AND DOUBLE THETA-FUNCTIONS. 
917 
where the numerator is a function not in anywise more important than any other 
linear function of A 2 u and Wu. 
38. The function Da 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'O = (a 3 -f-/3 3 ) = co 0 ~.3l (a — d ), 
B~ 0 = 2a/3 = co 2 }$(b-d), 
C' 0 = or — /3 3 =o) 0 2 (E(c — d), 
where aq 3 is what ah becomes on writing therein x—cl. It is convenient to omit 
altogether these factors ah and oq 3 ; it being understood that without them, the equa¬ 
tions denote not absolute equalities, but only equalities of ratios : thus, without the 
oq 3 , the last-mentioned equations would denote A 3 0 : EhO : C 2 0 = a 3 -[-/3 3 : 2 a/I : or — /3 3 , 
=&(ci—d) : $$(b — d): Q£(c—d). The quantities .3, 13, Qt, ID only present themselves 
in the products Hah, &c., and their absolute magnitudes are therefore essentially 
indeterminate, but regarding ah as containing a constant factor of properly determined 
value, the absolute values of At, 23, £f\ ID may be regarded as determinate, and this 
is accordingly done in the formulae H 3 = — agh, &c., which follow. 
Relations between the constants. 
39. The formulae contain the differences of the quantities a, b, a, cl; denoting these 
differences in the usual manner 
by 
so that 
and aiso 
b — c, c—a, a — b, a —cl, b—d , c — d 
a, b, c, 1 , g, h 
. —h +g —a = 0 , 
h . —f — b= 0 , 
—g+f • —e = 0, 
a d - b —|— c . = 0 , 
af+bg+ch = 0 , 
and then assuming the absolute value of one of the quantities M, 23, (£, ID, we have 
the system of relations 
