[650 
650] 
ON A QUARTIC SURFACE WITH TWELVE NODES. 
61 
and also 
p ( YZ - X Vf + q (ZX - YVy + r (X V - Z W)= = 0. 
The more general equation 
(P> V, r, l, m, nJYZ + XW, ZX+YW, XY+ ZW) 2 = 0 
represents a quartic surface (octadic) having the 8 nodes 
(1, 0, 0, 0), (I, 1, 1, 1), 
(o, i, 0, 0), (1, T, 1, 1), 
(o, o, i, o), (i, i, t, i), 
(0, 0, 0, 1), (1, 1, 1, T). 
We have 
d x U = 
d Y U = 
ES. 
p. 
IF 5 +YZW 
p. 
YZ 2 +XZW 
q. 
YW 2 + YZW 
q- 
YW 2 + XZW 
(1877), 
r. 
ZW 2 + YZW 
r. 
YX 2 + XZW 
1 
2XYZ + W(Y 2 + Z 2 ) 
1. 
2XYW + Z(W 2 + X 2 ) 
m. 
2XYW+ Z (W 2 + Y 2 ) 
m. 
2 YZX + W(Z 2 + X 2 ) 
n. 
2XZW+ Y(W 2 + Z 2 ), 
n. 
2YZW + X(W 2 + Z 2 ), 
d z U = 
II 
p. 
Y 2 Z + XYW 
p. 
X 2 W +XYZ 
q- 
X 2 Z + XYW 
q- 
Y 2 W +XYZ 
ope of the 
r. 
W 2 Z +XYW 
r. 
Z 2 W +XYZ 
1. 
2ZXW+ Y(W 2 + X 2 ) 
1 
2WYZ + X (Y 2 + Z 2 ) 
m. 
2YZW + X (W 2 + Y 2 ) 
m. 
2WZX + Y (Z 2 +X 2 ) 
n. 
2 ZXY + W(X 2 +Y 2 ), 
n. 
2WXY + Z(X 2 + Y 2 ). 
Hence there will be a node 
1, I, I, 1, if p + q + r + 21- 2m - 2n = 0. 
I, 1, I, 1, ... p + q + r — 21 + 2m — 2n = 0, 
I, T, 1, 1, ... p + q + r-21 — 2m + 2n = 0, 
1,. 1, 1, 1, ... p + q + r+ 2Z + 2m + 2ii=0; 
or say there will be 
1 of these nodes \ip-\-q + r + 2l + 2m + 2n = 0, 
2 p + q + r + 2l=0, m + n = 0, 
3 p + q + r = 21 = — 2m = — 2n, 
4 p + q + r = 0, 1 = 0, m = 0, n = 0; 
result is