ISOPERIMETRICAL PROBLEMS. 



35 



lniiliin plied it, ( in torn. vii. Comm. Petrop. and in his Meeha- 

 nia.) In relative maxima and minima, however, it 

 may happen, that the integral involved in V may be 

 the' very one which is supposed the same in all the 

 curves ; and its succeeding values remaining in conse- 

 quence unchanged, the foregoing reasoning will still 

 apply. Thus, if the condition of equal length be su- 

 peradded, the part BCDE (Fig. 2.) being equal to 

 IJt-rfE, any succeeding arc all will be the same, whe- 

 ther it coii-i-t of aBCDEH, or aBcJEH; and the suc- 

 ceeding value of t being unaffected by the variation of 

 the element BE, B is now invariable, so that here Ber- 

 noulli's principle holds good, even when V is a func- 

 tion of *, as in- the second case of his own programma. 



PROP. 



The tame supposition retpecting V being made, to de- 

 termine the relation between y and x, tekic/i reiulert 



J 



We hare here to make the element of the integral 

 maximum ; or, which comes to the same thing, the 

 variation of the whole' integral due to that of one ordi. 

 note eajnal to zero ; that is, (in the case we have been 



considering), 



) 



{\ 



or, 



JV, +>V, =ro. 



Now, we have 



whatever be the variation* of *, y, p, q, (being infi- 

 nitely small.) Substituting, then, for ) r, ) y, Ip, 



J q , and their consecutive values, the expressions before 

 Mid, we get 



V = +o 



3V,= -r-P.^-Q ' 



JV > = X,,_P.^ + Qi _L_ ; 



and the sura of these put equal to tero gives 



_P.-P. Q.-gQ.+Q) , = c 

 1 ' d x d x I 



or, dividing ofl the arbitrary variation )y, or , 

 P. P. = </P,, Q, 2Q, + Q = 



and 



In this equation, we may now write X and P for 

 N, and P,, from which they differ only by infinitesi- 

 mals, when we obtain 



Had we considered differential co-efficients of a higher 

 order, by vary ing y,, or^ 4 , See. or at once analysed 

 the genera] case, the process would have been precisely 

 similar, and we should have arrived at the equation 



- 



Problems- 



to resolve any problem of absolute maxima, however Isoperime- 

 complicated, where V involves no integral sign. As trlcal 

 an example of the manner of its application, we will 

 take the solid of least resistance of Newton. 

 The resistance being represented by 



v _ 



Ndj+Prfjh 

 Consequently, M=0, N - 



Now, the equation of the minimum ( A) gives 

 which, combined with 



gives, by eliminating Nrfy, d V=P dp+pd P=d (Pp), 

 whence, V = Pp+c (B) 



that is, by substituting for V and P their values, and 

 reducing 



c c ep 

 *~ ~ *p s ~~ p ~~ 2' 



Hence, since d* = , or x = I--, we find 



P J P 



Sc 



and the equation between r and y results from the eli- 

 mination ofp between these two. The equation (B) is 

 of considerable use in ordinary problems of this kind, 

 and is generally applicable when V is a function only 

 of y and />, affording immediately one integral of the 

 equation (A.) 



PROF. 



Required the nature of a curve which th all give IV dx 

 a greater or leu value than any other curve in tuhich that 

 of I W d x it the tame. 



Here two ordinatcs must be made to vary ; but their 

 variations being independent, and infinitely small, the 

 variations they cause in the two integrals may be com- 

 puted separately, and their sum taken for the joint ef- 

 fect. Now, let > and be the variations of the two 

 consecutive ordinates (ty and Jy,), and let P )y or P 

 be the variation produced in f\ dxby the variation of 

 y, then, by reason of the uniformity which must sub. 

 list between all operations which relate to the conse- 

 cutive points of the same curve, P, Jy ,, or P,. will 

 be the variation produced in it by the variation of the 

 consecutive ordinate, and therefore P.+P,.* will be 

 the whole variation ofJ'V dx owing to the simulta- 

 neous change f both ordinates. In like manner, if Q. i 

 be the variation of f*V> dx, caused by that of y, 

 Q + Qi * wll be the joint effect of those of y and y,, 

 The condition of maximum then gives 



P..+P,..=0, or P 



which ia the general formula of Euler. It enable* us And the invariability ofjW d x affords another equa. 



