364 
ON THE TRANSFORMATION OF THE DOUBLE-THETA FUNCTIONS. [848 
17. Starting with any two characteristics a, b at pleasure, the remaining charac 
teristics form seven pairs, such that 
a + b = c + d = e+ f=g + h = i + j = Jc+l=m + n = o+p; 
but among the seven pairs we have only three, suppose c + d, e +f g + h, which are 
such that (a, b, c, d,), (a, b, e, /), (a, b, g, h) are each of them either all even or 
else two even and two odd; that is, starting with any pair (a, b), we have these 
three tetrads having each of them the required property. The number of pairs (a, b) 
is 16.15, = 120; and we thence derive 120 x 3, = 360 tetrads; but each such tetrad 
is of course derivable from any one of the six pairs contained in it; or the number 
of distinct tetrads is £360, = 60, viz. we have, as mentioned above, 60 Gopel-tetrads. 
The four functions IT 0 , Eb, IT, n 3 . 
18. We consider four theta-functions 6 0 , 8 lf 6. 2 , 0 3 , which are such that to the 
modulus 2, the sum of the characters is = 0, and also the sum of the indices is 
= 0; taking the characters to be 
and writing throughout = for = (mod. 2), we have 
+ + f + g" + f" — 0, 
v + v + v" + v" = 0, 
q + q' + q" + f" = 0, 
r + / + r" + r" = 0, 
gq + vr + fq + vV + f'q" + v"r" + f'q"' + v"r" = 0. 
Writing for shortness (01) = gq +fq + vr + v'r, and so in other cases; and further 
(01)+(02) +(12) = (012), «fee., then substituting for f", v", q", r'" their values from 
the first four equations, we deduce (012) = 0; and similarly (013) = 0, (023) = 0, 
(123) = 0. 
19. Consider now a product d o a # 1 b 0 2 c 0 3 d , where a + b+c + d is = a given odd 
number h\ the characteristic is 
and it hence follows that the index is 
= a(gq + vr) + b(gf + v'r) + c (g'q" + v'r") + d+ v"r"). 
In fact, forming the index in question, we have first terms in a 2 , b 2 , c 2 , d 2 , which 
upon writing therein a, b, c, d for these values respectively (a 2 = a, <fec.) give the