GENERATING FUNCTIONS IN THE THEORY OF NUMBERS. 
181 
which involve only the three undetermined quantities 
YiS’ Yin 724' 
From these five equations we can eliminate the three quantities 
713> 7i4> 724j 
and thus obtain two independent relations between the coefficients of V 4 . These are 
the two conditions that the coefficients must satisfy in order that a redundant form 
may be possible. 
Since also these coefficients are the several co-axial minors of the determinant 
1 \ 
we establish the fact that these co-axial minors are connected by two relations or 
syzygies. Thus 
^■4 = 2 ; 
and assuming the satisfaction of these two conditions we can solve the equations so 
as to express 
7135 7i4’ 724 
as functions of the coefificients of V 4 . 
Solving these equations and writing 
P123 — Pm ~ PiPzz ~ PiPn PzPn H“ ^PiPilh^ 
we find 
7i3 — 2^3 (^123 i a/ (P''l23 ’ 
724 — 2 ^^ {^234 i \/ (P''2.84 “ ? 
^3 ~ 2^14 ^^134 i \/ (I^T34 ~ 5 
724 ~ 2 ^ (^124 i \/ (P^124 ~ ^9.l29.2ph'^} ’ 
and assuming these four equations, as well as the fifth ecpiation, consistent, there are 
just two systems of values of 
7i3’ 7]45 724> 
which satisfy all the equations. 
Let the two values of be 
l/cj 3 and I/C 31 , 
