386 Proceedings of the Royal Irish Academy. 



Therefore, adding, 



I{ABCP) - 1{AP) =[ Edx + % {Jc),. (19) 



Similarly, we may show 



I {DEAF) - I{BP) 



E 



JEdx + 2 (Jc\. (20) 



Adding (19) and (20), and cancelling I{AP) wMcli appears in 

 both, we get 



I^ODEABCP) - I{OBP) = liPEABCP - I{BP) 



E 



Edx 



D 



C 



Edx^l,(Jc\ (21) 



B 



where the term 2 {h)^ may become of the order (Jc), i.e. linear in h, as 

 is evident, because some at least of the terms in (Jc)^ arose from terms 

 linear in Tc being multiplied by Dx^ so that when we take their sum, 

 we get an integral linear in h. Hence replacing, in (21), 2 (^)2 by (^)i, 



/( OBEABCP) - /( OP) = 2 [ Edx + (^)i, (22) 



where I {OP) may be regarded either as the integral taken along the 

 stationary solution, or along ODP, as these integrals only differ one 

 from the other by quantities of the order (^)2. 



From (21) or (22), it is evident that the condition Edx > for all 

 values of x in the integration, whatever the values of y^ and 1/2 is 

 sufficient, provided only that Z; be sufficiently small. 



§ 9. The general method is as follows : — 

 Let the integjal be 



... Fdxidxodxz . . . 



Let us write ds for dxidx^dx^,, . . . and consider first the case where 

 the integration with regard to ds extends over two portions, one repre- 

 sented by %, where aU the variations are small, and the other represented 

 by 0-, where some are finite, as in fig. 5. Then we may write 



/=[[[... Fdx^dxodx^ . . . = [ Fds = pi^rfS + V Fdxx. 



Let rJso the general variation 8 be replaced by A + 8', where A 

 refers only to the variations which may be large, and S' to those which 



