122 INVESTIGATION OF THE TRANSFORMATION OF CERTAIN ELLIPTIC FUNCTIONS. [19 
The above supposition, f — 0, is, however, only a particular one ; omitting it, the 
conditions to be satisfied by a, b, a', b', may be written under the form 
ab' — a'b = v, 
ag — bg' = 0 [mod. y], > (8) 
a'g — b'g' = 0 [mod. v\,) 
to which we may join the equations before obtained, 
vH = b'K - a'K'i, j 
- vH'i =bK - aKi, I 
(9) 
which contain the theory of the modular equation. This, however, involves some 
further investigations, which are not sufficiently connected with the present subject to 
be attempted here.