PARTICULARLY THOSE OF TWO VARIABLES. 
795 
Q 1 = general term in 
and so on for the others ; and it follows that the expression of the product theorem is 
given by 
4IK> 
+ ( —1) A ' into 
+ 
— H cl) 
A ,P 
<7 + 
T J 
A+l, P 
CT + 
Y } 
-J- (— l) p ' into 
ra>j 
l 
—n$j 
!c::)+ 
!(r:4M 
— m>j 
+ (__1)A'+F'i nt0 
“{( 
-“{( 
-"*{( 
A , P ' 
cr + 1, cr'-h l y 
A ,P+1 
cr +1, cr' +1 
A + l, P 
cr +1, cr + 1 
+n<P 
/A+1, P +1 
\o- +1, cr' + l 
X, y} 
>x,y} 
|X,y' 
iX,y}. 
(23). 
7. This product theorem does not comprise 16 4 equations, as might be expected; in 
defining A, P, cr, cr' it was assumed that the sums of the four Vs, of the four /x’s, of the 
four p’s, and of the four vs, were each even. Thus when X L , X 2 , X 3 , are known, being 
limited to the values zero and unity, is also known; and similarly for the other 
numbers. Hence the number of equations comprised is 16 s , i.e., is 4096. 
8. If any uniform increase or decrease be made in a set of corresponding numbers 
