PARTICULARLY THOSE OP TWO VARIABLES. 
851 
Separate now tlie general term in (227) into parts corresponding to tlie particular 
cases of the values of the M’s (i.e., whether they are even or uneven), and denote them 
as follows:— 
2 0 when all the M’s are even, and general term ,P 0 : 
when all the M’s except the M/s are even, and general term P*: 
2 S|i when all the M’s except the M s , M* are even, and general term P M : 
and so on ; also let 
t—r 
S t t = sum of all the terms which have one set of M’s uneven and all the rest even, 
4=i 
s=r t=r 
S S 2^=sum of all the terms which have two sets of M’s uneven and all the rest 
*=14=1 
even, 
and so on; making the number of distinct terms on the right-hand side 
1 in which no sets of M are uneven 
-r in which one set is uneven 
’— 1 
in which two sets are uneven 
r.r — 1 
2! 
r.r — l.r—2 . 
3! 
+ 1 
viz.: =2 r in all; and then 
in which three sets are uneven 
in which all the sets are uneven 
$^^=S 0 P 0 +sVlW+s'sVl)^%ft4+. (228). 
4=1 *=14=1 
In this 
^ —y 
^ |{(2n 1 +N 1 ) 3 +. . . + (2^'" 1 +N'" 1 )4 j{(2/x,-+N,) + . . . + (2 / j L "V+N'",.)4 
Pq —(—Pi --'Vr 
Pi ^{ (2 ^ +1T l )( 2^ +N * )+ - • •+ ( 2^"' l +N Wi) (2,4'" 2 +N'" !! >} > 
^ i (2/x 1 +N 1 )X 1 +. . .+(2/i'",+N'" 1 )X'" I ^ ^ ^ ^(2 Mr +N,)X,+ . . .+(2 fi '" r +X'",)X"V 
jy _/ \ 2 n'Wt) l{(2 Ml +N 1 )a+ . .. +(2^'',+N"' 1 )2} i{(2 Mi +l+N,) 2 +. . .+$%"'<+1+N'",)®} 
fs,t— l)*- 1 Pi • • • Pt 
2 flt +l+N I ) 2 +. . .+(2/x ",+1+N"',) 2 } p i{(V+Nr) 2 +. . .+(2(U,"V+H w r) 2 } t 
p i ^{(2 f ti+N 1 )(2 f A ( +l+N. .+(2^" 1 +N« 1 )(2 M "' ( +1+N"'<)}^ ^{(2 / x,+ 1+N.)(2 / x,+ 1+N ( )+. . .+(2^"'.+1+N"',)(V"«+1+N'",)} 
^(2jot 1 +Ni)X 1 +...+(2 jU ."' 1 +N" 1 )X% > ^ . ^(2^+l+N,)X.+. . .+(2^",+ l+N'".)X'". 
^(2^+l+N,)Xt+.. .+ (2/x l "t +1+N'"()X"'i^ v (2iJ. r +K)^ r +. . .+(2^" 1 .+N" r )X"V 
and similarly for the others. Taking the terms in (228) separately we have 
