AND DOUBLE THETA-FUNCTIONS. 
989 
contains tlie factor ab. bc.bd.be. cf.df.ef, and throwing this out, the equation to be verified 
becomes 
cd.ae — ce.adfde.ac — Q 
which is true identically. The verification is thus made to depend upon that of 
c 6 4—c 2 4 + c i 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 inter¬ 
esting to show how these are satisfied identically by the values c 0 = fbet, &c. 
Th us one of these equations is c 13 4 +c 1 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. befcfef. ah.ad. bd + ab.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 -f- bd. cf. effbf. 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 cfc 9 2 — c 1 2 cf-\-cfc 6 2 = 0 . The 
term c 13 2 c 9 3 is here f aclf.bce f def.abc which is = v (bc) 2 (df) 2 .ab.ac.o,d.afbe.ce.de.ef 
which is taken = bc.dff ab.ac.ad.af.be.ce.de.ef; similarly the values of c 2 cf and c 3 3 c 6 3 
are — bd.cf and bf.cd each into the same radical, and the equation to be verified is 
be. df— bd. cf -j- If. cd = 0 , 
which is right: and the other equations may be verified in a similar manner. 
131. Coming next to the equations connecting the pairs of theta-functions, for 
instance 
C 3 Ci5d fr 9-i 3 — C 0 Ci 3 5-g^i 5 + C 4 C s ^ 7 Tn = 0, 
this is 
c 3 c Vct c 0 c u {fbd f ad—f bef ae\ -fc^cff n . fbf a— 0, 
the products fbd fad and fbefae contain besides a common term the terms 
^ 3 (dfc / e / d-d / f ce) v 7 aa^ly, and — (efc^+e^cd) f aa,bb,, hence their difference contains 
0 2 (de / — d / e)(fc / —fc)v // aa / bb / which is —defcf aa^b,, that is defcf afb : hence the 
equation to be verified is 
def c.c 3 c 15 c 0 c 13 + c^cffn = 0 ; 
c 3 c i 5 c u c i 2 4S = fbef.acd f aef.bcd. fbdface f adf.bce, where under the fourth root we 
have 24 factors, which are, in fact, 12 factors twice repeated; and if we write 
