GENERATING FUNCTIONS IN THE THEORY OF NUMBERS. 
117 
contains either + *^4.2, • • • ; viz., that every term contains an x with a suffix 
that does not occur in the s-procluct 
SyS .2 . . . 6 / ; 
for visibly the fraction contains neither 
Nf]5 + • • • nor s,i ; 
or, the same thing, the quantities s, occurring in the product 
. ■ . S/, 
are the only ones that are found in the fraction, the determinant should therefore 
vanish by putting 
^i+l — 'T^ + 2 — • • • — H • 
The determinant is 
hn'r.-), 
— CaXy 
~~~ h(X .2 
putting 
the first row is 
t^X/, tiXj, 
Xl^y — ^(' + 2 ^ • - Xu = 0 , 
^^2^2 ~1~ d” ■ ■ ' ®2^i’ ti^Xyi , . . rqx'j, 
and adding together, Xy times the ffi’st element, x^ times the second, . . . , &c., Xt times 
the element, we obtain zero. 
A similar operation, performed on the elements of all the other rows, likewise 
results in zero. 
Hence the determinant vanishes on the supposition 
^t+y — ^/+2) — • • • — — d, 
and accordingly every term, in its development, contains as factor one at least of the 
quantities 
^/+ 2 > • • • 
