569] 
ON THE CYCLIDE. 
73 
The Parabolic Cyclide. 
One of the circles C, G' and one of the circles D, D' may become each of them 
a line; the cyclide is in this case a cubic surface. The easier way would be to treat 
the case independently, but it is interesting to deduce it from the general case. For 
this purpose, starting from the equation 
(y 2 + z 2 ) 2 + 2ic 2 (y 2 + z 2 ) + Gy 2 + Hz 2 + (x —f)(x — g) (x — h)(x — k) = 0, 
where / -f g + h + k = 0, G =fg + hk, H=fh + gk, I write x — a for x, and assume a + f, 
a + g, a + h, a + k, equal to g', h', k' respectively ; whence 4a =f'+g' + li' + k'; and 
the equation is 
(y 2 + z 2 ) 2 + 2 {x — a) 2 (y 2 + z 2 ) + (f'g + h'k' — 2a 2 ) y 2 + (f'h' + g'k' — 2a 2 ) z 2 
+ (x — /') (x — g') (x — h') (x — k') = 0, 
or, what is the same thing, 
(y 2 + z 2 ) 2 -l- (2ic 2 — \ax) (y 2 + z 2 ) + (f'g' + h'k') y 2 + (f'h' + g'k') z 2 
+ {x — f) (x — g') (x — h') (x — k') = 0. 
Now assuming k' = oo , we have 4a = &' = oo , or writing 4a instead of k', and attending 
only to the terms which contain a, we have 
x (y 2 + z 2 ) — h'y 2 — g'z 2 + (x— f) (x — g') (x — li) = 0, 
or, what is the same thing, 
O -/) O “ 9) i x ~ h ') + ( x ~ h') y 2 + (x- g') z 2 = 0, 
where by altering the origin we may make f' = 0. 
It is somewhat more convenient to take the axis of z (instead of that of x) as 
the axis of the cyclide; making this change, and writing also 0, ¡3, <y in place of the 
original constants, I take the equation to be 
Z (Z - 0) (z - 7) + (z - 7) y 2 + (z - ¡3) X 2 = 0, 
viz. this is a cubic surface having upon it the right lines (z = 7, x = 0), (z = /3, y = 0); 
the section by a plane through either of these lines is the line itself and a circle; 
and in particular the circle in the plane x = 0 is z (z — ¡3) + y 2 = 0, and that in the 
plane y — 0 is z (z — 7) + x 2 = 0. And it is easy to see how the surface is generated: 
if, to fix the ideas, we take /3 positive, 7 negative, the lines and circles are as shown 
in fig. 3; and if we draw through Cy a plane cutting the circle GO and the line 
Bx in P, Q respectively, then the section is a circle on the diameter PQ; and 
similarly for the sections by the planes through Bx. It is easy to see that the whole 
surface is included between the planes z—j3, z = 7; considering the sections parallel to 
these planes (that is, to the plane of xy) z = /3, the section is the two-fold line y = 0; 
z = any smaller positive value, it is a hyperbola having the axis of y for its transverse 
axis; z— 0, it is the pair of real lines yy 2 + ¡3x 2 = 0; z negative and less in absolute 
c. ix. 10