1002 
ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
and then operating with b on the equations A= a>k ] x y/ ab, &c., we have for instance 
AbB — BbA=o>'^ 11 A; 7 { pv'){\/ ctb^'b — \/bb 1 y/a) 
+ (o-w'+rA)( x / cib. 2V /b— *ybb. 2X /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 S- 0 b3- 1 — S-^bS-Q, &c., each as a sum of products of 
pairs of functions, only on the right-hand sides we have expressions such as 
y/o b ix /b— y/7> b , s/a, &c., which present themselves as perfectly determinate functions 
of x, y : so that regarding zsu' -\-pv, cru'-j-rv' as given linear functions of the arbitrary 
quantities u', v', there is no longer anything indeterminate in the form of the equations. 
