AND DOUBLE THETA-FUNCTIONS. 
987 
being =0. We lienee liave the theorem that each square can be expressed as a linear 
function of any four (properly selected) squares. 
127. But we have also the theorem of the 1 6 Kummer hexads. 
Obviously the six squares 
Wa)\ (Vbf, (Xcf, (Vef, (Vff 
are a hexad, viz. : each of these is a linear function of 1, x-\-y, xy, and therefore 
selecting any three of them, each of the remaining three can be expressed as a linear 
function of these. 
But further the squares (\/ a) 3 , ( \/-b ) 3 , ( \/~ab) 2 , ( x/cd ) 3 , ( \/~ce) 2 , (\Zde) 2 form a 
hexad. For reverting to the expression obtained for (v ab)~— (x/ ccl)' 2 , the determinant 
contained therein is a linear function of aa / and bb /5 that is of (\/ft) 3 an d (vlf ; we 
in fact have 
(a-b) 
1, x+y, xy 
1 , a+b, cib 
1, c -j - d, cd 
= (b—c)(b—d)(a—x)(a— y) — (a —c) (ci—d)(b—x) ( b—y ). 
Hence (\Zab)' 2 — (\/cd) 2 is a linear function of (x/n) 3 , (v^h) 3 ; and by a mere inter¬ 
change of letters (\/ ab ) 3 — (\Zce) 2 , (x/ct6) 3 —(x/de) 3 , are each of them also a linear 
function of (x/n) 3 and (x/6) 3 ; whence the theorem. And we have thus all the 
remaining 15 hexads. 
128. We have a like theory as regards the products of pairs of functions ; a tetrad 
of pairs is of one of the two forms 
x/«x/&, \/ac\/bc, v ad\/bd, \Zae\/be and \/fy/ah, \/ c\/cle, \Ad\/ce, \/e\/cd ; 
in the first case the terms are 
x/aa^bbj, 
^{(a^+a^b) x/'edefb'd^f -j-(cfd,e,+c / f delx/aa^bb,}, 
„ » J r(dfc,e q-d/ye) „ }, 
k 
k 
+ (efc,d 4-c,fce) „ }, 
and as regards the last three terms the difference of any two of them is a mere 
constant multiple of x/aa^b,; for instance, the second term — the third term is 
6 l 2 
