THE lOTES SYMBOL O' THE CALCULUS OE OPERATIONS. 
19 
= (VV.I«,/3.) 
(W.IaA) 1 ^ 13 ) 
(VV.I«AI (VV,Ia./3.) X«/3) 
(VV,X«AX (VV.X^AX (vv.x^fi) 
- (W,VI«AI«A) I«/3) 
(w.vi«.AiaAI (vv,l«./3.) 
(VV,Vla,ftIa.AI (VV.I«/3) 
- (WJaA) I«/3) 
(w,X<^AX (VV,VX<‘AX‘‘A) I<«/3) 
(VV.Xi^AX (VV,VX«,/33X«.|3.X (VV,X«(3) 
- (vv,X“ 3 ft) Xa,/ 3 .) X“.AX»/ 3 ) 
(vv.XaAX (VV,X«./ 3 i) X“AX“=/ 3 ) 
I (VV,X«AX (VV,X<«,/ 3 .X (vv.vx«AX“/ 3 ) 
Transposing the columns of the third of these determinants, the sign becomes changed, 
and the sum 
(w,X«A) 
X'^A) 
X«A) 
I«/3) 
(W,X“AX 
(vv.XccA) 
X^A) 
Xaf3) 
{W,X“AX 
(vv,X«AX 
(vv.X'^A) 
(VV,X“J3,X 
(VV,X«AX 
(VV,I»,/3,X 
(VV:Xccl3) 
And as the above process is perfectly general in principle, we may conclude the general 
expression 
(VV.X«,/3,X (VV.X«„-./3„_,) .. X«./3.) 
' (WjaAX (VV,Xa„-./3„_.X . . (VV.Xa,i3,) 
In order to alculate the effect of the operation Vi, and so that of the operation 
(a/j . )" upon a given function, let 
Then 
N ow 
= 2 {(«-?■+ 1 >/,_ , + (f + 1 )«,.+ , ) x ’'-' y '. 
