296 
ON THE THEORY OF ELLIPTIC FUNCTIONS. 
[45 
and each of these values is an algebraical function of sn u, (viz. either a rational 
function or a rational function multiplied by cnitdnw). Also, in the transformation of 
the № th order, 
so that it is clear that the above values of z may be taken for the denominator and 
numerator respectively of sn / u; i.e. these quantities each of them satisfy the equa 
tion (7). 
By assuming 
n (n — 1) x 2 z + (n — 1) (ax — 2a?) ^ + (1 — ax 2 + ¿c 4 ) — 2 n (a 2 — 4) ^ = 0 
(8); 
which is therefore satisfied by assuming for £ either the numerator or the denominator 
of \/A,sn t u (the transformation of the n th order), which is the form in which the 
property is given by Jacobi. 
In the case where n is odd, the denominator is of the form 
B 0 + B ja? ... + 5 i(n _ 1( x n ~', 
and then the numerator is 
x (B iin _ 1) ... + Bx n ~ z + Brfc n ~ l ), 
where 
and all the remaining coefficients may be determined from these, the modular equation 
being supposed known. But the principal use of the formula is for the multiplication 
of elliptic functions, which it is well known corresponds to the case where n is a 
square number. Writing n = v 2 , when v is odd, the denominator is 
1 + B 2 x* ... + B±( v 2_ 3) x v2 3 + vx v ' 1 1 
(the ± sign according as v=(4>p +1) or (4p — 1)); and the numerator is obtained from 
this by multiplying by x and reversing the order of the coefficients. When v is even 
the denominator is 
1 + B-pf ... ± B 2 x v *- i ± x v<l , 
(+ or —, according as v = 4>p or v = 4>p + 2), so that there are only half as many co 
efficients to be determined; but then the numerator must be separately investigated.