TRANSACTIONS OF SECTION A. 599 



Tariation can only differ infinitely little from the value we obtain by neglecting all 

 the variations but that of the highest differential coefficient of y in the integral. It 

 is necessary to justify this reasoning by inquiring into the conditions of continuity 

 which must be satisfied by the variations. The conditions are not explicitly given 

 in the statement of the problem, but are implied in the method of obtaining and 

 reducing the first variation. The problem fully stated is, to make an integral greater 

 than any other integral which can be derived from it by a change in which the 

 variations of all the dependent variables and their fluxions appearing in the 

 integral are infinitely small, and all but the hiyhest Jluxions of the variables are 

 continuous. 



The application of this principle to II. is as follows. Suppo.se it is required to 

 make X = F(U, V) a maximum, where 



with similar expressions for V. Then, S"X. being the second variation of X 



in this expression hu includes such terms as ~-M'-''\ and taking an infinitely short 



range of integTation, we have proved that we may neglect dy'''> in comparison with 

 %"'', where r<n. Hence we retain only the terms 



Now since the range of integration is infinitely small --^^^^di/"^d.v is infinitely 



small compared to [fij/W^di, the former being of the order V'''(^'i--'''o)^ ^^^^ ^^i® 

 latter of the order di/<^"^'\a.\ -.fo), -''i and .Iq being the limits of integration. Hence 

 we need only retain the terms 



d¥( d-a 55 ,„.o, dF(d-v s„f„,o,^ 



the sign of which wheu the integration is small is evidently the same as that of 

 dF d-u dF d-v 

 7/0 dy<-")' "*" TV dy"-"'-' 



Therefore, for a sufiiciently short range of integration X is a maximum or mini- 

 mum, according as this quantity is negative or positive. 



This result can be extended so as to apply to any case, however complicated. 



The simplest case of problem III. is to make Ua maximum subject to V = 0, U and 

 V having the meanings above given. The ordinary method of obtaining the equa- 

 tion giving y in terms of x is to equate S(U + /xU) to zero, fi being determined from 

 the condition V = (J. A process just similar to that employed in II. leads to the 

 result that, when the range of integration is small, the integral U is a maximum or 



a minimum according as the sign of -r^, + M -,,-., is negative or positive. The 



dy^">- dy'-'"- 



general result is similar in character. 



The simplest case of problem IV. is to make U a maximum subject to the con- 

 dition v = o where 



u4*.j/(.,„.|.|,...g,-).» 



and D is a function similar to u. To find y and s as functions of x, the ordinary 

 method is to equate Ubu + fiSv^dx to cipher. By this means the value of /^ is 



