11 
II 
p 
1—1 
OO 
487] ON THE QUARTIC SURFACES (*$U, V, W) 2 = 0. 11 
If 2 ] + k 2 W 2 = 0, 
It is remarked (Salmon, [Ed. 2], p. 148 [Ed. 4, p. 176]) that there is in the plane 
z = 0, a nodal conic |-4 (1 —)w 2 = 0, the complete section being made up 
of this conic twice, and of the curve of the eighth order which is the parallel curve 
r 2 + F 2 + Z 2 ) = 0, 
of the ellipse — + — w 2 = 0; the like is of course the case as to the sections by 
= 0. 
the other two principal planes x=0 and y = 0. For the section by the plane w = 0 
(or plane infinity) we have at once p 2 r 2 (4pr — q 2 ) = 0, where observe that 
= 12, as it should be. 
itangent or node-couple 
le is imaginary for the 
ace in the case of the 
(f — 4pr = {(b 2 + c 2 ) x 2 + (c 2 + a 2 ) y 2 + (a 2 + b 2 ) z 2 ) 2 — 4 (x 2 + y 2 + z 2 ) (b 2 c 2 x 2 + c 2 a 2 y 2 + a 2 b 2 z 2 ), 
= (1, 1, 1, —1, —1, —1$(& 2 — c 2 ) x 2 , (c 2 — a 2 )y 2 , (a 2 — b 2 )z 2 ) 2 
= norm, [x \/(b 2 — c 2 ) + y V(c 2 — a 2 ) + z \/{a 2 — 6 2 )}. 
The section is thus made up of two conics, each twice, and of four right lines: viz. 
the conics are x 2 + y 2 + z 2 = 0, the circle at infinity and ■ + %- + — a = 0, the section at 
° J a 2 b 2 c 2 
infinity of the ellipsoid ; and the lines are 
x V(6 2 — c 2 ) ± y V(c 2 — ci 2 )±z \J{a 2 — b 2 ) = 0, 
') (6 s + 0)0 = 0, 
viz. these are the common tangents of the two conics. The circle at infinity is a nodal 
conic on the surface, which has thus 4 nodal conics. 
- a 3 ) 2 = o. 
2—2