578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS 137 
where 0 U 0 2 , 0 3 , are the roots of the equation in 0, viz. we have 
64 + 2 ^ 3 - 20« - w 4 = (0- 0\) (0 - 0 2 ) (0 - 0 3 ) (0 _ 0 4 ) ; 
and it was in this way that the equation for the case n — 15 was calculated. Observe 
that writing u= 1, we have (0 + l) 3 (0 - 1) = 0, or say 0 1 = 0 2 = 0 s = -i > 6> 4 = + 1. The 
equation in v thus becomes {(y - l) 5 (y + l)} 3 (y + l) 5 (y - 1) = 0, that is, (y - l) 16 (y +1) 8 = 0. 
28. The case where n has a square factor is a little different; thus n = 9, the 
values are 
^2qfy(q 9 ) } -V2V2 q h *f(q«), 
1 > 3 , 9 , roots; 
but here <u being an imaginary cube root of unity, the second term denotes the three 
^2q*f (q), V2 (qco)*f (coq), V2 (wtqff (co 2 q), 
the first of which is =u, and is to be rejected; there remain 1 + 2+9, =12 roots, or 
the equation is of the order 12. 
Considering the equation as compounded of two cubic transformations, if the value 
of v for the first of these be 0, then we have 
0* + 20 a u 3 — 20u — = 0 ; 
and to the four values of 0 correspond severally the four values of v given by the 
equation 
v 4 + 2v s 0 3 — 2v0 — 0 4 = 0. 
One of these values is however v = — u, since the y#-equation is satisfied on writing 
therein v = — u ; hence, writing 
0* + 20 3 u 3 - 20u - u 4 = (0 - 0 1 ) (0 - 0 a ) (0 - 0 3 ) (0 - 0*), 
we have an equation 
(y 4 + 2y 3 ^ 3 - 2v0, - 0, 4 ) (y 4 +.. - 0J) (y 4 +.. — 0 3 4 ) (v 4 +.. - 6> 4 4 ) = 0, 
which contains the factor (y + u) 4 and, divested hereof, gives the requiied modular 
equation of the order 12 ; it was in fact obtained in this manner. 
Observe that writing u= 1, we have (0+ l) 3 (0—1) = 0> or say 0i = 0 2 = 0 3 =~ 1> ^4—1 > 
the modular equation then becomes 
|(y — l) 3 (y + l)} 3 (y "h l) 3 l)-r(y + 1) 4 = 0, 
that is, 
(y-1) 10 (y + l) 2 = 0. 
C. IX. 
18