360] 
465 
360. 
NOTE ON A QUARTIC SURFACE. 
[From the Philosophical Magazine, vol. xxix. (1865), pp. 19—22.] 
It would, I think, be worth while to study in detail the quartic surface which 
is the envelope of a sphere having its centre on a given conic, and passing through 
X“ V* 
a given point. The equations of the conic being z = 0, 2 +^o = l> the coordinates of a 
point on the conic may be taken to be x = a cos 6, y = h sin 6, z = 0, whence, if (a, ft, 7) 
be the coordinates of the given point, the equation of the sphere is 
(x — a cos 6) 2 + (y — b sin 6)- + z 2 = (a — a cos 6) 2 + (/3 — b sin 6) 2 + y 2 , 
or, what is the same thing, 
x 2 + y 2 + z 2 — a 2 — ft 2 — 7 2 — 2 (x — a) a cos 6 — 2 (y — ft)b sin 6 = 0; 
and hence the equation of the surface is at once seen to be 
(x 2 + y 2 + z 2 — a- — /3- — y 2 ) 2 = 4a 2 (x — a) 2 + 4b 2 (y — ft) 2 . 
If a = b (that is, if the conic be a circle), then we may without loss of generality 
write ft = 0, and the equation then is 
(x 2 + y 2 + z 2 — a 2 — y 2 ) 2 = 4a 2 {(x — a) 2 + y 2 \. 
This may be written 
which, considering z as a constant, is of the form 
(x 2 + y 2 — a) 2 = 16 A (x — m); 
that is, the section of the surface by a plane parallel to the plane of the conic is a 
Cartesian. 
C. V. 
59