169 
578J A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
Hence, replacing M, N by their values, this is 
- 4uv (1 — u 8 ) (1 — v 8 ) 
- (u 2 - v 2 ) (1 v? u 6 v 2 - u 4 v 4 - u 2 v s - v 8 ) (u 4 + Qu 2 v 2 + v 4 ) 
+ 4u 3 v 8 (u 2 + v 2 ) 2 {1 - u 2 v 2 (u 4 - u 2 v 2 + v 4 )} = 0; 
viz. writing u 2 — v n - = A, uv = B, this is 
- 45 {1 - A 4 — 4A 2 B 2 - 2B 4 + B 8 } 
- A [l-A 4 - 5A 2 B 2 - 35 4 } (A 2 + 8 5 2 ) 
+ 4B 3 (A 2 + 4B 2 ) {1 - A 2 B 2 - 5 4 } = 0, 
that is, 
- 45 {(1 -A 4 - 4M 2 5 2 - 2B 4 + B 8 )-B 2 (A 2 + 4 5 2 ) (1 - A 2 B 2 - 5 4 )} 
- A (1 — A 4 — 5A 2 B 2 — SB 4 )(A 2 + 85 2 ) = 0; 
viz. 
-45 (l-M 4 -5M 2 5 2 -35 4 )(l-5 4 ) 
- A (1 — A 4 — 5A 2 B 2 — 35 4 ) (A 2 + SB 2 ) = 0 ; 
or throwing out the factor - (1 - A 4 - 5 A 2 B 2 - SB 4 ), this is 
A (A 2 + 85 2 ) + 45 (1 - 5 4 ) = 0, 
the modular equation, which is right. 
The four forms of the modular equation, and the curves represented thereby. 
Art. Nos. 72 to 79. 
72. The modular equation for any value of n has the property that it may be 
represented as an equation of the same order (=w+l, when n is prime) between 
u, v: or between u 2 , v 2 : or between u 4 , v 4 : or between u 8 , v 8 . As to this, remark that 
in general an equation (u, v, l) m = 0 of the order m gives rise to an equation 
(u 2 , v 2 , 1) 2?№ = () of the order 2m between u 2 , v 2 ; viz. the required equation is 
(u, v, 1 ) m (u, —v, 1 ) m (—u, v, 1 ) m {—u, -v, l) wl =0, 
where the left-hand side is a rational function of u 2 , v 2 of the form (u 2 , v 2 , l) 2m ; or 
again starting from a given equation (u, v, w) m — 0, and transforming by the equations 
x : y : z = u 2 : v 2 : w 2 , the curve in (x, y, z) is of the order 2m; in fact, the inter 
sections of the curve by the arbitrary line ax + by + cz = 0 are given by the equations 
(u, v, w) m = 0, au 2 + bv 2 + cw 2 = 0, and the number of them is thus = 2m. Moreover, by 
the general theory of rational transformation, the new curve of the order 2m has the 
same deficiency as the original curve of the order m. The transformed curve in 
x, y, z, = u 2 , v 2 , w 2 may in particular cases reduce itself to a curve of the order m 
twice repeated; but it is important to observe that here, taking the single curve of 
the order m as the transformed curve, this has no longer the same deficiency as the 
original curve; and in particular the curves represented by the modular equation in 
its four several forms, writing therein successively u, v; u~, v; u 4 , v 4 ; u 8 , v 8 , = x, y, 
are not curves of the same deficiency. 
73. The question may be looked at as follows: the quantities which enter 
rationally into the elliptic-function formulae are k 2 , A 2 = u 8 , v 8 ; if a modulai equation 
(u, v) v = 0 led to the transformed equation (u 8 , <T = 0, then to a given value of w 8