1SOPERIMETRICAL PROBLEMS. 



SE, (Fig. *.) being drawn, let the extreme points BE of the three properties, 



323 



(A) that of the maxi- Isopmme- 



Fro ;..-,.%. 



Fig *. 



1 \ 



P \ B 



E 



remain fixed, while C, D, vary along the lines QC, RD, 

 by the infinitely small elements or variations Cc, D d. 

 Then, to determine their positions, or the relation be- 

 tween the differentials dy,dx, the first condition is, 



t h ;i t 



BC + CD + DE = Constant, 

 and is therefore equal to its consecutive value, 



Be + cd + dE, 



which, treated analytically, leads to one relation be- 

 tween the variables C c, D d, or to an equation of the 



P. C c + Q. D d = 0. 



The maximum or minimum property affords *nother 

 condition, by which the mm of He tlrmtnli of the inte- 

 gral yVrfz exprewing that property, u itself to be 

 made a maximum or minimum. This must be treated 

 to as to afford another relation of the same form 



R.C c+ S. l)rf = o, 



between which, and the foregoing eliminating one of 

 the variations Dd, we get 



(PS QR). Cc = 0,orPS QR = 0, 

 Cc being arbitrary and independent. A similar rea- 

 soning was aAerwards applied by Euler to the varia- 

 tion of three, or any number of onlinates to satisfy as 

 many condition* ; but to such questions the Demoullis 

 did not extend their researches. Thus, in Fig. 3. each 



mum, and (B) and (C), the two common properties, 

 affords an equation of the above form between the 

 three variations Cc, Dd, Ee; from which, eliminating 

 any two of them, the co-efficient of the other, put equal 

 to zero, gives the differential equation of the curve. 



Tin's is, in fact, the identical process by which an or- 

 dinary question of maxima and minima would be treat- 

 ed, in the case where a certain function P of two (or 

 any number) of variables u and v is to be made a maxi- 

 mum, the variables being related to each other by n 

 given equation Q = 0. The differentiation of the lat- 

 ter assigns one relation between their variations, viz. 



dQ >- rfQ ^ = o, 



tru.,1 

 Piobknif. 



-r- 



tin 



- 

 dv 



while the condition of the maximum gives 3 P =r 0, or 



from which eliminating one of the arbitrary variations, 

 the resulting equation will have the other for a divisor; 

 and the remaining factor put equal to zero, gives that 

 equation which holds good between the variables only 

 in the case of the maximum or minimum. These va- 

 riables, in the present case, are the consecutive ordi- 

 nates y', y", &c. of the curve sought, and thus we see 

 the reason why the final equation so obtained is a dif- 

 ferential equation of the curve. 



Such was the state, however rudely expressed in the 

 symbolic language and mixed geometry of those days, 



, 



in which the problem was left by James Bernoulli, 

 (in his Analytit ma&u problematic Itojirrimetrici, I.eip. 

 tic act*, 1701) ; and although his brother, in a Memoir 

 published by the French Academy in 1718, exhibited it 

 in a more compact and elegant form, and thence took oc- 

 casion, most unfairly, to arrogate to himself a large share 

 of its merit, .such was nearly the state in which it was 

 found by Euler in 1733. It is true, that John Ber- 

 noulli, in this Memoir, had remarked a certain symme- 

 try in the terms of his fundamental equations, which, 

 pursued, would have led him to anticipate one of I'.u- 

 ler's most elegant and general conclusions, hut he ap- 

 pears (as well as some subsequent writers who have 

 given an account of his labours,) to have confounded 

 tliii with a thing of the same kind which obtains in his 

 final, or i/ieci/ic equations, as he calls them, in virtue of 

 which they are complete differentials, (of which we 

 have an instance in the equation (a) of our first solu- 

 tion of the brachystochrone,) and which is not univer- 

 sal, or of any very extensive ue in the theory of these 

 problems. Our countryman Brook Taylor, too, in hia 

 Mrthodui Incrrmextomm (1715). considered the sub- 

 ject, and though he added little to the stock of know- 

 ledge, and nothing to that of facility or distinctness, 

 yet he there first employed a general mode of repre- 

 senting the maximum property, viz. byJ'Vdx, where 

 he takes dV Mt/r + X<ly + Ldt, and thus may 

 be considered as having afforded the first handle to a 

 general and systematic analysis. 



The progress of invention had hitherto been tardy. 

 and that of generalization next to nothing, when the 

 subject was resumed by Euler, in a series of Memoirs 

 in the Petropolitan Commentaries, and in a work ex- 

 pressly on the subject, entitled Methodui invent fnili U. 

 neat extra* maximi minimive proprietalc gaudentet, 

 (1744). To whatever part of the mathematics this 

 wonderful man turned his attention, obscurity seems 

 to have fled his presence. It is impossible to exhibit a 

 more laminous view of the principles of the subject 



