554 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704 
which is true identically. The verification is thus made to depend upon that of 
c 6 4 — c 2 4 + Cj 4 — c 9 4 = 0 ; and similarly for the other relations between the squared functions, 
the verification depends upon relations containing the fourth powers, or the products 
of squares, of the constants c and k. 
130. Among these are included the before-mentioned system of equations involving 
the fourth powers or the products of squares of only the constants c ; and it is 
interesting to show how these are satisfied identically by the values c 0 = ^bd, &c. 
Thus one of these equations is Ci 2 4 + Cj 4 + c 6 4 = c 0 4 ; substituting the values, there is 
a factor ce which divides out, and the resulting equation is 
ad. af. df. be .be + cf. ef. ab. ad. bd + ah . af. bf. cd. de — ac. ae .bd. bf. df= 0. 
There are here terms in a 2 , a, a 0 which should separately vanish; for the terms 
in a 2 , the equation becomes 
df. be .be + bd.cf. ef+ bf. cd.de — bd.bf. df= 0, 
which is easily verified; and the equations in a and a 0 may also be verified. 
An equation involving products of the squares is c r /c 9 2 — c 2 c 2 + c 3 2 c 6 2 = 0. The 
term c 12 2 c 9 2 is here Vadf. bee Vdef. abc which is = V(6c) 2 (df) 2 .ab .ac. ad .af .be .ce .de .ef, 
which is taken = be. df*Jab. ac. ad . af. be. ce . de . ef\ similarly the values of Ci 2 c 4 2 and 
c 3 2 c 6 2 are —bd.cf and bf. cd each multiplied by the same radical, and the equation to be 
verified is 
be. df— bd. cf+ bf.cd=0, 
which is right: the other equations may be verified in a similar manner. 
131. Coming next to the equations connecting the pairs of theta-functions, for 
instance 
CjCisAuAia c 0 c 12 A 3 A ]5 + = 0, 
this is 
c :j Cj 5 CoC 12 {\- bd V ad — \-be V ae] + c 4 c 8 k 7 k n . \'b \! a = 0, 
the products VbdVad and VfteVae contain besides a common term the terms 
p (dfc / e / -f d^ce) Vaa / bb / , and (efc/1, + e / f / cd) Vaa / bb / , 
hence their difference contains ^ (de x — d,e)(fc, — f c) Vaa^b which is = de.fc Vaa^bb,, 
that is, de.fc\/afb: hence the equation to be verified is 
de .fc. c 3 c 13 c 0 c 12 + c 4 c s k 7 k n — 0 , 
c 3 c 15 c 0 Ci 2 is =\/bef .acd x^aef.bcd Vbdf.ace \/adf bee, where under the fourth root we 
have 24 factors, which are, in fact, 12 factors twice repeated; and if we write 
II, = ab. ac . ad .ae .af .be .bd .be .bf. cd . ce. cf. de . df. ef, for the product of all the 15 
factors, then the 12 factors are in fact all those of IT, except ab, cf de; viz. we have 
c 3 c 15 c 0 Ci 2 = \/U. 2 + (ab) 2 (cf) 2 (de) 2 .