565 
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS, 
then, substituting for P and Q their values, we have 
0 - -1 < w+ ^ l + ^ |) -1 + «0 (* V]r !+*t) • 
= + pv') 01 + (<nt' + TV') 02, 
if for shortness 
then operating with 0 on the equations A = wA? u Va&, &c., we have for instance 
.4.0.B — BdA = &r-ìc n k 7 { (W + pv') (V«. d-pJb — \/b 0jVa) 
+ (crii' + tv') (V a. 0 2 Vò — V& 0 2 V a)}, 
which is one of a system of 120 equations, the A, B being in fact any two of the 
16 functions. 
These are in fact nothing else than the foregoing system of 120 equations giving 
the values of the differential combinations So0Si — & c -> each as a sum of products 
of pairs of functions, only on the right-hand sides we have expressions such as 
Va d^b — V& 0 X Va, &c., which present themselves as perfectly determinate functions of 
x, y : so that, regarding «' + pv', au' + tv as given linear functions of the arbitrary 
quantities u, v', there is no longer anything indeterminate in the form of the equations.