162 
ON THE DOUBLE ©-FUNCTIONS IN CONNEXION 
[662 
then the result is 
LMN (X 4 + Y 4 + Z 4 + W 4 ) 
+ MN(FA + 2L ) (Y 2 Z 2 + X 2 W 2 ) 
+ NL (GA + 2M) (,Z 2 X 2 + F 2 TF 2 ) 
+ LM(HA + 2N) (X 2 Y 2 + Z 2 W 2 ) 
- 2cx/3y8FGHA XYZW=0. 
It may be easily verified that any one of the sixteen points, for instance (a, /3, y, 8), 
is a node of the surface. Thus to show that the derived function in respect to X, 
vanishes for X, Y, Z, W = a, /3, 7, 8; the derived function here divides by 2a, and 
omitting this factor, the equation to be verified is 
LMN. 2a 2 + MN (FA + 2L) 8 2 + NL(GA + 2M) 7 2 + LM (HA + 2N) /3 2 - ¡3 2 y-8-FGHA = 0, 
viz. the whole coefficient of LMN is 2 (a 2 + /3 2 + y 2 4- S 2 ), = 2A ; hence throwing out the 
factor A, the equation becomes 
2LMN + MNF8 2 + NLGy 2 + LMH/3 2 - 0 2 y 2 8 2 FGH= 0. 
Writing this in the form 
L (2MN + NGy 2 + MH/3 2 ) = F8 2 (GHfi 2 y- - MN), 
we find without difficulty GH/3 2 y 2 — MN = — (/3 2 — y' 2 ) 2 L; hence throwing out the factor L, 
the equation becomes 
N (2M + Cry 2 ) + MH/3 2 + F8 2 ((3 2 — y 2 ) 2 = 0; 
we find 
MH/3 2 + F8 2 (¡3 2 — y 2 ) 2 = (a 2 /3 2 — y 2 8' 2 ) (2/3 2 8 2 — y 2 (a 2 + /3 2 + 8 2 ) + y 4 ) 
= N (2/3 2 8 2 - y 2 (a 2 + y3 2 + 8 2 ) + y 4 ), 
or throwing out the factor N, the equation becomes 
2M + Gy 2 + 2/3 2 8 2 - y 2 (a 2 + /3 2 + 8 2 ) + y 4 = 0, 
which is at once verified: and similarly it can be shown that the other derived 
functions vanish, and the point (a, /3, y, 8) is thus a node. 
The surface seems to be the general 16-nodal surface, viz. replacing X, Y, Z, W 
by any linear functions of four coordinates, we have thus 4.4 — 1, =15 constants, and 
the equation contains besides the three ratios a : /3 : y : 8, that is, in all 18 constants: 
the general quartic surface has 34 constants, and therefore the general 16-nodal surface 
34— 16, =18 constants: but the conclusion requires further examination. 
Gopel and Rosenhain each connect the theory with that of the ultra-elliptic 
functions involving the radical VX, = V#.l — ¡».1— lx. 1 — mx. 1 — nx\ viz. it appears by 
their formulae (more completely by those of Rosenhain) that the ratios of the 16 squares 
can be expressed rationally in terms of the two variables x, x', and the radicals