472 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704 
which expresses that the four functions are the coordinates of a point on a quadri- 
quadric curve in ordinary space. 
17. The remaining 12 of the 16 equations then contain on the left-hand products 
such as 
A (u + u') . B (u — u); 
and by suitably combining them we obtain equations such as 
u+u u-u' u+u u-u' 
B.A-A.B 
C .D + D.C 
= function 
O'), 
where for brevity the arguments are written above; viz. the numerator of the 
fraction is 
B (u + u) A (u — u') — A (u + u') B (u — u), 
and its denominator is 
C(u + u')D (u — u') + D (u + u') G(u — u). 
Admitting the form of the equation, the value of the function of u' is at once found 
by writing in the equation u = 0; it is, as it ought to be, a function vanishing for 
u = 0. 
18. Take in this equation u indefinitely small; each side divides by ii, and 
the resulting equation is 
AuB'u - BuA'u 
—• = const., 
GuJJu 
where A'u, B'u are the derived functions, or differential coefficients in regard to u. 
It thus appears that the combination AuB'u—BuA'u is a constant multiple of 
CuDu : or, what is the same thing, that the differential coefficient of the quotient- 
function ^ is a constant multiple of the product of the two quotient-functions ™ 
and 
Du 
Au ' 
19. And then substituting for the several quotient-functions their values in terms 
of x, we obtain a differential relation between x, u; viz. the form hereof is 
du = 
Mdx 
\/a 
b — x.c — x.d — x’ 
and it thus appears that the quotient-functions are in fact elliptic-functions: the 
actual values as obtained in the sequel are