548 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
This is 
(2M 0 — 2il 0 ) (2M + 20) 
- (2M 0 + 2H 0 ) (2M - 20), = 8 (M o n - MCI»), 
if for a moment 
M — Q cos \ttvl — 2Q 5 cos ^7m cos ttu + Q a cos §7ru, M n = Q — 2Q 5 + Q\ 
O = 2QS 4 cos 7tu cos 7tv + A cos (u + 2v) + A' cos \nr (u — 2v), O 0 = 2QS 4 + A + A', 
or substituting and reducing, the value of 8 (M 0 VL — Mfl 0 ) to the proper degree of 
approximation is found to be 
= — 8Q (2QS i + A + A') cos tu 
+ 8 (Q-S i + 8(^d) cos \nt (u + 2v) + 8 (Q 2 S 4 + 8Q^1 / ) cos r (u — 2v), 
which in virtue of the relations QA — A 2 S 2 , QA' = A' 2 S 2 , Q 2 S 2 = A A', is equal to the 
foregoing value of CgC^^n. I have thought it worth while to give this somewhat 
elaborate verification. 
Resume of the foregoing results. 
120. In what precedes we have all the quadric relations between the 16 double 
theta-functions: or say we have the linear relations between squares (squared functions) 
and the linear relations between pairs (products of two functions): the number of 
the asyzygetic linear relations between squares is obviously =12; and that of the 
asyzygetic linear relations between pairs is = 60 (since each of the 30 tetrads of 
pairs gives two asyzygetic relations): there are thus in all 12 + 60, =72, asyzygetic 
linear relations. But these constitute only a 13-fold relation between the functions, 
viz. they are such as to give for the ratios of the 16 functions expressions depending 
upon two arbitrary parameters, x, y. Or taking the 16 functions as the coordinates of 
a point in 15-dimensional space, these coordinates are connected by a 13-fold relation 
(expressed by means of the foregoing system of 72 quadric equations), and the locus 
is thus a 13-fold, or two-dimensional, locus in 15-dimensional space. 
Hence, taking any four of the functions, these are connected by a single equation; 
that is, regarding the four functions as the coordinates of a point in ordinary space, 
the locus of the point is a surface. 
In particular, the four functions may be any four functions belonging to a hexad : 
by what precedes there is then a linear relation between the squares of the four 
functions: or the locus is a quadric surface. Each hexad gives 15 such surfaces, or 
the number of quadric surfaces is (16x 15=) 240. 
The 16-nodal quartic surfaces. 
121. If the four functions are those contained in any two pairs out of a tetrad 
of pairs (see the foregoing “Table of the 120 pairs”), then the locus is a quartic