522 
[616 
and the theory of the curve is connected with that of a quasi-elliptic integral 
depending on this radical. 
616. 
A GEOMETRICAL ILLUSTRATION OF THE CUBIC TRANSFORMA 
TION IN ELLIPTIC FUNCTIONS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xm. (1875), 
pp. 211—216.] 
Consider the cubic curve 
a? + y 3 + z s + 6lxyz = 0. 
If through one of the inflexions z = 0, x + y — 0, we draw an arbitrary line 
z — u (x + y), we have at the other intersections of this line with the curve 
u [id (x 4- y)- + 6lxy| + a? — xy + y' 2 = 0 ; 
that is, 
(w 3 + 1) {a? + y‘ 2 ) + 2xy (w 3 4- 3 la — 0 = 0; 
and from this equation it appears that the ratio x : y is given as a function involving 
the square root of 
( u 3 4- 3 lu — — (u 3 4-1) 2 , 
which, rejecting a factor 3, is 
= (2u 3 + 3 lu + U (lu — 0- 
It may be noticed that lu — ^ — 0 gives the value of u, which in the equation z — u(x+y) 
belongs to the tangent at the inflexion; and 2u 3 + 3lu + ^ = 0 gives the values which 
belong to the three tangents from the inflexion. 
It thus appears that the coordinates x, y, z of any point of the curve can be 
Expressed as proportional to functions of u involving the radical 
f{(lu — •§■) (2 u u 4- 3 lu 4- 2 )}>