231 
35] ON HOMOGENEOUS FUNCTIONS OF THE THIRD ORDER &C. 
so that a single equation more, such as 
3> = 0 (6), 
where d? is homogeneous and of the second order in x, y, z, would, in conjunction with 
the equations (3) and (5), enable us to eliminate linearly the seven quantities x 2 , y\ z\ 
yz, zx, xy, A. Such an equation may be thus obtained. 
Let L, M, N, R, S, T, be the second differential coefficients of U, each of them 
divided by two. The equations (3) may be written 
Lx+ Ty + Sz + A£ = 0, (7). 
Tx -f- My -f- Rz -f- A?y = 0, 
Sx + Ry + Nz + A£ = 0. 
And joining to these the equation (4), 
gx + y y + £z = 0, 
we have, by the elimination of x, y, z, in so far as they explicitly appear, and X, an 
equation <1> = 0 of the required form. Hence we may write 
0) = - 
L, 
T, 
s, 
1 
T, 
M, 
R, 
V 
s, 
R, 
X, 
K 
V > 
r, 
or substituting for L, M, N, R, S, T, and expanding, 
<E> = Ax 2 + By- + Gz- + 2 Fyz + 2 Gzx + 2 Hxy 
where the values of A, B, C, F, G, H are 
(8); 
..(9), 
(10). 
[I omit these values (10), and the values in the subsequent equations (13), (14), 
(20): the values (10) and (13) serve for the calculation of (14), FU, the expression 
for which with the letters (a, b, c, f g, h, i, j, k, l) in place of {a, b, c, i, j, k, i lt j 1; k 1} l) 
is reproduced in my “ Third Memoir on Quantics,” Phil. Trans, vol. cxlvi. (1856) 
pp. 627—647: the values (20) serve for finding that of (21), K{U), but the developed 
expression of this has not been calculated.] 
Performing the elimination indicated, the result may be represented by 
a , h, j, 
l, 
k, 
1 
k , b , ?i, 
i, 
l, 
k\, 
V 
ji > *? c > 
^19 
j> 
l , 
r 
n, • • 
K, 
V 
. 2 V . 
K 
z 
. . 2^ 
V 
e 
ABC 
F 
G 
H 
(11). 
Partially expanding, 
F U = Aa + Bb + Cc + 2Fi + 2Gg + 2Hh 
(12).