384 Proceedings of the Royal Irish Academy. 



The remaining terms at the ^-limit are 



/^(,F„^8'y"i-^^ + {Xn-^) - ^Yn,)^y''^-^' + &c.), (17) 

 where the last term implied by the &c. is 



with corresponding terms for yo. 



Hence the right-hand side of (11) is expressed as 



/ {F-F)Dx 



plus (16) plus (17) plus the terms in ^2 corresponding to (17). 



But, by taking 8y = 0, i.e. by taking A on OF, and by adding to 

 the left hand of (11) the terms 



I{OA) - I{OA), 



we see at once that either side of (11) is 



I{OA£F)-I{OF), 



A being on OP. But, in that case, the con- 

 ditions of a permissible Tariation are evidently 

 fulfilled by OAFF, so that if the integral 

 along the stationary solution be a true minimum for such variations, 

 we must have the right-hand side of (11) always positive, whatever 

 be the relative magnitude of Fx, S"yi^"T^\ Sec, the arbitrary quantities 

 which appear in (11). JSTow the variations which appear in (17), which 

 has been shown to be part of right-hand side of (11), can obviously 

 have either sign, and therefore an expression containing them cannot 

 be always positive, unless the coefficients of each of these variations 

 vanishes; and since F is an arbitrary point, these coefficients must 

 vanish for every point on the curve. Hence we at once get the series 

 of equations 



,Y„^^0, iF(„^_„-0.. ., iF(„^,i)-0,i (18) 



with corresponding identities for ^2- 



^ These conditions follow immediately from the fact that a variation which shall 

 be zero from to A, and have at A small arbitrary values of Sp''"-\ is a permissible 

 one. For the integral is then an integral from A to Pwith arbitrary values for the 

 limits of the y^'^'> fluxions at A. 







