DE, J. CASEY ON CYCL1DES AND SPHEEOQUAETICS. 
then the equation of the cone is 
633 
^>+c- 
b 2 
: 0 . 
Now the equation of a tangent plane to the quaclric is 
«2T ^“T „2 — L - 
Hence the equation of a parallel tangent plane through the centre is 
^ i yy[ i z A —(\ . 
a 2 + i 2 + c 2 — U ’ 
and the condition that this should be a tangent plane to the cone is 
10 
y,l - 
a 2 ( d 1 —r 2 ) ' b 2 (b 2 — r 2 ) ' c~ (e 2 — r 2 )' 
: 0 . 
op" y* 
Hence x' y' z 1 is a point on the intersection of the confocals — \ 
and 
V 
<2 
c — r 
, — 1. Hence the proposition is proved. 
a 2_ r 2W 6 2_ 
101. If tangent planes be drawn to the quadric parallel to the tangent planes of the 
four cones through the sphero-quartic XJV, the locus of their points of contact are four 
lines of curvature on Y. 
Demonstration. Let OH be a central vector of V parallel to an edge of one of the 
cones, then OE is constant, and the proposition is evident from the last article. 
Cor. If a developable be described about V by drawing tangent planes to it along the 
sphero-quartic UV, the four cones whose common vertex is at the centre of V, and which 
stand on the double lines of the developable, inter sect the quadric in the lines of curvature 
stated in the proposition. 
102. If tangent planes be drawn to U parallel to the tangent planes to the cones, the 
loci of the points of contact are sphero-conics ; these sphero-conics are the focal sphero- 
conics of the sphero-quartic. This proposition is evident. 
103. If through any tangent line of a sphero-quartic four planes be drawn passing 
through its four centres of inversion, the anharmonic ratio of these four planes is constant. 
Demonstration. Let \, X 2 , \ 3 , /, 4 be the four values of X, for which U+aY represents 
a cone ; then if we represent by Uj and V, the tangent planes to U and V through the 
given line, the four planes in question are evidently tangent planes to the four cones, 
and their equations are Uj-j-AiVj, Uj-fNjVj, Uj-J-YVj, and the anharmonic 
ratio is 
(A] — A 2 )(A 3 — X 4 ) /'qo\ 
(Aj A 3 ) (A 2 A 4 ) 
Cor. It is easy to see that the theorem, “ that the anharmonic ratio is constant of the 
pencil formed by the four tangents which may be drawn from any point of a plane curve 
