PARTICULARLY THOSE OE TWO VARIABLES. 
The substitution of these in (20) gives 
4H<£ j p ^j x , y j =4^(—l)^i (MlAi+ • • • +ma*)+(n 1 p 1 + ... +n 4 p 4 )}^i{(m 1 w+... +(m 4 w} 
^i{(Ni + <r' i ) a + ... +(N 4 +cr' 4 ) 2 |^(M 1 +tri)X 1 + . . . +(M 4 + t r 4 )X 4 ^(N' 1 +o'i)Y'i+ . . . +(N 4 +a' 4 )Y 4 
r i{(M 1 + <r 1 )(N 1 +«r' 1 )+ . . . +(M 4 + t r 4 )(N 4 +<r' 4 )} 
the summation being taken for al] values of the M’s and N’s defined by the equations, 
i.e., for all integral values which give integral values to the ms and ns. Now the 
difference between any two of the M’s is even, so that they are either all even, or all 
uneven. Taking the first of these cases, let 
M 1 =2M' 1 , M 3 =2M' 3 , M s =2M' 3 , M 4 =2M' 4 
then since 
4 m x — — M 1 d-M 2 +M 3 4-M 4 
and similar expressions hold for 4 m 2 , 4 m 3> 4m 4 it is sufficient that 
M' 1 +M' 3 +M’ 3 +M / 4 = even. 
Taking the second case, let 
M 1 =2M' 1 + 1, M 3 =2M' 3 +1, M 3 =2M' 3 +1, M 4 =2M' 4 +1 
it is sufficient that 
M / 1 +M , 3 +M / 3 +M , 4 = uneven. 
With corresponding quantities substituted for the N’s in the two cases exactly 
similar relations hold. Separate the terms in (21) and denote by 
those in which M' x + • • • -f M' 4 = even, and N'j-f . . . + N' 4 
^ 2 * a ,, ,, — even, ,, ,, 
^ 3 * » >> >> — odd, ., ,, 
^ 4 * >> 5 ? == odd, ,, ,, 
Also write 
^i+^2+^3H _ ^ =a i+A2+A 3 +A 4 =2A / 
and, for shortness, let Q x , Q 3 , Q 3 , Q 4 denote the general terms in 2^, £ 2 , ^s> ^ 4 respec¬ 
tively, so that 
= even; 
= odd; 
= even; 
= odd. 
793 
( 21 ) 
5 i 2 
