324 



ISOPERIMETRICAL PROBLEMS. 



I.pitne- than he has taken in this latter work. In his Memoirs, 

 irical (.lead. Pelropol. torn. vi. vii. and viii.) the novelty of 

 u the course he was pursuing, as well as the superfluous 

 "~~Y~-' difficulties which usually hang on the origin of inven- 

 tion, had betrayed him into some errors, arising chiefly 

 from too extensive an application of the principle of 

 James Bernoulli, which makes the maximum property 

 common to the curve and its elementary portions ; but 

 these he speedily rectified : and a sketch of his method, 

 while it will put the solution of all ordinary isoperime- 

 trical problems in the reader's power, will serve as the 

 best practical introduction to the calculus of variations, 

 which (however closely connected with the subject in 

 an historical point of view) we cannot help regarding 

 as a great deal too abstract to-be made the basis of an 

 explanation of its principles' without .some preparation 

 of the kind. , 



Isoperimetrical problems, then, are distributed by 

 Euler into classes, according to the number of proper. 

 ties they involve, or of ordinates, which must be made 

 to vary in their analysis. The first class consists of 

 absolute maxima and minima, where one integral Cv d x 

 is to be made a maximum or minimum, between limits 

 any how determined. The rest, where, besides this 

 condition, others are superadded, viz. that certain other 

 integrals /*W d x,C7, d x, &c. are to be given, are call- 

 ed, in general, questions of relative maxima or minima. 

 To begin with the former, to which, as we shall soon 

 see, all the rest are reducible. 



PROP. 



If CVA'x. be an absolute maximum or minimum for 



the whole curve, it mill be so for every infinitely small 



part of it, provided V be some determinate function of 



d y d 1 y d 3 y 



x > y P = -> q = "*"' r = 



ing invariable, so as to makeqne element mo of the Isoperime 



and .become tr ' ca ' 



Fig. 4. 



curve in the middle of the series vatf, 

 mio. Let V,, V,, &c. p,, &c. q,, &cTbe the conse- 

 cutive corresponding values of V, p, q, &c. To fix 

 our ideas, suppose n =r 2. Our series of ordinates will 



then be 



and we may suppose 



sup 

 =M 



Qdq. 



If we now conceive the integral J"V d x resolved into 

 its elements, thus, 



"ax* 



' &c< invohin s 



y* 



i . 



of which let the middle one y n vary by the quantity 

 n ' = %& ( Fl S- *) ^ thos e on each side of it remain. 



integral expressions. 



Whatever be the order (n) of the highest differen. 

 tial co-efficient contained in V, imagine twice that num. 

 ber of ordinates erected at equal distances (dx) from y 

 and from each other, so as to form a series of ordinates 



we must first inquire which of these elements are af- 

 fected by the variation of the element mno of the 

 curve. Now, we have, 



1st, 3x=o, )y = o, Jy, = o, Jy >= H(=:), tyj=o, &c. 



2dly> P=i(y -?>'*. =^(y-y.).p=^(*i-*0. 



&c. 



so that the only values of p, which are affected by the 

 variation of.y,, (being those which contain y,), are p, 

 and these give 



-. , = 



dx dx dx 



In like manner, the only values ofqf = -\ which 



vary, are those whose expressions in terms of three con- 

 secutive ordinates, contain the variable one y lt that is, 



dx* 

 give 



~ dx*' *' 



and, in the same way, the variations of the higher dif- 

 ferential co-efficients might be found, the numerical 

 co-efficients being those of a binomial 1 1 raised to 

 the successive powers 1, 2, 3, &c. 



Hence V, V,, V,, (or ), being the only values of 

 V which contain these quantities, are the only ones af- 

 fected by the variation of the ordinate y^, so that the 

 only portion of the integral fV d x, which varies by 

 the variation of m n o is that consisting of the elements 



Representing then the whole integral by 

 A 



B, 



A and B are invariable, and the condition of the maxi- 

 mum cannot therefore be satisfied unless the nature of 

 the curve be such as to render this elementary portion 

 a maximum. 



A train of reasoning precisely similar, only longer in 

 its details, is applicable to the -higher values of n. 

 When, however, V involves an integral, as, for instance, 

 the arc s rr fdx "/I -\- p 2 , since every succeeding va- 

 lue of this is affected by a variation in any part of its 

 extent, it will thus affect every succeeding value of V ; 

 and' the rest of the integral represented above by B, not 

 being invariable, the principle of James Bernoulli ceas- 

 es to be true in this case. Still less can it be extend- 

 ed to cases where V involves a quantity not given, but 

 implicitly, by an unintegrable differential equation, as 

 in the problem to Jind the curve of sniiflesl descent in a 

 resisting medium, to which Euler himself erroneously ap- 



