382 Proceedings of the Royal Irish Academy. 



where {h)^ is used to represent any function involving h in the second 

 degree at least. 



Since the variations of the lower fluxions in AB are small, i.e. less 

 than k, we may write 



j"" FQ„y.:)I)x = j'' FBx+iJc),, 



where Fh what F^y-^y^) becomes, when for y'"-'^ and the Tiiylier fluxions 

 only we write ^^"^ and the corresponding fluxions ; in other words, 

 ^is the i^-f unction which appears in (3). 



^^'^''^ I[AB) = j^ FLx + (X-)2. 



Again, if S and S' represent the small variations by which we pass 

 from the stationary solution to AP and BP, respectively, we may write 



I{AP) = { {F+ SF) dx 4- (Jc)^, I{BP) = [ (i^+ S'F) dx + {k),. 



Hence we may wi'ite 



I{ABP) - I{AP) = /^ FBx + [ {F+ h'F) dx 



- [ {F+ 8F) dx + {Ic)o. 

 Ifwewi'ite 8" = S' - S, (10) 



we may simplify the above expression by writing 



{F + S'F) dx-[ {F+ SF) dx = \ {I'F - IF) dx - f {F-^ 8F)dx 



J B ^ A J B J A 



rP lA 



= h"F- FDx + {k),. 



Writing I{ABP) - I{AP) as I(ABPA), we thus obtain 

 I{PABP) = 1^ {F-F)I)x+ r l"Fdx + (Z;)2. 



Since the solution OP is a stationary one, the integral of h"Fdx 

 depends only on the limiting variations, and we get, according to the 

 usual theory, 

 I{PABP) 



= j^ (F- F) Bx + /^ (i Y,,,S"y,<"i-'-^ + (, Y,„^_,, - , t^)S"y,i"r^'-> + &c.) 



+ r (, F„,8'V.("^-^) + (. r,.,.,) - , r;, J s'v,(»2-=^) + &c.), ( n ) 



