598 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
Let a, b, c, d be the centres of the spheres of reference, a, j3, y, h ; r', r", r'", r"" their 
radii; then, since U is the Jacobian of a, /3, y, S, the tangents from a, b, c, d to U are 
equal to r\ r", r'", r'"' respectively. Again, let perpendiculars from a, b, c, d on the 
tangent plane T be denoted by a, /a, v, g. Now the result of substituting the coordinates 
of P in a=P« 2 — r 2 = difference of squares of tangents from a to the limiting point P, 
and the orthogonal sphere U = 27.0P (O being the centre of U). Hence the results of 
substituting the coordinates of any point P of the cyclide in the equations of the spheres 
of reference are proportional to the perpendiculars X, v, g. Hence we have the fol- 
lowing theorem : — 
If ( a , b, c, d, l, m, n, p, q, rja, /3, y, o) 2 =0 be the equation of any cyclide , 
(a, b, c, d, l, m, n, p, q, rj\, p, v, §) 2 =0 
is the tangential equation of the corresponding focal quadric of the cyclide. 
Cor. 1. Hence if we are given the equation of the focal quadric, we are given the 
equation of the cyclide, and vice versa. 
Cor. 2. Hence, when the sphere of inversion U and the focal quadric F of a cyclide 
are given, the cyclide is determined ; but U is determined by four constants and F by nine. 
Hence a cyclide is determined by 4 + 9 = 13 constants. 
28. By means of the last article we are enabled to get a very important expression for 
the sphere U in terms of the four spheres of references. Thus, since a cyclide is the 
envelope of a variable sphere cutting U orthogonally, and whose centre moves along the 
surface of a given quadric F, now if the given quadric F he the sphere U itself, it is 
plain that the cyclide will in this case be the sphere U counted twice, that is U 2 . But 
the equation of U in tetrahedral coordinates, x. y , 2 , w being the coordinates, is (see art. 3) 
(r'x) 2 -\-(r"yy + (r"T) 2 + (r w w) 2 
— 2 'd'd'xy cos (a/3) — ‘ly J r"'xz cos (ay) — 2 r r r""xw cos (aS) 
— 2 r"r"'yz cos (/3y)— 2 i J, r""yw cos (/3d) — 2r"'r""zw cos (yo) = 0. 
Hence, forming the corresponding tangential equation, and substituting a, (3, y, S for 
the variables, we get the following determinant for the square of U 
rV cos (/3a), 
r"'r' cos (ya), 
r""r' cos (ha), 
a, 
r'r" cos (a/3), 
r"'r" cos (y/3), 
r""r " cos (cS/3), 
ft 
r'r 1 " cos (ay), 
r"r"' cos (/3y), 
r'"\ 
r'"W" cos (Sy), 
7 , 
r'r"" cos (ah), 
r"r"" cos ((3h), 
r"'r"" cos (qo), 
r llll2 
a, 
ft 
7 , 
0 . 
This determinant may be simplified as follows : — divide the first row by r', the second 
