GE^TERATIN^G FUN'CTIONS IN THE THEORY OF NUMBERS. 
140 
Taking always the positive sign, let Cj.,j ^ be the value deduced for y^-y. 
Then 
^xy — tOy/^.ry) 
and 
yxxljxyjyz — CxyCy^jCxz- 
Hence, the (^) equations are all satisfied by the system 
Jxy — C.ry 
Similarly, tlmy are all satisfied by the system 
where 
Jxy — Cyx ^ 
Cyx - (^yxj^yx' 
Art. 54. To show that each of these systems satisfies the remaining equations, it 
suffices to consider the typical determinant equation of the fourth order. 
We have— 
^^xx ^^xy^yx ' ^^xz^^zx 
7« 
7^y 7y-- 
a 
'yy 
1 
'^xy yyz 
7« 
^xy yyz yzw ^yz ^zw 
’Hyio 
ClyzO^zy 
cn 
Ixy Jyz Jz,o 
^xir^^ir.z 
lyz Izw 
^zw^wz 
^XV 3 
On the left-hand side put 
and the determinant becomes 
— 1 ^xx ^yy ^^zz ] • 
yxy — Cxy ^ — hxyjttxyi 
axx 
Ctxy(^yx 
^^xy^^yz ^^zx 
Cixy (^yz ^^XW^mX 
1 
^^yy 
Uy: O-zy 
Ctyx (^zw ^^ivy 
^xy ^^yz 
1 
(■('ZZ 
CtgiijCtiug 
Clxw 
CCyiQ 
1 
Ctwio 
Ctxij Oxyz ^zw ^ZIO 
In succession, multiply the first column by a^y, divide the first row by a^y', multiply 
