466 
NOTE ON A QUARTIC SURFACE. 
[360 
If a and b are unequal, but if we still have /3 = 0, the equation of the surface is 
(x 2 + y 2 + z 2 — a 2 — <y 2 ) 2 = 4a 2 (x — a) 2 + 4b 2 y 2 . 
There are here two planes parallel to the plane of the conic, each of them meeting 
the surface in a pair of circles. In fact, writing x 2 + y 2 = p, and therefore also y 2 = p — x 2 , 
putting moreover z 2 — a 2 — 7 2 = k, we have 
that is, 
(p + k) 2 = 4a, 2 « 2 — 8a 2 ax + 4 ara 2 + 4b 2 (p — x 2 ) ; 
p 2 + 4 (b 2 — a 2 ) x 2 4- k 2 — 4a 2 a? + 8a 2 ax + (2k — 4b 2 ) p — 0, 
or, as this may also be written, 
(1, 4(6 2 —a, 2 ), k 2 — 4a 2 a 2 , 4a 2 a, k — 2b 2 , 0«, 1) 2 =0, 
which is of the form 
(a, b c , / , g , 0#p, «, l) 2 = 0; 
and the left-hand side will break up into factors, each of the form p + Ax + B (so that, 
equating either factor to zero, we have p + Ax + B = Q, that is, x 2 + y 2 + Ax + B = 0, the 
equation of a circle), if only 
abc — af 2 — by 2 = 0. 
Writing this under the form b (cic — g 2 ) — af 2 = 0, and substituting for a, b, c, f, g their 
values, we have 
b = 4 (b 2 — a 2 ), ac — g 2 = k 2 — 4a 2 a 2 — (k— 2b 2 ) 2 , = 4 (b 2 k — b 4 — a 2 a 2 ), a/ 2 = 16a 4 a 2 , 
and therefore the condition is 
that is, 
(b 2 — a 2 ) (b 2 k -b 4 — a 2 ar) — a 4 ar = 0 ; 
b 2 {(6 2 — a 2 ) (k — b 2 ) — a 2 a 2 } = 0. 
If b 2 =0, the surface is a pair of spheres; rejecting this factor, we have (b 2 —a 2 )(k 2 —b 2 )—a 2 a 2 =0; 
or putting for k its value, the condition becomes 
(b 2 — a 2 ) (z 2 — a 2 — 7 2 — b 2 ) — a 2 a 2 = 0 ; 
that is, for each of the values of 0 given by this equation, the section by a plane 
parallel to the plane of the conic will be a pair of circles. 
The planes in question will coincide with the plane of the conic, if only 
(■b 2 — a 2 ) (a 2 + 7 2 + b 2 ) + a 2 a 2 = 0, 
or, what is the same thing, 
b 2 oi 2 — (a 2 — b 2 ) 7 2 = b 2 (a 2 — b 2 ) ; 
that is, if the point (a, 0, 
hyperbola in question and 
7) be situated on the hyperbola y = 0, ——^ 2 — p = -^ e 
the ellipse z= 0, ~r 2 + j^= 1, are, it is clear, conics in planes 
(XT' 0^