GEN^ERATING FUN-CTIONS IN THE THEORY OF NHMBERS. 
143 
Let be the value of obtained by always taking positive signs and that 
value obtained by always taking negative signs. 
We have the system and the system There are thus two representa¬ 
tions of the redundant form, each involving n — 1 undetermined quantities. 
Art. 49. Given a redundant form of order n, involving the matrix 
I I > 
we may exhibit its two representations, each involving n — 1 undetermined quantities. 
The coelEcients of the condensed form now necessarily satisfy the proper conditions, 
and passing through the condensed form we must, in the matrix of Art. 48, write 
~~ j ^xx^yy | — (-^xy^-^yxi ’ 
and then it only remains to find the values of c^y and Cy^ in terms of the elements of 
the determinant 
I «. 
Solving the quadratic equation 
'^xy 7yj; 
1 a 
Jx,, yy= 
'yy 
1 
CtyzCl-y 
a. 
^xz 
— I ^^xx^yy^zz \ ? 
transformed from Art. 48, we find 
^^^xy^yz^zx 4“ ^ys^zy^x^ i i^^xy^yz^^zx ^^yx^^zy^^'x^ 
'yxy '^yz 
2a^Mzx 
or taking the positive sign 
Jxz 
Clxy CCjfz 
Jxy Jyz 
CCxz ’ 
and taking the negative sign 
Jxz 
^yx 
yxy Jyz 
Hence, if be the value of y^y deduced by always taking positive signs and 
<^yx~^ that value arising from the negative signs, we find 
