54 REPORT—1846. 
After fully developing this part of the subject, he next treats of the nature 
of the modular equation, and shows that it possesses several remarkable 
properties. One is, that all modular equations, of whatever order, are pare 
ticular integrals of a differential equation of the third order, of which the 
general integral can be expressed by means of elliptic transcendents. 
16. We now enter on the second great division of M. Jacobi’s researches, 
the evolution of elliptic functions. 
The evolution of elliptic functions into continued products with an infinite 
number of factors presents itself as the limit towards which M. Jacobi’s 
theorem for the transformatlon of the mth order tends as ” increases sine 
limite. It is for this reason that we may look on the problem of transforma- 
tion as the leading idea in M. Jacobi’s researches. 
We may in some degree illustrate these evolutions by a reference to cir- 
cular functions. A sine is, as we know, an elliptic function whose modulus 
is zero. Now if & is zero, A is also zero. Thus if we apply a formula of 
transformation to a sine, we shall be led to another sine either of the same 
or of a multiple are. Accordingly the first real transformation degenerates 
in the case in question into the known formula for the sine of a multiple 
arc; while the second, leading us merely to the sine of the same are, becomes 
illusory. Thus in the case of a sine, transformation is merely multiplication ; 
but from the formula for multiplication, viz. 
i a : gee sin? 6 mos es sin? 6 
sin (2m+1)§=(2m-+1) sin 6 rere gre Fe cai To 2me ae 
Q2m+1 ; 2m+1 
we at once deduce, by making (2m + 1)4= @and 2m + 1 infinite, the 
common formula 
ES ¢° 9° 
sin 6 = ig Bye (2 geld 
Cue ? G 9) ¢ 4s x? 
This then is a formula of evolution deduced from the first real transfor- 
mation. It is however only when & is zero that the first transformation will 
give such a formula. In all other cases it is, for a reason which we cannot 
here enter on, impossible to derive from it a formula of this kind. M. Jacobi’s 
formule are accordingly derived from the second real transformation, and 
therefore are illusory when & is zero, or for the case of the sine. There is 
nothing therefore strictly analogous to them in the theory of angular sections. 
By means of them we express the function sin am 2 in terms of sin mx, m 
being a certain constant. 
From the fundamental expressions in continued produets, of which there 
are three, many important theorems may be derived, This part of the sub- 
ject seems to admit of almost infinite increase, and it is difficult to give any 
general view of it. I may, however, mention a remarkable transcendental 
similarly from { VI —# to VI — ae a transformation {a to kh. The first and last of 
these transformations correspond respectively to the differential equations— 
dy Ms 1 dx 
Vi-y)G—-x#y) MV/G—2)(1— a?) 
da’ 1 dy 
Ji — a) (1— a?) M/A —y)(1— ayy 
Hence, combining these equations and integrating, 
1 
Fike’) = ivivtag (ka) ; 
and it may also be shown that ats is an integer. 
MM’ 
