©# = 0, %y = 0, ©z = 0 (5); 
230 
[35 
ON HOMOGENEOUS FUNCTIONS OF THE THIRD ORDER WITH 
THREE VARIABLES. 
[From the Cambridge and Dublin Mathematical Journal, vol. i. (1846), pp. 97—104.] 
The following problem corresponds to the geometrical question of determining the 
polar reciprocal of a plane curve of the third order: the solution of it is also important, 
with reference to the linear transformations of homogeneous functions of three variables 
of the third order ; reasons for which it has appeared to me worth while to obtain the 
completely developed result. 
Let 
3 U — ax s + by 6 + cz 3 + Siy 2 z + 3jz 2 x + 3kx 2 y + ‘¿iyyz 2 + 2>j x za? + Sk^y 2 + 6lxyz (1). 
It is required to eliminate x, y, z, X from the equations 
U= 0 
clU 
dx 
dU 
dy 
dU 
dz 
+ = 0, 
+ Xy = 0, 
+ XÇ = 0, 
(2), 
(3). 
From the equations (2), (3), we obtain immediately 
© = %x + yy + £z = 0 (4); 
and thence