656 
DE. J. CASEY ON CYCLIDES AND SPIIEEO-QUAETICS. 
Cor. 1. By inversion we get the theorem that any circle in the plane of a bicircular 
is in general touched by 24 osculating circles of the bicircular . 
Cor. 2. Any line in the plane of a bicircular is in general touched by 24 of its osculating 
circles. 
Cor. 3. The line at infinity being touched by 24 osculating circles, shows that the line 
at infinity is counted 24 times in the evolute of a bicircular (see art. 174). 
Cor. 4. Chasles’s characteristics for the osculating circles of a bicircular guartic are 
fb= 12, *=24. 
Section II . — Locus of the Poles of the Osculating Circles of a Sphero-guartic. 
180. The equation (121) is the osculating plane of WU at the point x'y'z'w' ; and it 
the coordinates of the pole of this plane with respect to U be X, Y, Z, W, we get 
X = 
X 
4'(«) 
y= 
4' (b) 
&c. ; 
but ad, y\ z', w' satisfy the two equations 
ax 2 + by 2, + cz 2 + duf = 0 , x 2 -\-y 2 J r A -f w~ = 0 . 
Hence, by substitution and replacing X, Y, Z, W by x,y, z, w, we see that the locus of 
the poles of the osculating circles, or, what is equivalent, that the cuspidal edge of 2 is 
the intersection of the two surfaces 
x 
4'(«) 
+ 
’+5 
+ 
4'(c) 
w 
+ _0, * * 
. ( 122 ) 
_y_ \* 
4'0) 
' + o 
4'(e) 
w 
4V) 
= 0 (123) 
181. Since the equation (121) of the osculating plane is satisfied by the coordinates 
of any point in it, we must have 
and substituting as in the last article, we see that the cuspidal edge of 2 is a curve on 
the surface 
Xl 1* ■ I V 4 H I ** 1* , | 
'P'(ffl) / 
~4 i 
= 0 . 
(124) 
Or we may prove this theorem otherwise. The developable % is the envelope of the 
quadric 
+ 4”^ + /c + c + /v + d +/!:~~ 0 ( see art< 
