642 
DR. J. CASEY ON CYCLIDES AND SPHERO-QTJAETICS. 
where X, X 1 are small circles, whose centres are the double foci of the sphero-quartics, and 
that this is again equivalent to the equation in elliptic coordinates, 
lt*- V ' = ky/C, ( 101 ) 
135. If in the equation we put 
%'=2"±7i*, 
we get 
t(^"±h 2 )=7c 2 C, 
and the intersections of the circle %" with the sphero-quartic are also the points of inter- 
section of the circle JCCdnld% with the sphero-quartic. Hence the sphero-quartic meets 
the circle "S" only in two points. Hence we have the following theorem : — Any circle 
whose plane is perpendicular to a double focal line of a sphero-quartic meets the sphero- 
quartic only in two points. 
136. Let P, P' be the points in which the circle %" meets the sphero-quartic, then PP' 
will he a generator of a paraboloid passing through the quartic. Hence we easily infer 
the following theorem : — If II, H, the double foci of a sphero-quartic , be joined to any point 
P of the quartic , and circles described with radii IIP, H'P cutting the sphero-quartic 
again in P', P" respectively , the lines PP', PP" are parallel to the planes of circular sec- 
tions of quadrics passing through the quartic , and they are the generators through P of 
one of the paraboloids which can be drawn through the sphero-quartic. 
Cor. Since three paraboloids can he drawn through the sphero-quartic, this theorem 
affords another proof that a sphero-quartic has six double focal lines. 
137. If we take the canonical form of a sphero-quartic aa 2 -\- &/3 2 + cy 2 -f-(^ 2 =0, we 
get precisely, in the same way as in art. 117, the following system of equations, each 
denoting a sphero-quartic confocal with the given sphero-quartic, that is, each having a 
quartet of foci common with it : — 
/ \ O / 7\ <>0 
( 102 ) 
(103) 
(104) 
(105) 
In these equations the 7r's may have any value. 
138. As in art. 118, we can show that the reciprocals of two sphero-quartics having one 
quartet of foci common are two sphero-quartics having quartic contact at the points 
where they are intersected by their common circle of inversion a. 
139. The two sphero-quartics 
aT+bf+cy^ + dP, 
(a-b)P 
r 
1 
sf 
to 
1- 
{a—d) 8 2 
{a — b) +k 
1 ( a — c) + k 1 
(a — d) + k 
( b — c)y' 2 
(b-d) 8 2 
(, b—a )« 2 n 
{b — c) + k' 
|5i 
+ 
1 
r 
~T(b-a) + k' U 
(c-d) 8 2 
c 
1 
o» 
ox 
S' 
1 
+ 
^3 
1 
' (c — a) + k" 
j 
S' 
1 
+ 
1 
< 
( d — a)or 
, (d-b)p 
c 
c* 
JN 
S' 
1 
[d — a) + k'" 
1 {d-b)+k" r 
‘ (d— c ) + k" 1 
+ 
/ 3 ' 
- .2 S2 
j_7_ , L 
i ^ i ,t 
