdx.B 
 
 + *(*-£+ **) 
 
 + &c. 
 now the equation to the curve depends on the equation 
 
 N — -r- + -j-. - - &c. and the additional terms hifA dx, 
 
 SbfBdx, 0Y$aJ'-j- dx, ^bf-jj dx, will affect the equa- 
 tion of limits* 
 
 If x — x' — a, then Sx = <JVr' — hi ; therefore, if x' be the 
 same, <J\r= — Sa; consequently the term A hi, introduced 
 by reason of the variation of the origin of the co-ordinates, 
 will be affected with a negative sign. 
 
 The methods described in this Chapter comprehend 
 all cases belonging to absolute maxima and minima : 
 they extend also, if we employ Euler's reasoning *, to 
 relative maxima and minima; or independently of the 
 
 * See p, 78. 
 
Lagrange's general Method. 109 
 
 last author's method, which may he thought not perfectly 
 satisfactory, they may be extended by the following 
 process, which is Lagrange's. 
 
 Let u be the function, the integral of which is, to 
 have, within the assigned limits, a determinate value. 
 Let s be its integral, then ds- udx — ; consider this to 
 be an equation of condition, such as L = 0; then, since 
 X$L was added [p. 107.] to the variation of fVdx, we 
 must now add 
 
 \$(ds - udec) .... [a] 
 the first of this, \2ds=zxd2s = d(\$s) - dxjs ; a term 
 therefore — d\ . $s will be introduced under the integral 
 sign, which, since fs is an arbitrary quantity, must = 0; 
 .*. dh. = 0, or A = a, a constant quantity. The term d(\fo) 
 will solely affect the equation of limits, and since the 
 whole variation must be taken between s t and s Q , a$s I 
 — ats must be added to the equation of limits, which, by 
 the hypothesis of s having a determinate value, must = 0. 
 
 Hence there remains of [c/] I. 10, only — a$ (udx)\ 
 consequently, the whole variation is reduced to this 
 Sj \V - an) d,i : whence the rule before given, p. /8, 
 which reduces relative maxima and minima to absolute, is 
 derived ; for the formula directs ns to find the conditions 
 of the absolute maximum or minimum of j\V -an) d,r, 
 a being a constant quantity. 
 
 We now proceed to the last Chapter of this Work, 
 in which, the formulae previously established, will, with 
 some slight alteration, be applied to the solution of 
 problems. 
 
CHAP. VIII. 
 
 Particular Formulae deduced from the General one, for the Purpose 
 of facilitating the Solution of Problems — Problems solved. 
 
 Euler in his treatise has deduced from his general 
 formula, several subordinate ones, limited indeed, but 
 materially expediting the solution of problems. These 
 will be first described. 
 
 In the general formula, 
 dV=Mdx+Xdy + Pdp+Qdq+Rdr + kc. . .[J] 
 suppose .17, Q, and all coefficients excepting N, P, to 
 equal O ; then, dV=Ndy+Pdp\ but generally [p. 89.] 
 
 A 7 — -j-+ -} &c. — [a] ; in this case, therefore, 
 
 ax ax* 
 
 N- -j— = ; consequently, Ndy — dP .-— = ; or, since 
 
 p = -p, Ndy — p.dP = O; and substituting, in 1. 10, 
 
 dJ'=p.dP+Pdp = d{Pp) ; whence, by integration, 
 V = Pp + c [b~], c being the correction. 
 
 If 31 is not = 0, then F=f31dx + Pp+c . . . . [c]. 
 
 Let 31=0, A r =0, and all the coefficients after Q; 
 then dV—Pdp + Qdq, but the general formula [a] 1. 11. 
 in this case becomes 
 
 dP_ r?Q 
 
 dx + dx* ~ ° ; 
 
Formula adapted to the Solution of Problems. \\\ 
 
 whence, P z=J—— dx -f- c = ~ 7 \- c ; 
 
 ax* dx 
 
 d n 
 multiply this by dp, and since ~~ = q, we have 
 
 P.r/y? = q.dQ + c.efy? 
 hence, substituting, in the equation, [y/] p. 110. 
 
 dF = q.dQ + Q<ty -i- c.dp = d(Q^) + c.dp 
 and integrating, 
 
 F = Qq + cp + c ..... . [d] 
 
 c, c', being the corrections. 
 
 If 31 does not = 0, we must add, to the above fortifr, 
 the term fMdx. 
 
 If 31=0, but A T is not = 0, that is, if the form be 
 dF= Ndy + /V/> + Q<fy 
 
 then, since A — ; 1 -r— = 0, 
 
 dx dx 3 
 
 we have, multiplying by dy, which = pdx, 
 
 d z Q 
 Ndy -p.dP + p. j^-dx = 0. 
 
 « , d*Q 7 ( do, da 
 
 But, p ._^ = rf^_^^_ 
 
 Hence, substituting, in 1. 12. 
 dV=pdP + Pdp - d(p.^) -rqdQ + Qdq; 
 
 consequently, V =■ Pp + Qq - P-~f7 + c • • • W* 
 If dV =Pdp + /tar. 
 
112 Problem* . 
 
 then the formula \a\ p. 110. is reduced to 
 dP dm _ 
 dx dx 3 
 
 consequently, P = — -r- - + c, and Pdp — c . dp - dp. -7— 4 ; 
 
 hence, since dp = q. dx, we have, by substitution, 
 
 d~ R 
 dV — c . dp + Rdr — q. -j— % . dx 
 
 = c.dp+Rdr-d(q^)+d q .^ 
 
 [since dq = rdx] =c.dp+ Pid r + r d R — d(q .—- ) 
 Hence integrating, and adding the correction c', 
 V^cp + Rr-q.^ + d ' . ...if] 
 
 These forms are sufficient for the solution of the fol- 
 lowing problems. 
 
 Prob. 1. 
 Required the relation of x and y, such that 
 f{ax—y 2 ).ydx shall be a maximum or minimum. 
 
 Comparing this expression WiihfVdx, V—axy —y 3 ; 
 con sequently, dV— ay . dx -\-{ax — 3y ) dy ; 
 which, compared with the general formula [A] p. 110. 
 that is, with dV = Mdx + Ndy + Pdp + Qdq + &c. 
 gives M = ay, N=ax-3y\ P = 0,(1 = 0, 
 
 dP 
 
 consequently, since A T — -r- + &c. = O, in this case 
 
 , /ax 
 
 ix - 3y* = 0, and y = y — . 
 
becomes - ^( ^^ )=0; .'. p = «%/(! + /r) 
 
 Problems. 113 
 
 Prob. 2. 
 
 Required the shortest curve that can be drawn be- 
 tween two points, or between two curves. 
 
 Here fds or fdxs/{\ +p*) =a minimum; /. V— 
 V(l +P % ) and dV = P dp ■ ; ilf, A 7 , &c. =0: and 
 
 P = — ■— sv ; consequently, the formula [a] p. 110, 
 
 i 
 
 , dy a .... 
 
 and 7? or -~ = . -, whence, by integration, 
 
 &X \/ \L~~Cl) 
 
 y \/ ( 1 — a 2 ) = ax -j- c, an equation to a right line. 
 
 Prob. 3. 
 
 Required the curve of quickest descent between two 
 given points. 
 
 The time */"* =/^±M=/^il^.<fc 
 
 = /^ — -— — '—L.dx, which compared with fVdx gives 
 
 ,r = vOi±£l . whence <*F = - ^±£) ^ + ' 
 y/y 2y* 
 
 f- dp, which compared with Form \A\ p. 1 10, 
 
 gives M= 0, iV= - ^ ( * +f \ P = -_/-_, Q^o. 
 Now by the Form [ft] p. 1 10, V— Pp+c; /. in this case 
 
 Q, 
 
H4 Prbblans. 
 
 \/(l+/v 2 ) p 
 
 s/y vA/.v/ci+r) 4 " c ' 
 
 . 1 dx 
 whence — ; — . = c, or — ; ., . , t — : = c 
 
 and finally. fifo = —7- — ¥-—. dy. an equation to a cv- 
 \/(l-<ty) <-" ^ 
 
 cloid. See p. 5. 
 
 Prob. 4. 
 
 Required the relation of x and y when/^ar 1 + y z ) n .ds — 
 a maximum [see Euler's Methodus Inveniendi, &c. p. 52.] 
 
 Here r= (#' + yj l x %/ ( 1 + jb 1 ), since ds = dx </ ( 1 +f) 
 and, dV — 
 
 (2-nx dx + 2nydy){x* +y a )"" V(l +p') + ^ +^ <fr ; 
 
 .-. M=2nx (^ 2 +i/T~ 1 x /( 1 + / , *)> ^= 2w # (** + 3/ s ) n " 1 
 
 /a 4- ^ and P - &+tf) n 'P 
 J(l+p) 9 and P- y (1+ ^ } . 
 
 Here it is more convenient to employ the general 
 formula, 
 
 jV - ^ + &c. = O. 
 dx 
 
 which, since 
 
 dP = v ./ „ .(2npxdx + 2npi/dy+± — ,y ' l ) 
 
 by reduction becomes 
 
 2n{ydx — x dy) dp 
 
 x*+y z 1+p* 
 
 m> 
 
 .. . , # , .. ydx— xdy 
 
 if 9 = arc, whose tangent = - , then dv = "2 — —- ; 
 
 y x* + y l 
 
Problems. \ \,$ 
 
 consequently, integrating the equation [/?.], p. 114, 
 2nQ = A + c, A being an arc, whose tangent=jp, or -~- 
 
 x 1 
 
 .*. - =tan. 9 = tan. — (A 4- c). 
 y 3» V 
 
 Prob. 5. 
 
 Required the curve which by a revolution round its 
 axis generates the solid of least resistance, [see Newton, 
 p. 324, 3 d ed. Euler's Methodus Inveniendi, &c. p. 51, 
 Simpson's Fluxions, p. 487, Ed. 1750. Emerson's 
 Fluxions, p. 183, 3 d ed. Lacroix's, Calcul. Diff. torn. II. 
 p. 698.] 
 
 The resistance =«/ V V , ' j, * » which, compared with 
 fVdx, gives 
 
 V = ■ „ 1 since p = -r- 1 , whence 
 1 +/?* V ^ dx/ 
 
 1 +i» (1 -\-p ) 
 
 iir tkt P 3 n 3l/P* + VP 4 
 
 consequently, M= 0, N=-—; '*> F = (1 +/*)* * 
 
 Hence by the Form [fe] p. 1 10, using -c for the correct", 
 
 whence, by reduction, 
 
 c , c cp 
 
116 Problems. 
 
 but, pdx = dy = - ^±dp - — dp + Up; 
 
 3c 7 c T c dp 
 2j» s ' /> 3 ^ 2 p 
 
 , Of 7 V j u uu 
 
 and * = 8^" + i^ 1 + 5' hyp ' log '^ + c ' 
 
 which equation must be combined with c(l +p 2 Y = 2yp\ 
 
 Prob. 6. 
 Required the curve in which fyxds is a maximum, 
 [see Euler, Methodus Inveniendi, &c. p. 52.] 
 
 Since, yxds = yx*J(\ -f p 2 ).dx, V = yx^J (\ +/>*} 
 and dV =.y ^/(l + /»»)«&? + * </(l + F*) <fy + 
 
 
 In this case therefore, we must use the Form [c] p. 110, 
 and then yxj(l +p*) zz/y^/O +p*).dx+ /*f+ pa) + c 
 
 which by reduction leads to a differential equation of 
 the second order. 
 
 Prob. Jr. 
 Required the curve, or in other terms, the relation of 
 
 /d % v 
 ■j^=a maximum. (See Euler, Comm. 
 
 Acad. Petrop. torn. VIII. p. 1/1.) 
 
 d 2 y _^ dhf _d z y dx q 
 
 Ts ~dxj(l+f) - S ' VI 1 +P 9 ) ' '"' ~ -s/0+1*") 
 
Problems. ] \ 7 
 
 and dV = H .dp -\ 7 —— i — -. ,dn ; 
 
 /. M= o, N = o, />= 2£_, a = , * ,. ; 
 
 (1+F)" v/(l+/r) 
 
 ,\ by the Form [d] p. Ill, 
 9 _ _JL 
 
 ^/ c 
 
 c 
 
 - + c/> -f c\ 
 
 whence, p or ~- = ■ , and ci/ = </' — c\v. an equa- 
 
 tion to a right line. 
 
 Prob. 8. 
 
 Required the curve in which J — ~ = a maximum. 
 ("see Comm. Acad. Petrop. torn. VIII. p. 1/1-] 
 
 d z y __ d z y 
 
 -.dx; .-. F = 
 
 y.ds dx\y\/{\ + p 1 )' ys/{ l +/> 2 ) 
 and dV = — JL -r.dy ^— 7 .dp+ 
 
 dq; 
 
 Vs/( l +f) 
 
 • M-o N-- q P= K 
 
 Q = —tt^ jr; /. by the Form [e] p. HI, 
 
 r • ^n d y pdp - 1 
 
 since aU= . ,/; jr - — - tt 
 
 L ifJ( l +P) y{i+P~Y J 
 
 __ 9 W % + _JL_ + 
 
 ^7(1+/) 3/( 1 +/) i) yVC 1 *?*) 
 
 yVO +^' 2 ) 3/(1 +/^ 
 
1 1 8 Problems. 
 
 consequently, ~- — — — - — = c, and 
 and since p — -j- , dx = ; — — ^— ; . 
 
 F dx y N /ey±cv r (4+cy) 
 
 Prob. 9. 
 
 Required the curve which, within its own arc, its 
 evolute, and radius of curvature, shall contain the least 
 area*, [see Euler, Comm. Acad. Petrop. torn. VIII. 
 p. 169 ; also Methodus Invemendi, &c. p. 64.] 
 
 T , v , . ds* (dx* + df)± 
 
 I he radius of curvature = j r- f = j jr-^ 
 
 — dx.d'y — dx.dy 
 
 . !l\ 
 
 dx.d'y q 
 
 radius x ds 
 
 +J»*)* A 3 i 1 +P 2 )^ since „ - d P-^L 
 
 Hence, the differential of the area = 
 
 2 
 
 corn- 
 
 _ (l+y*)\ /(l + f)d.r- -Il±gy Jr ; which 
 
 pared with ^af^r, gives V— — — — ; consequently, 
 
 * The area is A OR in Simpson's Biugram, p. 7S; Fluxions, 
 ed. 1750. 
 
 + Simpson, p. 72. Vince, p. 149, First Edit. Woodhouse Anal. 
 Calc. p. 180. 
 
Problems. 1 19 
 
 .-. 1/ = 0, N= o, P = - ^(1+p 3 ), Q = ( i±£T 
 
 ,\ by Form [d] p. Ill, that is, f^=Qq + cp + c, we have 
 
 (l±p 2 Y (14-p'Y 
 - v ' — = - l —!- + cp + c', 
 
 or i l_jL_ _ c ^ .|_ c ' . but q = y- , 
 
 .'. - 2(1 + p a )*\/(l + p*).dx = cpdp + c'.dp; 
 or, since, 6^ = ^/(1 +p*).dx 
 
 -ad*= cpdp , + c '^ ; 
 
 (1 +/7 2 )^ (1 + />*)* 
 
 
 cp —c 
 
 v/(i +p 8 ) 
 
 If c, c" = O, — 2 s = ,, ^ — , , which shews the 
 
 curve to be a cycloid : for by p. 5. -j- orp — \f ( J 
 
 .-. . . ? = \/^ — , and the arc of a cvcloid 
 
 measured from its vertex = 2 chord of generating circle 
 = 2.7a (a — ?/) = -77: iv • 
 
 Prob. 10. 
 Required the curve in whichy -5 — ^ — is a minimum, 
 [see Euler, Comm. Acad. Petrop. torn. VIII. p. 185.] 
 
1 20 Problems. 
 
 a dhj dhf 1 __ r 
 
 Here —j — ^- = -—- x -.dx\ ,\ V = - 
 
 ax. ay ax" p p 
 
 and dV — .£?» -| — r/r ; 
 
 and by the Form \j*~\ p. 1 1 2, 
 
 - = cp H H <?.-/-• -7 ± c'. 
 
 /? L p M ax p z 
 
 Hence, O = cjt> -f-^ — c', and q=p s /{c' — cp) 
 
 or ' £ = aA'-«jO an<1 ••• dx = ? V( ?- cp) 
 
 and ,-i_hl ( V<?-JV- r P) \ 
 
 PROB. 11. 
 
 Required the relation of j and y wheny — ^T" * s a 
 
 maximum, [see Lacroix, Diff. Calc. p. 704 : also Borda 
 Mem. Acad, des Scieyices, 1767. p. 06*0.] 
 
 and dV-2q.dq, r.M = 0, N=0, JP = 0, a=2ry, 
 R = o, &c. 
 
 .-. by the Form [a] p. 1 10, -j^ = o, or 2.j± = 0; 
 and integrating 2.-7+ = c, 2q or 2^- = c# + c 
 
 2» or 2-r = — + dx + c", and 
 1 dx 2 
 
Problems. 121 
 
 C/V»3 r\ yi* 
 
 2// = — + h C'# + c" . 
 
 2.3 1.2 
 
 The final equation contains four arbitrary quantities, 
 c, c', &c. : in order to determine them, suppose the values 
 of y to be given at two points of the abscissa (x = O f 
 x = a), and also the angles in which the tangents to the 
 curve at those points * are inclined to the axis ; for the 
 sake of simplicity, suppose y = 0, when x = 0; and y = b, 
 whena? = «: also the values of the two tangents of the 
 angles in which the curve cuts the axis to be t, t' : then 
 we have, for the determination of c, c', &c. the following- 
 equations : 
 
 x = 0, y = ; 
 
 x ~ a, y = b; 
 
 chi 
 ax 
 
 dy p 
 
 x — a : .\ -~ ■=. r ; 
 
 ax * 
 
 whence the values of c, c '. 
 
 The preceding problems involve one property onlv ; 
 that of the maximum or minimum ; and therefore, in 
 strictness, ought not to be classed amongst Isoperimetri- 
 cal problems, since they involve neither the Isoperi- 
 metrical property, properly so called, nor any other 
 equally affecting the theory and the analytical pro- 
 cesses. The following problems involve more than one 
 property. 
 
 c"' = 0, 
 
 
 , ca 3 d.a % 
 
 2b = -\ 
 
 2.3 2 
 
 + c"a, 
 
 2t = c", 
 
 
 2 if = hc'«4 
 
 ■ d\ 
 
 * This is only one of the many hypotheses that may be framed for 
 the determination of the arbitrary quantities, 
 
 R 
 
122 Problems. 
 
 Prob. 12. 
 
 Required the brachystochrone, that is, the curve of 
 quickest descent, when the length of the curve is given. 
 [See John Bernoulli's Works ; torn. II. p. 255. Acad, des 
 Sciences, 1/18. p. 120; also of this Work, pp. 18, &c] 
 
 Here, by the formulae, pp. 78, 109, we must find the 
 variation of fdx {V — au), putting V — au instead 
 
 of V\ now V = ^ (1 t^ and u = ^/(l + f) ; 
 .-. V-au } or V = ^SL^£i - aj(\+f); 
 
 s/y 
 
 .-. by the Form [b] p. 110. 
 or {-7- - «) = c x /(l + P% a "tl by reduction, 
 
 If, instead of the length, the area had been given, 
 then j'udx =J)jdx; and consequently, we should have 
 
 V 
 
 in which case, P — —. — . , 
 
Problems. ] 23 
 
 whence, -5 77 -~ = c + tf?/, and by reduction, 
 
 dx (c+ay) s /y.dy 
 
 If c = 0, dx = — — - — '—-, which agrees with Simp- 
 v/(l-a l y 3 ) & ^ 
 
 son's result, Fluxions, p, 492. But, since c is not neces- 
 sarily = 0, it follows, that Simpson's solution is not so 
 general as it ought to be. 
 
 Prob. 13. 
 
 Required the solid of least resistance amongst all the 
 solids of equal capacity. [See Maclaurin's Fluxions, 
 p. 751. Emerson's Fluxions, p. 188.] 
 
 Here, [see Prob. 5. p. 155.], V— ^J' „ , wdfudx 
 = J^vy % dx ; 
 
 .-. V—au, or V =~~~~ z — ay" (including *r in the 
 quantity a.) 
 Hence, P = -fl*. ^f > and by the Form [6] p. 110, 
 
 ■ • ^ , - ay 7 - = ~^ — ~- — c, whence, by reduction, 
 l+/r * (1+P) 
 
 (c — ay 1 ) (1 +p*) a = 2«/p 3 , or (c - ay 2 ) ds* — 2ydy\dx. 
 
 If instead of the condition of equal capacity, that of 
 equal superficies be substituted, we have 
 
124 Problems. 
 
 F-au, or V ' =~^ - ay J{\ + V % 
 
 and P = ^yf±lt _ a riP f 
 
 (i+ry v/(i+/> 2 ) 
 
 whence, by the Form [6] p. no, 
 
 and, by reduction, 
 
 c.(l + /? 2 ) 2 = 2yp 3 +ay.(l +p 2 )^ 
 or c.rfs 4 = 2y,dy 3 ,dx -+- ay.ds*.dx. 
 
 If the solid of least resistance be required with both 
 the specified conditions, then we must use a formula 
 f{V—au — b\i) dx. See p. 78, and if u=y 2 , and v = 
 
 y s/( l +2 f )> we liave 
 
 V-au-bv, or ^={£fi ~ ay* - bi Js J(\ +p*), 
 
 and consequently, by the Form [6] p. 110. 
 
 7 3 yf + z/p* hijf 
 
 * — ~T7~; — w TTTn — *\ - c > whence 
 
 c(l+p*)* = 2yf + by(\ +/>*)- -f ^/-(i + jr) 2 , 
 or (c — ay")d6 A = 2ydy 3 dx + byds 3 .dx. 
 
 Prob. 14. 
 
 Given the length of the curve, to determine its nature 
 when the solid generated by its rotation is a maximum. 
 [See Simpson, p. 486 : Maclaurin's Fluxions, p. 749 ; 
 Euler, Methodus Tnveniendi, &c. p. 196V] 
 J Vdx =firy\dx [tt = 3.14159 &c] and fudx = 
 
Problems. J 25 
 
 and dV = 2-m/. dy -r-J— — . x . whence P= -. — ' „ ; 
 
 ,\ by the Form [&] p. 110. 
 
 try x -ay/\ < \ + p) = c - 
 
 and .*. ^/(l + /; 3 ) = — ; , and by reduction, 
 
 7T U — C 
 
 TT if 
 
 If c — O, f/.r == ~rrir — r~ i\ dy, which is Simpson's 
 y/(a —v~y*) v 
 
 result, and, like the preceding, restricted. 
 
 Euler says, that the curve is the elastic ; and the 
 curve determined by Simpson is the rectangular elastic 
 curve. 
 
 Prob. 15. 
 
 Required the curve that generates the solid of the 
 least surface, the area being given. [See Euler, Metlindus 
 Lnveniendi, &c. p. 198.] 
 
 V = 2vys/{l +p% it = y; 
 .*. V\ or V - au—2vyy/{\ + p z ) — ay, 
 
 and dV> = l2wy/(\ + f) - a] dy + -^E—.dp, 
 whence F - 
 
 ZJ — ■ • .•. by the Form 17/1 p. 110, 
 
 2 TT//// 
 
 and, by reduction, 
 
F.r- . 
 
 -- " - - 
 
 P ,« - = _ 
 
 j —ay 
 
 — ■ ; ! ^.d.u. 
 
 I: length is g 
 
 v — = / I — 
 
 .-./"=;- % l-rp*)— a I - 
 
 P = ' ! . 
 
 I - - - - • 
 
 = 
 
 -" - 
 
 
 _ 
 = — ~ - -" ~ — s -- - - 1 - 
 
 5 the " [See Eoler, 3Iei 
 
 J 
 
 Pb 
 
 - 
 
 Here r =,/ _- - P= 
 
 I 
 
 n -- " = — — - — — = - 
 
 = N = = . 
 
- z ----- 
 - ■-- 
 
 
 
 _ 
 
 
 V — = > * 
 
 ;~ = 
 
 
 •i 
 
 = 
 
 
 - 
 
 - 
 
 j E 
 
 , 
 
 " 
 
 
 
 - 
 
 a 
 
 , 
 
 - v- 
 
 
 
 
 ~ - 
 
 H, , : 
 
 
 N. 
 
 : - 
 
 ■ VTf- - 
 
 : -~ 
 
"128 Problem. 
 
 - ; the fixed point, from which lines such as y are drawn, 
 being in one extremity of the diameter. 
 
 Prob. 18. 
 
 Required the curve that, by a revolution round its 
 axis, generates the greatest solid under a given surface. 
 [See Euler, Methodus lnveniendi, p. 194.] 
 
 Here V\ or V -an = mf — 2 airy ^/ (l + p 2 ) ; 
 
 •'• p = " y/ii+py andb > 7the Form W p- 110 > 
 
 putting C7r for the correction : hence, by reduction, 
 
 . or & = VC^V/'-fa'-O 1 ] . 
 ' */# ?/ a - c 
 
 , (ir — c).du . . . 
 
 .*. dx = , r , , — V~r — w=! the equation to the curve. 
 v/[4ay— (y»-c)»] 
 
 U c = 0. dx = , ; , ', — . an equation to a circle : 
 
 v/( 4 «-#) 
 
 which Simpson [Fluxions, p. 487], determines it to be; 
 bu+, it is plain that this is only a particular case of the 
 general solution. Lyons's Solution, p. 100, is also equally 
 restricted with Simpson's. 
 
 Prob. 19. 
 
 Of all Isochronous curves to find AB such, that the 
 space included between the arc AB, and a chord drawn 
 from A to B shall be a maximum. [See John Bernoulli's 
 Works, torn. II. p. 263 ; also Acad. Roy. des Sciences, 
 1718, p. 132.] 
 
Problems. 1^9 
 
 Here the time, a given quantity represented hy fudx a 
 
 answers 
 
 ./^O. 
 
 and JVdx, the maximum, to 
 
 rr i i /%v\~\ r/ydx xdy\ 
 
 J \_yd* - d (f )J , «/(«5 f ; , 
 
 °'/(f -f)dx, since p = |f. 
 
 Hence, F- aw, or ^'=5 (if — X P) _ g / > 
 
 a 
 
 «7? 
 
 /#>/(!+/>*) 
 
 a/r 
 
 /. by the Form [c] p. 110, 
 1 , v v/(l+K) * jo 
 
 But p = -f^ ; 
 
 ,\ /Wa? = y. Hence, by reduction, 
 
130 
 
 Problems. 
 
 „_ a c , nd p or ^ - n/ ["' - y'y ~ c )'l 
 
 and consequently, 
 
 y/lcP-y.iy-cY]' 
 the equation to the curve. 
 
 Prob. 20. 
 Of all I soperi metrical curves drawn between B and 
 N, to find JSjF/Vsuch, that BZN shall contain the great- 
 
 est area. PZ being a function of PF. [See James and 
 John Bernoulli's Works, p. 909, and p. 8l6 ; also pp. 12, 
 25, of this Work.] 
 
 Let PF=y, and let PZ= a function of y = Y\ then 
 fVdx corresponds tofYdoc, axidfudx to/v/( 1 +_//) . </x. 
 
 Hence, F-f ««, or F' = Y + a*J(l + p*) 
 
 dy J v/(i+f) '' s/ii+P 1 ) 
 
 Hence by the Form [//] p. 110, 
 
 i' + fls/fl +/r) = 
 
 #/?* 
 
 %/(!+//) 
 
 K + C > 
 
 or. 
 
 — 7- — = c — I~ and bv reduction, 
 
 \/{l-t/r) 
 
Problems. 131 
 
 p } or -~ = **-*• ^-p= — ^-J-, and consequently, 
 
 dx = - v r „ — '"'. -, the equation to the 
 curve. 
 
 In the equation, as it stands, there are two arbitrary 
 undetermined quantities a and c. The integration of 
 the equation would introduce a third : and to determine 
 these three, we have given, the two points B and iV, and 
 the length of the curve. 
 
 Prob. 21. 
 Required the curve which generates by its rotation 
 the solid of the greatest volume ; the length of the curve 
 and its area being given. [See Lacroix, Calc. Diff\ et 
 Int. vol. II. p. 713.] 
 
 Here we must use the form [See p. 78.] 
 f{V — au — bv).dx, corresponding to which is 
 fit ~ "y-b.Jil +fj\dx, 
 a and b involving, in their values, n and other given 
 quantities. 
 
 Hence, the differential of the quantity corresponding 
 to V is 
 
 {2 y-a)Jy -bp.-j^—ry, 
 
 C, by the Form [li] p. 110, 
 
 .% y' — aw - c = ,,, , — rr, and by reduction, 
 */(l+F*) 
 
132 Problems. 
 
 r.vlL-Jp-y-'V-W, and 
 
 1 7 ax y - ay — c 
 
 aX ~ s/\b*-{y*-ay-cyy 
 an equation to the elastic curve. 
 
 Prob. 22. 
 
 Let now PZ, instead of being a function of y, (see 
 Fig. p. 130.) be a function of the arc BF (s). [See 
 James and John Bernoulli's Works, p. 912, and p. 91 7 : 
 MaclaurirCs Fluxions, p. 508, also p. 25. of this Work.] 
 
 Here fVclx corresponds to f Zdx, Z a function of s, 
 and fudx corresponds to f (1x^/(1 +p 2 ) ; 
 
 ,\ V - au, or F'= - a^/(l + p*) + Z, 
 
 and dV — — ., — rr-dp + -j—.ds. 
 ^/(1+p) 1 ds 
 
 This form therefore is to be solved by the method 
 given in p. 93 : and, on comparison, we have /* = 0, 
 
 u = 0, tt — — ., — r, T = -j— , ds — dt. The formula 
 */{l+f)' ds' 
 
 of solution then, p. 93, is 
 
 k being — H - f Tdx. Hence, integrating 
 
 ... ft= cjd+rt-ap an(J rfft= «*J> = _ Td 
 
 V }>\/( l +P) 
 
 .: Tdx^{l +p>), or Tils = - c.%: .-.fTds, that is, 
 
 P 
 
Problems. loo 
 
 c du c 
 
 t'dZ or Z — d + - . Hence » or ~- = r j — 
 J p l dx 6 — 
 
 dx = dy — . 
 
 -, and 
 
 c 
 
 lfdx, dy are to be expressed in terms of the ares, 
 then since dx = dy . &^1 , ds = \f [l + (— ~) ] . dy 
 
 consequently, dy = ^ / ^ + [ Z ~ df\ 
 
 (Z-c').ds 
 
 ' Ll '~x/[c*+(z- c ry 
 
 Prob. 23. 
 
 Given the length of the curve; required its nature 
 when its center of gravity is most remote from the axis 
 (y). [See Euler, Comm. Acad. Petrop. torn. VI. p. 146: 
 Simpson's Fluxions, p. 498. 
 
 Suppose y to be horizontal, and x vertical, then the 
 
 distance of the center of gravity from y, 
 
 fxds 
 
 (distance kji liic cciilci ui giavii.^ uum if, — /V/«r 
 
 But J'ds — s — b by hypothesis ; 
 ... V - an, or V = x 'S(i+P 9 ) _ a j {l +}n> 
 
 and P = (| - a ) X ^jt_. . 
 
 dP 
 Hence, by the general formula, N j^ + &c TV, Q, 
 
 Simpson's Fluxions, p. 205. ed. 1750/ 
 
134 Problems. 
 
 &c. being = 0, we have 
 
 .-. 1 -a = c —i £■', and d# = be x a ,, , — jr. 
 
 Hence, c/.t\/(1 +/> 2 ), or r/s = - cb.-—, and integrating, 
 
 cb t , cb . , , , x ^s 
 
 v = hf 3 or ^ = , ; .-. since */ (1 -fp 3 ) = -y , 
 
 dr~ (s-<!)ds 
 
 ■. 7 cb.ds 
 
 a " d ^ = v/ [('-*') W] ' 
 equations to the catenary. 
 
 If the condition of equal length be omitted, that is, 
 
 if « = O, i* will equal T ., r ; and the nature of 
 
 the resulting curve will still remain the same, s being 
 represented by b : and if s be not so represented, 
 but be supposed to vary, still the resulting curve will be 
 the catenary, as will appear, with the reason thereof, in 
 the following problem. 
 
 Prob. 24. 
 
 Suppose a chain or cord, of variable thickness, to be 
 attached to two fixed points ; it is required to find the 
 nature of the curve which it ought to form, so that its 
 center of gravity shall be the lowest. 
 
 Let the weight of an element of the curve be ex- 
 pressed by Q.ds, 6 being a function of* dependent on the 
 
Problems. 135 
 
 law of the chain's thickness ; then, the distance of the 
 
 / T it S / 30 9 ft S 
 
 center of gravity, instead of being ' . / , is J . * , which 
 
 is to be a maximum. 
 
 tt .v /fxQds\ /'0 . ds Jf xkds- fxQ.dsJ / 9 , ds 
 
 Hence, $ ( J -——)=J 1 — J 1 =0, 
 
 ' \Jbdss {jb.dsf 
 
 now $fx . ds = Sfx §J{\+if).dx; /. V = x 0^/ ( 1 +^ 2 ) 
 
 and d^ 7 " = 
 
 ey(i+r)-&+~f^ } +.,«•(! +>).$.*■ 
 
 Hence by the Form [C] p. 93, supposing <h? = 0, 
 and not regarding S, the sum of the definite terms, 
 
 , / ^ =/ ^ y [_ rf (-»fE_)£] 
 
 + /^"<</fe))i] L " J 
 
 sincev=0 , *= jjjfe, N -°' p= v^wr a,ul 
 
 h, which equals H-fTdx, being in this case = H - 
 
 rib 
 
 fxV (1 +p*).j^.dx = H -fxM = if - *0 +/Q.dx 
 
 substituting this value in the expression [m] ; the 
 equation becomes 
 
 With regard to the other variation SfQ.ds, /^answers 
 to 8.v/(l +P% and rfr=^^+ ^/(l +f)-f/»i 
 
136 Problems. 
 
 .*. by Form [C] p. 93, supposing $x = 0, and not 
 regarding S the sum of the definite parts, SfFdx = 
 
 fdx.s,,[-d. n7i ^ ~ - a. ^fp) j£ ■ ■ W 
 
 v being = O, v = -y— — , and A* in this case being 
 
 equal to K — f \/ (\ + p*).-?-.dx; that is. K - fdQ, 
 
 or A'- 8; K is analogous to jFf, and equals the entire 
 value of between the two limiting values of x. Hence, 
 substituting in [11] 1. 3, we have 
 
 tfi.ds = (dxJij\-d( , Kp nt \ 4-1 . 
 
 If we now substitute in the expression for 
 
 ^ (-«'.'," ) p. 135. 1. 4, and represent the two integrals 
 
 of xQds, and Of/*', taken between the two limiting values 
 [x Q and a?,] of x, by ^ and 5, there results 
 
 4A^/[-^( 7 ^ ) )-a=o, 
 
 and putting — = A, 
 
 ,/f — / — LL — _i — X.d{ —r, — — ; x J, and integrating 
 i//> + pf'Qdx = xKp + c^/(l -f;r), whence 
 
Problems. 137 
 
 //- \K+fldx = cvSL±JLl 9 and taking the differential 
 
 UX = ~ C P\J^W) ' ° r °^v/(l+? 2 ) = - c&, 
 or Qds = — c — ^ ; whence integrating 
 
 fUs = f + <f, and j. or g = f% £ _ c „ 
 
 and since yd +K)=v/(l + g) = £ 
 
 , 7 fUds-c' j 
 
 we have «# = /r /;., , ^ — rr«"*> 
 
 eds 
 
 ancl ^ = v/[(/e^— c7+c x j ■ 
 
 If the chain be uniformly thick, = 1 and 
 
 (* — </) ^£ 
 
 * = ^[(.v-cT + <•"-]' 
 which equations are the same, in fact, as those in the 
 preceding problem, [p. 134. 1. $7~], and belong to the 
 curve called the catenary. 
 
 In problem the twenty- third, where s is a given quan- 
 tity, the solution was effected by means of the common 
 
 dP 
 
 formula N j— + &c. ; and the same formula is suf- 
 
 ax 
 
 ficient, if instead of the process there used, we institute 
 one similar to that in Prob. 24, p. 134, that is, if we 
 
 T 
 
138 Problems. 
 
 f i 7 
 
 investigate the variation of ' . ■ — a fds, which equals 
 
 J as J * 
 
 s x Sfxds — fx ds 9 /' (Is . r , 
 ^ i £ — - aSfds 
 
 for, it is manifest, that tliis expression can be expanded 
 by the common formula, p. 88 *. 
 
 The condition however of a variable thickness intro- 
 duces the quantity 6, a function of s ; for instance, instead 
 of<T/' l r v /(l +7? 2 ) . dx - &c, we have, [seep. 135. 1.5.] 
 $J I'Q^/il +P 1 ) — &c. ; and therefore when the differential 
 of the quantity corresponding to V is taken, a term 
 
 (19 
 3^/(1 -f- p 2 ) -y,'d s ls introduced, which obliges us to 
 
 have recourse to the formula [Cc] p. 93. 
 
 The next instance will illustrate the use of Lagrange's 
 formula in his third problem, p. 185, in which, the quan- 
 tity that is to be a maximum is expressed under the 
 form of a differential equation. 
 
 Prob. 25. 
 
 Required the curve, down which a body falling, in a 
 resisting medium, shall acquire the greatest velocity. 
 
 * Enlcr, Comm. Acad. torn. VI. shews, that the catenary equally 
 
 results, whether the length of the curve be taken into the computation 
 
 or not. " Potuisset quidem eadem aequatio multo- facilius inveniri, si 
 
 . fxds .... . . ,. ■ 
 
 in* neglexissem deiroiwnatorem, qmppe per priorem conditionem 
 
 debet in omnibus curvis esse idem : Verum quia hoc fortuito accidit, &c. 
 
Problems. 139 
 
 Let z represent the square of the velocity, g the force 
 
 of gravity, p a function of z, the resistance ; and let x bo 
 
 supposed vertical : then, 
 
 dz - 2g.dx + 2p s /(l+p*).dx=0, . . . [l] 
 
 [by the principles of mechanics.] 
 
 Compare this with the formula p. 94. 1. 1 2, and we have 
 
 V— z, X= 2g, U— 2^^/(1 + p z ) a function of p and p ; 
 
 7rr </£/ 7 dU , 2pp , 
 
 .-. dU = -j-c?/? + -i-.dp = —-lf-;dp + 
 
 2 Tz' dz J( l +f) 
 
 hence, the equations are, 
 
 dh dp . r _ 
 
 and ^For ^ f= = e- Jrdx fe rdx J(p = 0, 
 
 or 2X P -j-1- = -t~ [3] 
 
 V(!+P ) y/ a 
 
 In equation [2] substitute from [l] the value of dz ; 
 
 .'. 2gd\ = 27^/(1 +/r).r/ e + 2 fv /(l + p*).d* 
 
 = 2 s /(l+p*)[*de + p.dx']=2 s /(l+p*)d(\t)^ 
 
 2.</j> fV /(i + />*)] - 2x P . y^l f) = (byW) 
 
 2.d\Kps/{l+p)-]- d ^ 
 Hence, integrating, 2g \ = 2\p V ( \ +p z ) y- -\-c — 
 
 s/a 
 
 1 +p~ ^ 1 
 
 v , e 
 
 and .'. from [31, - p —~ rv H ; % t: = ~r~ '• 
 
140 Problems. 
 
 whence, by the solution of a quadratic the value 
 of p. 
 
 This problem may be also solved, and readily, by 
 means of the general method and formula given in p. 104, 
 thus : since z is to be a maximum, f Vdx corresponds to 
 
 / d z\ 
 ,\ dV '= d(-f-) i •*• D y comparison, tr = 1, all the other 
 
 coefficients being = O ; .-. Y ~ 0, Z = 0, V = o, 
 Z = 1. Again, 
 
 f IV dx corresponds to J l-j- — 2g -}- 2^.^/(1 +p 2 )\dx 
 .'. by comparison, 
 
 (2) =x/-%3» 3V ( 1+p .).|-^ 
 
 (Z') = At/ = A. 
 
 Hence, taking from the expanded form, for 
 §jVdx + $f\JVdx, or 
 
 *flH dX + ^[ X ^- 3 ^ A + 2 ^v/(l+P 8 )] rf*, that 
 
Problems. ] 4 1 
 
 part which is under the sign j\ and on which the nature 
 of the curve depends, we have 
 
 (F).«to + (Z)te = O, and (F) = 0, (Z)=0, 
 since $u, <Jw' are independent of each other; hence, 
 
 <^))=°> and3 ^(^)-£=o 
 
 as before, p. 139. 1. 12, 14. 
 
 The following problem will illustrate the use of the 
 general method given in p. 104, in which the maximum 
 or minimum property involves a quantity not expressed 
 except by means of a differential equation ; which equa- 
 tion, in the subjoined case, is called an equation of 
 condition. 
 
 Prob. 26. 
 
 Required the brachystochrone, or the curve of quick- 
 est descent, in a medium resisting as any function of 
 the velocity. 
 
 Let z represent the square of the velocity, *? the force 
 of gravity, ^ a function of z, the resistance ; and let y be 
 horizontal and x vertical : then by the principles of , 
 mechanics, 
 
 dz - 2g.dx + 2f.ds = O, 
 
 or [~-2£ + 2 fx /(l+7r)] dx = o 
 
 the differential equation of condition. 
 
 If we compare this with Wdx^ p. 104, we have 
 
 OTm 2.^/(1 +,f) M + d(Pf + o. / d >> . 
 
1 42 Problems. 
 
 *'• V ' =2 -^f^/ (l +p ~^ *** l * since ^(zD = ^ 
 
 and P' = , 2 W ; 
 '• (/ ) a dx ~ V(l +f)) dx> 
 
 The equation of minimum corresponding tofldx \*> 
 and by comparison, 
 
 andr=i\r-.§? + & c . = - df-j-Jt — -A4- 
 
 Hence, substituting in the expression for ifVdx + 
 i j\xW)dx p. 104, and retaining only the parts 
 under the integral sign, which must separately equal 0, 
 we have 
 
 r+(F)=o,or~d( -, — I }-d( -?*££_J)=o, 
 
Problems. ] 4,3 
 
 If we put t = -j- + 2\o, and integrate the first of 
 these two equations, it hecomes 
 
 -Jfc = -L rii 
 
 «/o+p*) jet L1J 
 
 the second is ^- ( 2 X^ --1^)^(1 + //) = O . . [2] 
 
 Add to these dz-2g.cLv -f- 2^.^^/(1+7/) =° • • [ 3 ] 
 and the equation to the curve is to be determined from 
 these three. 
 
 _. . . _ , dt 1 d\ do 
 
 From the value of t, -7 = + 2—. + 2X-i ; 
 
 a £ 2^4 a 3 s rf.s 
 
 and from this equation substitute the value of — f- 
 
 2* 4 
 
 2A-~, into [2], and there results 
 
 az L J 
 
 In this, substitute the value of rfs derived from [3j, 
 and there results (two quantities destroying each other) 
 
 d\ dt . . . _ _ 
 
 '^■dl- ~ Zv^ (l + ?) = ° M 
 
 but g v /(l +/). r/., = rf[ V(l +?'")] -'jffifi 
 
 (from the value of £ in equation [l] ; ) 
 hence, the equation [a] is 
 
 2g,~£.- c/LA- (1+ ^.Ji+ 'P ; and integrating 
 
1 44 Problems. 
 
 2£X-^/(l+p 9 ) + J^ = c; 
 
 or, substituting the value of t from [l] p. 143, 
 (l+p 2 ) v 
 
 i 
 
 from equation [l], and consequently, 
 
 J_ = s/( l +P*) -<! ( l +c ). 
 
 from this equation obtain the value of dz, which will 
 involve the differentials dp, and -f- .dz. By means of 
 equation [3], eliminate dz, and there will result a dif- 
 ferential equation involving dxand dp, or-p- 
 will be the equation of solution. 
 
 In the case of a void, or non-resisting medium £ = 0; 
 
 . j_ _ xAi+p 2 ) v L. 
 
 or i — ^— = -,-. an equation to the cycloid. 
 ds\/z ^J a 
 
 [See pp. 5,21.] 
 
 The preceding instances are sufficient, it is hoped, 
 fully to illustrate the nature and use of those formula? of 
 solution invented by Euler, on which the nature of the 
 curve, or the relation of x to y, depends. But hitherto, 
 no illustration has been given of those other formula?. 
 
 -~ .dx\ which 
 
Problems. 1 4-5 
 
 that, on assigning the initial and final variations, have 
 definite and absolute values *. These formulae, it has been 
 already said, were invented by Lagrange, and their use 
 will be shewn in the following cases, which may be 
 viewed, as unresolved parts, or undetermined conditions 
 belonging to certain of the preceding problems. 
 
 Prob. 27. 
 
 If the brachystochrone is to lie between a point and 
 a curve, required the angle in which the cycloid must 
 cut the curve. 
 
 By Form [Act] p. 89, and since the quantity under 
 the integral sign f, = 0, 
 
 V X .*X % + P x .U t =0; 
 
 for, the initial point is given ; .*. 8x oi Sy Q = O, and <$w = 
 $ij — p $x , = O, and since 
 
 V^Jll+f) Q = , and P= , f ( . 
 
 [see p. 113]. 
 
 Hence, (V, - P t p t )Jx t + P x .*y x = O, 
 
 but F= Pp + c [p. 110] ; /. c.3x 1 + P 1 Jy [ =0 . . [m]. 
 
 Let now M(a function of x t and ?/,)=(); then, [seep. lOf .] 
 
 < ni * dM % , dii ' 
 
 3F'^ + % J ^° s and Jfc/ # *' - *■ = °- 
 
 Eliminate fx lS $y x by means of this and of the preceding 
 equation [/«], and there results 
 
 P. 89,90,91. 
 U 
 
1 46 Problems. 
 
 dy J _ c V— Pp i 
 
 deer ~ i\ ~~~Fr~ ~p7* 
 
 Now /;=—-, y and x belonging to the cycloid,, and p. 
 
 is what -y- becomes at the limiting curve, when for x 
 
 dy 
 and y, #, and y t are substituted : but, -~ is the tangent 
 
 of the angle which a tangent line to the curve makes with 
 
 dx 1 
 
 the axis *, and -r- or - is the co-tangent of the same 
 ay p 
 
 angle t, and — is the co-tangent at the limiting curve : 
 Vi 
 
 dy 
 for the same reason —i, is the tangent of the angle 
 
 which a tangent line to the limiting curve makes with the 
 
 axis : and since -~ = — — , or since, the tangent of the 
 dx, p t 
 
 latter angle equals the co-tangent of the former, one angle 
 is the complement of the other : their sum therefore is 
 equal to a right angle : consequently, the remaining 
 an^le of the triangle, or that which the two tangents 
 form, is a right angle ; or, the two curves intersect each 
 other at right angles. [See p. II.] 
 
 The same conclusion will follow from the expression 
 * for the tangent of the difference of two angles [A, B] the 
 
 * Piinc. Anal, Calc. p. 173. f Trig. p. 9. 
 
Problems. 147 
 
 denominator of which expression is 1 + tan. A. tan. B *, and 
 
 in the symbols previously employed, 1 +;?, . y^ = ; 
 
 .*. the fraction or the tangent (A — B), is x ; or the differ- 
 ence of the two angles is a right angle. 
 
 Prob. 28. 
 
 Suppose now the brachystochrone to lie, not between 
 a point and a curve, but between two curves. Required 
 the angles in which it must cut thern. 
 
 By Form \_Aa~] p. 89. 
 V X .$X X - V Jx + P,.*., - P J» = O; 
 /. as before c$x t + P x .tyj — (cJx + P .ty ) = O, 
 and this equation, if there be no relation between the two 
 limiting curves, or between x Q , y o9 and x 1} y ti resolves 
 itself into these two, 
 cJx t + P z .#y,»o [l],andc.#i + P .#y = o . . [2] 
 
 If therefore L, (a function of x oi y ) — O, be the equa- 
 tion to the initial limiting curve, by combining it with 
 [2] we shall arrive at the same results as were obtained 
 in p. 145, by combining M=Q with [lj ; since the re- 
 spective equations are similar. Hence, the brachysto- 
 chrone, which here is a cycloid, must cut the first curve 
 also at right angles. [See Misc. Taurin. torn. IV. p. I87.] 
 
 In this last case the origin of the abscissas is supposed 
 to be fixed, and consequently, the velocity with which the 
 
 * Woodhouse's Trigonometry, p. 23. 
 
148 Problems. 
 
 body leaves the first curve is variable, varying as \/y - If 
 the origin of the abscissas is supposed to vary, then, 
 
 Tsee p. 108.1 terms such as ~-$aj'— r , — ih I — - - will 
 L r J da db 
 
 be introduced into the equation of limits : thus, instead 
 of the differential of the time being — 7- , let it be 
 
 & Vv 
 
 ds 
 -7 — — - , and let h be a function ot a TrJ, and b f~ 1/ 1 
 
 V(j/- h) L J L,yoJ 
 
 solely ; then, 
 
 • " rr < /(l+P l dF s/(l+p 2 ) dh 
 
 since, v = -77 W , ~r- = • T~> ana 
 
 */(#-//) da 2.{y-hy da 
 
 dV _ J(\ +p) dh 
 
 db ' <2(y-hf'db' 
 
 In this case therefore, the equation of limits, at the 
 
 . . , , . dh . dh 
 
 last point where x = i. lt or = a , since -j-, = 0, and -rj, — o 
 
 \Ii not involving a' and Z/] becomes, 
 
 Cix t + P x .^ = [3] 
 
 but, the equation of limits, at the first point, becomes 
 n * /^ k dh * \ f* (Is - -, 
 
 c lte . + p.. *„+ (35 »,. .-;- ^ . *y yj 2 —— p =0, [4] 
 
 Since by the gen 1 , form 3 . N- -r- = ; • = - d/\ 
 
 and/ =— P + corr. Now at the beginning, 
 
 2.(y-h)* 
 
 the integral = 0, and P = P Q ; .-. corr. = P . Hence, 
 
 ^T^Tay* =p °- p=p °- p > f from * = *« t0 * = J ■ j 
 
 the equation [4] therefore becomes 
 
Problems. ] if) 
 
 = o. 
 
 This last result, expressed in the equations [3] and 
 [4], comprehends, as particular cases, the preceding ones 
 p. 146, 1 47 ; for if h = 0, the equations [3] and [4] become, 
 
 cJx x + iVty, = 0, and cj'x + Pji/ = O 
 
 as before ; which equations shew that the cycloid must 
 cut the two curves at right angles. This was Lagrange's 
 original determination of the conditions. [See Misc. 
 Taurin. torn. II. p. 180.] 
 
 If h = b, or b — e, [e a constant quantity;) that is, if 
 y - //, at the point of departure, = or =e; or if, in other 
 words, the velocity of the body when it quits the first 
 curve is either nothing or constant ; or, if the origin of 
 the abscissas, according to M. Borda's expression, be sup- 
 posed to be situated in the first curve : then, — = 1, and 
 
 du 
 
 dh . . . _ . 
 
 -r- = O ; .'. the equation [4] becomes 
 
 c.$x Q + P x .ty — 0. But the equation [3] is 
 
 cJx, + P t .ty t = 0; 
 
 consequently, if from the equations L = 0, 31=0, we 
 
 eliminate, as before p. 145, $r , <F.r l3 &c. there will result 
 
 from the first ~° — —-— and from the second -~ = 
 ax P t dx x 
 
 c .i dy dir . 
 
 — -p. ; consequently, ~~ = -p- 1 , or the tangent at 
 
] 50 Problems. 
 
 the point of intersection of the cycloid and first curve, is 
 parallel to the tangent, at the intersection of the cycloid 
 and second curve. This is the result which M. Borda 
 first arrived at [Mem. Acad, des Sciences, 176*7, p. 558,] 
 and which was afterwards confirmed by Lagrange, and 
 Legendre. [Misc. Taurin. torn. IV. p. 187; a »d Mem. 
 Acad, des Sciences, IJ&6, p. 30.] 
 
 This completes the solution of the brachystochrone, 
 of which there are six cases, [see pp. 113, 122, 141, 145, 
 147, 14 9-] and this curve which first called the attention 
 of mathematicians to the connected doctrine and calculus, 
 has been also the object of their latest researches. 
 
 Newton, in the Acta Eruditorum, Mai 1697, p. 223 * 
 gave, without proof or the authority of his name, a method 
 of describing the cycloid. But John Bernoulli, from the 
 Work recognised its author : " ex ungue Leonem " f. 
 
 As additional instances of the use of the equation of 
 limits, we will solve the following problems. 
 
 * See also Phil. Trans. No. 224. p. 3S4 : his Works by Horsley, 
 vol. IV. p. 415 : and his Opuscula, vol. I. p. 2S9. 
 
 f Quoique l'auteur de cette construction par un exces de modestie 
 ne se nomme pas, nous savons pourtant indubitablement par plusieurs 
 circumstances que e'est le celebre Newton, et quand meme nous ne le 
 saurions point d'ailleurs, ce seroit assez de le connoitre par un 
 echantillon, comme ex iwgiie Leonem, Joann. Bernoulli, Opera, torn. I. 
 p. 197. 
 
Problems. 151 
 
 PROB. 29. 
 
 Required the conditions for drawing a right line that 
 is to be, the shortest distance, between two given curves. 
 
 1. By Prob, 2. p. 113, the curve was determined 
 to be a straight line; also V — \/{\ + P 2 ), and 
 
 v/(l +p % )' 
 
 Hence, by the form, p. 89, the equation of limits 
 becomes 
 
 Fi j Xj _ V M + P x M n - P .J« - o, 
 or, since $u = $y — pjx [p. 88.], and 
 
 r-Pp- JJT+pry we have 
 
 3x t + P x .iy x - jr^T- * *x +P .iy = o. 
 
 If x ti x Q , &c. are independent of each other, this 
 equation resolves itself into 
 
 If M = 0, and L = 0, are the two equations to the two 
 limiting curves, there results, as before, p. 145. 
 
 g.^,-^0, g£ *.-*.- a. 
 
 Hence, by means of these, and of the two preceding 
 equations, eliminating <$\r,, $x Q , &c. there results 
 
 ^ = -- and ^= - — 
 dx x p x ' d.r Po ' 
 
 and consequently, for the reasons assigned in p. 1 46, tin 
 
 line must intersect each curve at right angles. 
 
159 Problems. 
 
 Prob. 30. 
 
 It is required to investigate the conditions for the 
 curve, down which, a body, in a resisting medium, 
 acquires the greatest velocity. [Seep. 138.] 
 
 We have, p. 104, the equation of limits, 
 = (F+ \W).lx + [V + {¥')] x {iy - pix) + 
 [Z f + (Z')] (Sz - p'lv), 
 in which [see p. 140.] 
 
 y '=°> {r) =jWTfy z = h (2 ' ) = A - 
 
 Hence, the equation of limits is 
 
 = [^( 1 " f ^- 2 5^+ 2 e A v/( 1 +F)] ** + 
 
 and, by reduction, 
 
 and if from p. 139, we substitute the values of 2gA and 2 A^ 
 there results 
 
 - e<to + -^- -f- (1 + a).J« = 0, 
 
 v « 
 
 which is the general expression : and consequently, the 
 two equations at the limits, x , x lt become 
 
 - cix + ^r a -ly + (l + \)>** 9 = °* 
 
Problems. 153 
 
 -.c.rtr, + -7^-^r + (l + *i)'**« = °- 
 
 Now z is the square of the velocity ; and if the initial 
 velocity be given, J« = 0; in the second equation, 
 since $z 9 is indeterminate, make (1 + A x ) ^ = 0, or 
 A, = - 1, and then the two equations are reduced to 
 
 ~ CJX + ^-tyo = O, 
 1 * 
 
 If L = 0, and 31= 0, are the two equations belonging 
 to the limiting curves ; then there results, as before, 
 p. 145. 
 
 %•»' ~ **° = °' 
 
 From these two last, and the two preceding, eliminating 
 Jj? , fy , &c. there results, 
 
 consequently, the tangents at the respective points of 
 intersection of the curve, and the two limiting curves, 
 are parallel to each other. 
 
 The conditions relative to the curve of quickest 
 descent may be similarly determined. The preceding 
 instances are, however, sufficient to explain the use of 
 Lagrange's determinate formulae. 
 
 x 
 
154 Problems. 
 
 The purpose of the present treatise, is now accom- 
 plished. There are, however, several curious results 
 deducible from Euler's formula ; but, to these, since they 
 are not closely connected with the matter of the preced- 
 ing pages, the reader is merely referred*. 
 
 * Lngrange, Fonctions Analy. p. 217, &c. Lecons sur le Calcul 
 des Fonctions, p. 401, &c. Lacroix, Calcul. Diff. &c. p. 660, 667, &c. 
 Lexell. Novi Comm. Pdrop. torn. XV. p. 127, &c. 
 
 THE END. 
 
RETURN CIRCULATION DEPARTMENT 
 
 TO— ^ 202 Main Library 
 
 LOAN PERIOD 1 
 HOME USE 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 
 
 1 -month loans may be renewed by calling 642-3405 
 
 6-month loans may be recharged by bringing books to Circulation Desk 
 
 Renewals and recharges may be made 4 days prior to due date 
 
 DUE AS STAMPED BELOW 
 
 Z 
 
 O 
 
 
 
 xi- 
 
 < 
 
 -a 
 
 
 TX 
 
 -ft- 
 
 LU 
 
 I- 
 
 ^2- 
 
 
 r:r , era . AUG 1 7 1981 
 
 UNIVERSITY OF CALIFORNIA, BERKELEY 
 FORM NO. DD6, 60m, 3/80 BERKELEY, CA 94720 
 
 PS 
 
 1,1)21-20™*: 39 (9269s) 
 
U.C. BERKELEY LIBRARIES / 
 
 CDbl3SED^^ 
 
 
 \ 
 
 V 
 
 V 
 
 i/ 
 
 u-y 
 
 
 
 
 '■ 
 
 ?-■' 
 
 I* 
 j 
 
 
* 
 
 
 
 
 
 ^ t:^ 
 
 ,-^Gl 
 
 
 •^ - i. 
 
 
 -* 
 
 -*-" 
 
 J# 
 
 
 TT 
 
 
 V 
 
 vr*v 
 
 <%>. 
 
 -j 
 
 
 ^ -42