[704 
any 
?, for 
mt p 
inear 
■efore 
inear 
exad. 
ained 
ave 
i. 
nter- 
ction 
S 15 
A 
mere 
we have thus a tetrad such that, selecting any two terms, each of the remaining 
terms is a linear function of these. 
In the second case, the terms are 
^ {f Vabqd^f, + f Va^tycdef}, 
whence clearly the four terms are a tetrad as above. And it may be added that 
any linear function of the four terms is of the form 
ì {(X + pix) fabc / d / e / f / + (X + /¿y) Va^cdef}. 
129. Considering next the actual equations between the squared theta-functions, 
take as a specimen 
/>2C V .2_|_ r »2^2_ r ,2(X. 2 _ A 
C/0 rOfi rJo I 0/f rJ l U) rJg \J, 
that is, 
c 6 4 (V ab) 2 — c 2 4 (V cd) 2 + c x 4 (V ce) 2 — c 9 4 (V rfe) 2 = 0, 
where c 6 , c 2 , c l5 = ab, v 7 cc£, v^ce, respectively. Since the functions ab) 2 , 
2 
&c., contain the same irrational term -^fXY, it is clear that the equation can only 
be true if 
Cg 4 — Co 4 + Ci c 9 4 = 0 : 
and, this being so, it will be true if 
c 2 4 {(Vab) 2 — (Vcd) 2 } — cf {(Vab) 2 — (Vce) 2 } + c 9 4 {(Vab) 2 — (Vde) 2 } = 0, 
where, by what precedes, each of the terms in { } is a linear function of (fa) 2 and 
(fb) 2 . Attending first to the term in (fa) 2 , the coefficient hereof is 
ef. bc.bd. c 2 4 — df. bc.be. c x 4 + cf. bd .be. c 9 4 , 
where for shortness be, bd, &c., are written to denote the differences b — c, b — d, &c.: 
substituting for c 2 4 its value (v 7 cd) 4 , = cd. cf. df. ab. ae. be, and similarly for c x 4 and c 9 4 
their values, =ce.cf. ef. ab .ad. bd, and de. df. ef. ab .ac .be respectively, the whole ex 
pression contains the factor ab .be .bd .be .cf. df. ef, and throwing this out, the equation 
to be verified becomes 
cd .ae — ce .ad + de .ac = 0, 
C. X.