G50 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUAETICS. 
x, y, z shall lie on the line joining x, y, z to a given point x\ y', z' ; denoting the coordi- 
nates of any point on this latter line by x' — XX, y'—xy, z'—Xz , we find (as in Salmon, 
art. 472) that the generalized evolnte of ax 2 -{- by 2 -j- cz 2 is the discriminant of the conic 
ax 2 by 2 cz 2 
(« + Ap+(U+U) + (7+Ap ~ 
with respect to X, and therefore the required evolnte is the curve of the sixth degree 
ai(b — cfx ?s + b ts (c — iify ?t -f- c\a — =0; (114) 
and the reciprocal of this with respect to the circle x 2 -\-y 2 -\-z 2 = 0 is the quartic curve 
(b — c ) 2 (c — a ) 2 (a — b ) 2 
hex 2 
car/ 
abz 2 
= 0 . 
(115) 
1G5. The equation (115) occurs so frequently in subsequent articles that we shall 
examine its properties with some detail. If in the equation of the developable A formed 
by the tangent lines of WU we makew = 0, the result will be the square of (115). 
Hence we infer the following theorem : — The nodal lines of the developable A are the 
reciprocals of the generalized evolnte of the conics in which the reciprocals of the focal 
quadric are cut by the faces of the tetrahedron. 
166. If we invert the sphere U from one of the eight centres of inversion (see art. 83) 
into one of the faces of the tetrahedron, the sphero-quartic WU will invert into a bicir- 
cular ; and it is easy to see that the nodal line of A in that face of the tetrahedron will 
be the locus of the intersection T of tangents to the bicircular at a pair of inverse 
points P, P' (see art. 43, ‘ Bicirculars’) ; but the point T is evidently the centre of simi- 
litude of two consecutive generating circles of the bicircular. Hence the locus of T is 
the envelope of the axis of similitude of three consecutive generating circles of the 
bicircular. Hence we infer the following theorem : — If a sphero-quartic WU be 
inverted into a bicircular on the plane of one of the faces of the tetrahedron, the nodal 
line of the developable A formed by the tangent lines of WU is the envelope of the radical 
axis of a pair of inverse osculating circles of the bicircular. 
167. The equation (115) may, by incorporating constants with the variables, be written 
in the form 
1 1 1 
and in this form it will be satisfied by the coordinate of the point common to the system 
of determinants 
x, y, z , 
sec®, cosec ip, 1. 
If we call this the point p, then the equation of the chord joining the points p, p' will 
be the determinant 
a, y , z, 
seep, cosecp, 1, 
seep', cosecp', 1, 
= 0 . 
(116) 
