64 REPORT—1846. 
to the imegral | Foy and ¢, the corresponding function for 
the modulus ec, then, on introducing the inverse notation, the differential 
equation 
dy dz 
= 24 
v7G—y*) iy) V(1—a*) (1—e? 2°) 
becomes of course d§' = ad, with c= 6,4 and y= ¢,6'. Hence for a 
given increment « of 6, that of 6! isaa. 
Let us take the simplest case, and suppose y to be a rational function of 
x; then, as 2 or 6,9 remains unchanged when 9 increases by a period of the 
function ¢,, ¥ does so too; that is ¢,9' remains unchanged when 6! increases 
by a times a period of ¢,, or in other words, a times a period of ¢, is neces- 
sarily one of ¢,. 
Suppose now & and ¢ to be both real and less than unity; then ¢, and ¢, 
have each a real period, here denoted by 2w, and 2w, respectively, and each 
an imaginary period w,2 and @w,7 respectively, @, and w, being both real. 
Let 6 receive first the increment 2w,, and secondly the increment @, 7, then, 
by what has been said, 
2aw,=2mw,+ nwt 
aa.i=2pu, + qa,t*, 
m, n, p, g being certain integers. But can these two equations subsist simul- 
taneously? Not generally, since if we eliminate a and equate possible and 
impossible parts, we get éwo relations among w, w, w, @;,, Which are con- 
tinuous functions of the éwo quantities k and c. Hence both are determinate ; 
and if we wish c to remain indeterminate, we must either make m and ¢g 
equal to zero, in which case a is impossible, or, making and p equal to zero, 
assign a real value to it. When a is real we have 
a=m = = q ait 
W, ors 
and hence the remarkable conclusion, that 
wy. Ww, 
san ee SY Ns 
Dy, @, 
m and q being integers. 
The commensurability of the transcendental functions “, — is therefore 
k G 
a necessary condition, in order that an integral with modulus e¢ can be trans- 
formed into one with modulus &, the regulator a being real and c indeterminate. 
And it may be shown that this condition is not only necessary but sufficient. 
Similar considerations apply to the case in which a is impossible. 
Simple as this view is, it leads to many consequences of great interest. 
The function g, of which we have already spoken (p. 55), is merely e- a 
and as we know for the first real transformation of the mth order, it becomes 
nw E : wD oT : 
e—*-~@. Hence in this case we have Frye ped keg according to the 
e 
general law. It may be well to remark, that if k = c¢ we have a=m“t=m 
Ww, 
* here is in M. Jacobi’s notation 2K’, so that 9p¢=9(4+2ma+nwi), mand n 
being any integers. 
