PARTICULARLY THOSE OF TWO VARIABLES. 
845 
^A ^0 ^5 +^1 A 1^10^15 : 
$A $4 &- + $iAA Al5 : 
^AAA) "1"A MlA : 
^wVl ~\~@3 AAA : 
#3^8 ^1(A + ^4 ^15^13^6 : 
$A ^1(A +^6 ^15^13^ : 
^0^15^1 A H"^3 ^lA ^13 : 
Vl Vl5+^2 ^3 AA.S : 
^(A AA H - ^7 AAA : 
$(A AA H~ ^7 ^5 AA : 
^(A ^3 A + AAA A . 3 : 
$(A A A + AA4A A.0 : 
: 0 3 0 q S -% S-rf - f - 0 $ A Al 3 
: 03 @2 A A + A A A 2 A 3 
A A A A A Ao A A Al 
: A A A A “1“ Al A A Ao 
: 0 2 A A Al + A A 4 A 2 A 
: 0 Q 0g S-% Al -f- 0rj A4A2A 
A A A A A A AoA Al 
A A A A + AoAlA A 
: A A A Ao+A 0 i5 A & 13 
: A A A Ao+A A AfiAl3 
: A A 2 A 1 A 0 AA A A A 
: A 5 A 2 A A A A @1 A A 
and many of a similar form. 
57. If there be four pairs of arguments x 1} y la x 2 , y 2 , x 3 , y 3 , x 4 , y 4 , such that 
x i+%2+ x 3+x4=o=yi+y-2+!/3+y4 
then with the notation of the first section we have 
Xi-f-a?!—X 3 -|-a?2—Xg-J-a’g—X 4 fi-a: 4 =0 
Xi+yi=Y 2 +2/a= Y 3 +?/ 3 =Y 4 +y 4 = 0 
and the product theorem (23) will give results similar to (205), (206), (207). As 
examples, if we put 
n=$(x 1} yj)S(x 8, y 2 )S(x 3 , y 3 )$(x 4 , y 4 ) 
the equations corresponding to (205) will be 
n 0 + n 6 + ni 4 = n l + n 0 + n 19 
n 0 + n i 0 + n i 3 = n 3 + n 9 + n i 2 
and so on ; and if 
n /= A (« i 5 Vi ) H ®$ | i ) A (%- y %) H x ^ 2 / 4 ) 
the first equations of the sets corresponding to (206), (207) will he respectively 
n 5 °+ n 7 2 + n 6 3 + n / = n 0 s + n a 7 + n 3 6 + n ^ 
n 2 8 + n 6 12 + n 4 14 + n 0 10 = n 8 2 + n 13 6 4 - n 14 4 + n 10 0 
