450 
PROFESSOR CAYLEY ON THE TRANSFORMATION 
or throwing out the factor — (1A 4 — 5A 2 B 2 — 3B 4 ), this is 
A(A 2 + 8B 2 ) + 4B(1 — B 4 ) = 0 , 
the modular equation, which is right. 
The four forms of the modular equation , and the curves represented thereby. 
Article Nos. 72 to 79. 
72. The modular equation for any value of n has the property that it may be repre- 
sented as an equation of the same order (=n-\- 1, when n is prime) between u, v or 
between u 2 , v 2 , or between id, id, or between u s , v 8 . As to this, remark that in general 
an equation (u, v, l) ro =0 of the order m gives rise to an equation {id, v 2 , T) 2m =0 of 
the order 2m between u 2 , v 2 ; viz. the required equation is 
(u, v, 1 ) m (u, —v, 1 ) m (—u, v, l) m (—u, —v, l) m — 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 intersections of the 
curve by the arbitrary line ax-\-by-{-cz—0 are given by the equations (u, v, w) rn = 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 , uf 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 2 , v 2 ; 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 formulee are 1c 2 , 7d=u 8 , v 8 ; if a modular equation (u, v) v =0 
led to the transformed equation ( u 8 , v 8 ) 8v =D, then to a given value of u 8 would corre- 
spond 8 values of u, therefore 8v values of v, giving the same number, 8v, values of v 8 ; 
that is, the values of v 8 corresponding to a given value of u 8 would group themselves in 
eights corresponding to the 8 values of u. There is in fact no such grouping; the 
equations are (u, v) v =0, (u 8 , v 8 ) v =0 ; to a given value of u 8 correspond 8 values of u, 
and therefore 8v values of v, but these give in eights the same value of v 8 , so that the 
number of values of v 8 is =v. 
74. I consider the case n= 3 : here, writing x, y for u, v, we have here the sextic curve 
I. y 4 — x 4 -\-2xy{x 2 y 2 — 1)=0 ; 
and it is easy to see that the remaining forms wherein x,y denote u 2 , v 2 ; u 4 ,v 4 ; and u 8 , v 8 
respectively, are derived herefrom as follows ; viz. 
II. {if — x 1 ) 2 — ixy{xy — 1 ) 2 = 0 , that is 
f+e>xy + x 4 -±xy(xy + 1) =o ; 
