128 
[739 
irf’ /- 4d’ 
739. 
NOTE ON THE OCTAHEDRON FUNCTION. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879), 
pp. 280, 281.] 
A sextic function 
U = (a, b, c, d, e, f gjx, y) 6 , 
such that its fourth derivative 
(U, U) 4 , = (ae — Abd + 3c 2 )(P 
+ 2 (af — 3be + 2cd) ct?y 
4- (ag — 9ce + 8d 2 ) x*y- 
+ 2 (bg — 3cf + 2de) ccy 3 
+ (eg — 4df+ 3e 2 ) y i 
is identically = 0, is considered by Dr Klein, and is called by him the octahedron 
function. Supposing that by a linear transformation the function is made to contain 
the factors x, y, or what is the same thing assuming a = 0, g = 0, then the equations 
to be satisfied become 
— 4bd + Sc 2 = 0, —3be-\-%cd = 0, —9ce + 8d 3 —0, —3cf+2de = 0, — 4c?/+ 3e- = 0, 
which are all satisfied if only c = d = e = 0; and then assuming, as is allowable, 
& = -/=!, 
we have his canonical form xy (x 4 — y 4 ) of the octahedron function. 
But the equations may be satisfied in a different manner; viz. the first and last 
equations give