704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
481 
The square set, u' — 0 ; and x-formulce. 
37. We use the square-set, in the first instance by writing therein u' = 0; the 
equations become 
A 2 u = aX +/3 Y, = &> 2 2l (a — x), 
Bhi — (3X 4- aF, = a> 2 23 (h — x), 
C 2 u = oX — /3 Y, = co 2 ($ (c — x), 
D 2 u = /3X — aF, = or¡2) (d — x), 
viz. the equations without their last members show that there exist functions to 2 and 
xu> 2 , linear functions of X and F, such that 2i, 23, (£, 2), 2la, 236, (Sc, 2)c£, being 
constants, the squared functions may be assumed equal to 2la. x> 2 — 21. co 2 x, fee., that 
is, or21 (a — x), &c., respectively: the squared functions are then proportional to the 
values 21 (a — x), «fee. 
To show the meaning of the factor co 2 , observe that, from any two of the equations, 
for instance from the first and second, we have an equation without rw, 
A 2 u 4- Bhi = 21 (a — x) 4- 23 (6 — x); 
and using this to determine x, and then substituting in &> 2 = A 2 u 4- 21 (a — x), we find 
2 %A 2 u-№hi 
W = (a — 6) 2123 ’ 
where the numerator is a function not in anywise more important than any other 
linear function of A 2 u and Bhi. 
38. The function Du vanishes for u — 0, and we may assume that the corresponding 
value of x is = d. Writing in the other equations u— 0, they become 
A 2 0 = (a 2 4- fi 2 ) = <w 0 2 2l (a — d), 
' B 2 0 = 2afi = &v23 (6 — d), 
C 2 0 = a 2 — fi 2 = &) 0 2 Gt (c — d), 
where &> 0 2 is what w 2 becomes on writing therein x — d. It is convenient to omit 
altogether these factors &r and &) 0 2 ; it being understood that, without them, the 
equations denote not absolute equalities but only equalities of ratios: thus, without 
the w 0 2 , the last-mentioned equations would denote 
^1 2 0 : B 2 0 : C 2 0 = a 2 4-/3 2 : 2a/3 : a 2 -/3 2 , =2l(a-d) : 23 (6-d) : ®(c-d). 
The quantities 21, 23, (£, 2) only present themselves in the products 2hw 2 , fee., and 
their absolute magnitudes are therefore essentially indeterminate: but regarding w 2 as 
containing a constant factor of properly determined value, the absolute values of 
21, 23, (£, 2) may be regarded as determinate, and this is accordingly done in the 
formulse 2l 2 = —agh, &c., which follow. 
C. X. 
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