232 
ON HOMOGENEOUS FUNCTIONS OF THE THIRD 
[35 
The values of the coefficients a, b, c, f, g, h may be useful on other occasions: they 
are as follows (13). 
Substituting these values the result after all reductions becomes 
0 = F(U)= (14). 
It would be desirable, in conjunction with the above, to obtain the equation 
K(U) = 0, 
which results from the elimination of x, y, z from the equations 
^ = 0 
dx ’ 
du =o, d A = o 
(13), 
dy dz 
(i.e. the condition of a curve of the third order having a multiple point), but to effect 
this would be exceedingly laborious. The following is the process of the elimination as 
given by Dr Hesse, Crelle, t. xxvm. (and which applies also to the case of any three 
equations of the second order). Forming the function V U, of the third order in x, y, z, 
by means of the equation 
L, T, S (16), 
T, M, R ' 
S, R, N 
(L, M, N, R, S, T, the same as before). 
Then, in consequence of the equations (15), we have not only 
VH= 0 
which is very easily proved to be the case, but also 
Avh=o, 4-vu= 0, ~VU = 0 
dx dy dz 
•(U), 
.(18), 
as will be shown in a subsequent paper “ On Points of Inflection.” [I think never 
written.] 
And from the six equations (15), (18), the six quantities x 2 , y 2 , z 2 , yz, zx, xy, may 
be linearly eliminated: we have 
V U — Ax? + By 3 + Cz 3 + 3 Iy 2 z + 3 Jz 2 x + 3 Kx 2 y + 2>I{yz 2 + oJ x zx 2 + SK v xy 2 + 6 A xyz.. .(19), 
where the values of the coefficients A, B, ... A are 
and the result of the elimination is 
K(U) = 
a , 
ki , 
j > 
l , 
ii ’ 
k 
k, 
b , 
h > 
i , 
l , 
h 
h > 
i , 
c , 
h. > 
j . 
l 
A, 
K x , 
J, 
L, 
Ju 
K 
K, 
B , 
h, 
I, 
L, 
K x 
Ju 
I , 
c, 
¿i, 
J, 
L 
= 0 
.(20), 
.(21). 
[K (U) is consequently, as is well known, a function of the twelfth order in a, b, c, 
i, j> k, i 1 , ji, k\, 1].