DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 599 
by r", See. Again divide the first column by r', the second by r" See., and we get the fol- 
lowing result : 
U 2 = 
- 1 , 
cos (/3a), 
cos (ya), 
cos (bcc), 
a-^-r , 
cos (a/3), 
- 1 , 
cos (y/3), 
cos (ci/3), 
/3-hr" 
cos (ay), 
cos (/3y), 
- 1 , 
cos (iy), 
yh* 
V" 
cos(aci), a 
cos(/3S), /3 H-r", 
cos(y^), y~r"', 
-1 
S+V'" 0. 
(29) 
Cor. Hence, if the four spheres of reference a, /3, y, b be mutually orthogonal, the 
equation becomes 
+ 
+ ( jii ) + 
= 0 ; 
and by incorporating constants with the variables it becomes a 2 -f-/3 2 -f-y 2 -j-cd:=0. 
We shall find the value of U 2 in this latter form very important. 
29. If the tetrahedron to which F is referred be inscribed in F, the coefficients a, b, c , d 
vanish ; then forming the tangential equation and replacing /., y,, v, § by a, /3, y, 6, we 
have the following theorem. If the equation of a cyclide be in the form 
0, 
n, 
m , 
ib 
a, 
n, 
o, 
l , 
<b 
P> 
m, 
l , 
0, 
r, 
7, 
jp> 
£> 
r , 
o, 
1, 
a , 
P> 
r > 
0 
= 0, 
(30) 
that is, of a symmetrical determinant bordered with the variables whose diagonal terms 
are each zero, the spheres of reference have each double contact with the cyclide ; in 
other words, they are generating spheres. 
Cor. From this theorem, combined with article 3, we can easily get the equation of a 
sphere circumscribing a tetrahedron. 
30. If the equation of a cyclide be given in the form 
a 2 +/3 2 + y 2 + § 2 + 2/(/3y + aS) ] 
(°-*-) 
+ 2 m(ay -j- /3!$ ) -j- 2n (a/3 -j- yc$) = 0 J 
where 1+2 lmn=l 2 -\-rrd -{-n 2 , it can be proved, precisely as in Salmon’s ‘Geometry of 
Three Dimensions,’ p. 153, that each of the spheres of reference cuts the cyclide in two 
circles. Hence (see art. 19, Cor. 1) each of the spheres of references is a generating 
sphere. 
31. If the coefficients of a 2 , /3 2 , y 2 , ci 2 in the general equation of a cyclide vanish, then 
the coefficients of A 2 , yd, v 2 , g 2 vanish in the tangential equation of the focal quadric ; 
and hence if the coefficients of the squares of the variables vanish in the equation of a 
