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method of generating quartics having two finite double points, and the transition from 
that was easy to the discovery of the method of generating surfaces of the fourth degree 
having a conic for a double line (see Chapter II., Section II. of this paper). 
63. Since the plane of each of the circles S J — L, S J — M, S^— N passes through the 
pole of the plane of the orthogonal circle J with respect to U, that is through the point 
((5’ H’ !?) ( see ai ^ ^3), ^ f°ll° ws that the plane of the generating circle 
- L) + ^(S* - M) + - N) 
passes through the same point, therefore the pole of the plane of the generating 
circle with respect to U lies in the plane Ia , + H < yd-Ks=G ; in other words, the pole 
of the generating circle is complanar with the poles of the planes of the circles of refer- 
ence, and therefore it describes a conic in space, namely, the conic in which the plane 
Is+Hy+K* = G intersects the cone reciprocal to 
(a, l, e,f, g, hJL, M, N) 3 =0. 
64. Since the planes of the circles S } — L, S 1 — M, S J — N are respectively parallel to 
the planes L, M, N, the envelope of the plane of the generating circle is a cone similar 
and similarly placed with the cone (a, h, c, f g, h\ L, M, N) 2 , and its vertex is at the 
I H K 
point g-. Hence we are led to the known proposition, that a seller o-quartic is the 
intersection of a sphere and a cone of the second degree , and therefore that it is the inter- 
section of a sphere and a quadric. 
65. If the poles of the planes of the generating circles S’ — L, &c. with respect to U 
be the centres of three spheres a, (3, y which cut the sphere U orthogonally, then «, (3, y 
will intersect U in the circles S 1 — L, S J — M, S 1 — N respectively, and the cyclide 
(«, b, C,f, g, hja, (3, y) 2 
will intersect U in the sphero-quartic 
(a, b, c,f g, h,X &-L, S*-M, S^-N) 2 . 
Hence we are led to the known theorem, that a sphero-quartic is the intersection of a 
sphere and a cyclide. 
66. The results we have arrived at in this Chapter may be projectively extended to 
curves described on quadrics, in other words, the analytical proof is the same for this more 
general case as for the particular one we have examined. Thus, instead of a sphere U 
of radius unity, let us take a quadric S — K 2 inscribed in S as the surface on which the 
curve is described. Now the quadric S — Lf intersects S — K 2 in two plane conics, and we 
may take the equations of these two conics to — L=0, and S^+L— 0, precisely similar 
to the method we have given of representing small circles on the surface of a sphere. It 
is plain that the equations S^—L, S--J-L have another interpretation, namely, they repre- 
sent respectively the two parts into which the surface S — L 2 is divided by the plane 
L=0; so that on the surface of the quadric S — K 2 =0 a plane conic is represented by 
an equation of the same form as that of a circle on the surface of a sphere. Again, if 
