OF ELLIPTIC FUNCTIONS. 
399 
3. To develop the condition, observe that the assumed equation gives 
^(P 2 + 2PQ + QV) 
P 2 + 2PQ* 2 +Q 2 a: 2 ’ “35 suppose, 
where £1, 33 are functions each of them of the degree ^(n — Y) in x 2 . (Hence, if with 
1 1 / 2Q\ 2/3 
Jacobi denotes the value {y-^-x) x=0 , we have ( l + -jr j , =l+” a , as mentioned.) 
Suppose in general that U being any integral function (1, x 2 ) p , we have 
u*=(*wr(i.^-)' s 
viz. let U* be what U becomes when x is changed into and the whole multiplied by 
(*"*T 
Let y* be the value of y obtained by writing for x ; then, observing that in the 
expression for y the degree of the numerator exceeds by unity that of the denominator, 
we have 
* 1 
y kx 33*’ 
whence 
yy*~ic 3333 * ’ 
and the functions 33 may be such that this shall be a constant value, ; viz. this 
will be the case if 
a 3333* 
which being so, the required condition is satisfied. 
4. I shall ultimately, instead of Jc , X introduce Jacobi’s u, v {u~^/lc, v=^/ 7 ,) ; but 
it is for the present convenient to retain Jc, and instead of X to introduce the quantity O 
connected with it by the equation x=Jc£l 2 ; or say the value of O is =v 2 -=rU 2 . The 
modular equation in its standard form is an equation between u, v, which, as will appear, 
gives rise to an equation of the same order between u 2 , v 2 : and writing herein v 2 =Q.u 2 , 
the resulting equation contains only integer powers of u 4 , that is of Jc, and we have an 
IM’-form of the modular equation, or say an fl^-modular equation, of the same order in 
O as the standard form is in v; these UMorms for n— 3, 5, 7, 11 will be given pro 
sently. 
5. Suppose then, H being a constant, that we have identically 
this implies 
*=A a*- 
3 g 2 
