[662 
WITH A 16-NODAL QUARTIC SURFACE. 
161 
f to the several 
X, W, — Z), Ave 
the same linear 
fibers in square 
662] 
and so (2) gives 
(/3, a, -8, -y)(F, -X, W, -Z), which is (-a, (3, 7, - 8) (X, Y, Z, W), 
or, reversing the signs, 
(a, -/3,- % 8)(X, F, If), =[2]. 
Comparing with the geometrical theory in Rummer’s Memoir, it appears that the 
several systems of values (1), (2), ...,(16) are the coordinates of the nodes of a 16-nodal 
quartic surface, Avhich nodes lie by sixes in the singular tangent planes, in the manner 
expressed by the foregoing scheme, wherein each top number may refer to a singular 
tangent plane, and then the numbers beloAv it show r the nodes in this plane: or 
else the top number may refer to a node, and then the numbers below it show the 
singular planes through this node. 
And, from what precedes, we have the general result: the 16 squared double 
0-functions correspond (one to one) to the nodes of a 16-nodal quartic surface, in 
such Avise that linearly connected squared functions correspond to nodes in the same 
singular tangent plane. 
The question arises, to find the equation of the 16-nodal quartic surface, having 
the foregoing nodes and singular tangent planes. Starting from one of the irrational 
forms, say 
VM [1] [5] + Vi? [2] [6] + VC [3] [7] = 0, 
the coefficients A, B, C are readily determined; and the result written at full length is 
V2 (a/3- 78) (a8 + /87) (/3X + ocY-8Z-yW) (8X + yY + /3Z + ctW) 
+ V(a 2 — /3 2 — 7 2 + 8 2 ) (07 — /38) (aX — /3 Y — 7 Z ■+ 8 If) (7X — 8 Y + <xZ — ¡3 If) 
+ V(a 2 4- /3 2 — 7 2 — 8 2 ) (ay + /38) (aX + ¡3 Y — 7Z — 8 If) (yX + 8 Y + aZ + ¡3 W) = 0. 
It is a somewhat long, but nevertheless interesting, piece of algebraical work to 
rationalise the foregoing equation : the result is 
(/3y - a 2 8 2 ) (y-a? - /3*8’-) (a 2 /3 2 - 7 2 8 2 ) (X 4 + Y 4 + Z 4 + W 4 ) 
+ (y 2 a 2 - /3 2 8 2 ) (a 2 /8 2 - 7 2 8 2 ) (a 4 + 8 4 - /3 4 - y 4 )(Y*Z* + X 2 Tf 2 ) 
+ (a 2 /3 2 - 7 2 8 2 ) (/3 2 7 2 - a 2 S 2 ) (/3 4 + 8 4 - y 4 - a 4 ) (XX 2 + F 2 If 2 ) 
+ (/3y - a 2 8 2 ) (7 2 a 2 - /3 2 8 2 ) ( 7 4 + S 4 - a 4 - /3 4 ) (X 2 F 2 +XTf 2 ) 
- 2a/378 (a 2 + /3 2 + 7 2 + 8 2 ) (a 2 + 8* - ¡3* - y*) (/3 2 + 8 2 - a 2 - 7 2 ) ( 7 2 + 8 2 - a 2 - /3 2 ) XF^lf = 0 ; 
or, if Ave Avrite for shortness 
L = /8y - a 2 8' 2 , 
ili = 7 2 a 2 — /3 2 8 2 , 
X = a 2 /3 2 — 7 2 8 2 , 
F = a 2 + 8 2 — /3 2 — t 8 , 
G = /3* + 8 2 - 7 2 - a 2 , 
H = 7 2 + 8 2 - a* - /3 2 , 
A = a 2 + /3 2 + 7 2 + 8 2 , 
If), = [1] ; 
C. X. 
21