PARTICULARLY THOSE OF TWO VARIABLES. 
853 
where in the third and fourth terms s is not to take the value t ; in the fifth term 
neither s nor u may take the value t, and so on ; and all restrictions on the values of 
the ju/s have been removed. In the expression the number of negative terms will be 
i.e., half the terms will be affected with a negative sign. Now 
P t =general term in 11$ 
(-1 )>•!>,= „ „ n<i> 
and so for the others : thus 
Ap Ag, . 
■M Aj ,. . 
Np Ng, . 
Ap Ag, . 
. . , A,+l, .. 
• » A i , .. ., A,. 
NpHg, . 
. ■, N s ,.. 
. , N ( +l, • ■ •, 
.,x. 
f / Ap Ag, . 
• • j A t ,. . 
1 • 5 A Ayr yr 
. . . , Xrl 
LWp • 
.., HV+1,. 
. . , N,/ 1 35 
* • * ? I 
, J/Ap A 3 , . . ., Aj + l, . . . , A^ , . . ., A r 
.fi \\X V Ng, ..., N, ,. . ., ^+1,. . . ,N, 
X 3 
s=r u=r 
+ t 2±im> 
s=l u= 1 
J /A-u A 3 , 
1 Wm N 3 , 
. ., A s +1,..., A# ,. . ., A„+l,. 
• • * Nj , . .. , +1, .. . . } . 
. , X 
> A 
, tf, 
x„ x* 
+ 
X,} 
. (232) 
an equal number (2 ?,_1 ) of terms being affected with a positive and with a negative 
sign; the rule by which it is applied will shortly be pointed out. Similarly 
■ ■ ■ *-} 
. J /A x> A 2 , . . . , Ajt + l, ... ? A, , ..., A t , , AA-y -y -y- 
+ A ±n4> lK, sr„,.., N. .N.+l,.... N,+l,..., nJ X * ’ x - 
I , yri J /Ap Ag, . . . , A K + 1, . . . , A/+ 1, . . . , A s ) * • . ) A l ) . . . ) A, 
\\ Xv n 2 ,... m K ,,x, ,...,n,+i,..n,+i,...,n 
X v X 2 ,.. ., X r j + . . . . (233) 
half the terms, as in (232), being affected with a negative sign. 
Proceeding in this manner, (228) gives 
2 > $$'$"$ w = sum of 4 r products of four functions $ . . . (234). 
