988 
PROFESSOR A. CAYLEY OR THE SINGLE 
= ^(cd—c / d)(fe / — f e) v / aa / bb / , = (c— d)(f—e)\/ aa / bb / ; we liave 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 
0 { l'\/abc / d / ef / + f a^ycdef), 
+ c / }f 
„ + d, 
h e » + e - 
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 
0{(A+/ne)\/abcy^ed’ +(X+/x^) v ajb/xlef }. 
129. Considering next the actual equations between the squared theta-functions, 
take as a specimen 
c 6 s V=0, 
that is 
c 6 4 {\/ ab )~— c £ 4 '(v / c‘d) : ' + c 1 l (\/ ce)~—cf(\/- de) 2 = 0 , 
where c 6 , c. 2i e v c (J = \/ub, \Z cd, \Zcc, v de respectively. Since the functions ( \/ab)~ , 
&c., contain the same irrational term |v XV, it is clear that the equation can only be 
true if 
c e 4 —c 3 4 + c i 4 —c 9 4 = 0, 
and this being so it will be true if 
%*{( Vabf- (x/XI) 3 } - Cj*{ (VabY- (v/5) 2 } +c*{(Vabf-(Vdef} = 0, 
where by what precedes each of the terms in {} is a linear function of ( \/ a) 3 and 
( x/b )' : : attending first to the term in {\f a) 3 , the coefficient hereof is 
ef. be. bd. c 2 4 — df. be. be. c/ffi cf. bd. be.c^> 
where for shortness be, bd, &c., are written to denote the differences b — c, b — d, &c.: 
substituting for cd its value (\/crf) 4 , =cd.cfdf.ab.ae.be, and similarly for c* and c 9 4 
their values, = ee.ef.ef.nb.ad.bd, and de.dfef.ab.ac.be respectively, the whole expression 
