74 



Proceedings of the Royal Irish Acaclcmi/. 



If we now write 



A = A + pU,^ + ^ ^ + ^ .-, „ + 



&c.. 



c^"«F= F + pQ„r + 



p"-Q,:Y p'Q,'Y 



+ ... &c. 



1.2 1.2. 



we have, on multiplying to^^A' by c^oY, 

 {cpD„Xj [c^QoY] = ZF + p(rQ„A' + XGo^O 



+ -^ ( FQo'.-r + '2Q,J:QoY + AQiY) + &c., . . 



= (.YF) + pQ,{Xl^ + -^-Q/(Xr)f, &c., ... 



where JT and F are any two functions whatever ; hence we see that 

 (25) {<f^A') (cf^Y) = C^'o^A'Y). 



And if we now let p = - yr> it follows that we have 

 B,'c ^' A,- = - (AJiX 



(26) 



B,'e (A^, - ^.') = (A,B,) {A,B,) - (A,B,y, 



A, 



Bi 



no 



£.'« - A,' = {AJi,)\ 



Bi 



flo 



&C., 



B,'c A,A,^iA,B,)(A,B,). 



- — flo 

 I B,c "' A, = (A^,). 



Wo ai-e consequently led to the following value of JE(A,, B,) : — 



(27) E (A„ B,) = (A^,y + (A^^y {A,B,) + (AyB.y (A,Bo) 



+ (A,B,y(A^,) -2(A,B„){A^0(^2B,:) 

 - {A,B,){A,B,)(A,B,) - {A,B,){A^^){A^B,). 



Tlic six functions {AtUi){AiBi), &c., are not independent, but are connected 

 by the relations 



(28) {A^B,) (A^B,) - (A,B,) {A,B,) + {A^,) {A,B,) = 0. 



The value of E(Aj, B,) should be such that when operated upon by 8 or A, 

 as defined in the earlier part of this paper, it should vanish identically ; and 

 this we find to be the case which the reader can easily verify for himself. 



