672 
DR. J. CASEY ON CYCLIDES AND SPHERO-QTJARTICS. 
which UY inverts ; consequently the evolute has the same singularities as the cone (see 
art. 51, ‘ Bicircular Quartics ’). 
227. If U and Y touch, the singularities of A are (see Salmon’s 4 Geometry of Three 
Dimensions,’ art. 342) 
-^4 
II 
II 
(3=0, 
n =6, 
II 
ss 
II 
05 
II 
^4 
II 
y= 4- 
Hence, by the method of the last two articles, we get for the bicircular and its evolute 
the following singularities : — 
Bicircular, y>= 4, j= 6, &=3, r= 4, z=0, t= 6. 
Evolute, v= 6, c)=G, r=6, z=4, ;=4. 
This bicircular is the inverse of an ellipse or hyperbola ; and the characteristics of the 
bicircular are the reciprocals of the characteristics of the evolute of an ellipse or 
hyperbola. 
228. If U and V osculate, we have (see Salmon’s 4 Geometry of Three Dimensions,’ 
art. 342) the singularities 
m=4:, <7=2, (3=1, 
n =1, h= 2, oc=2, 
r = 5, a = 1, y= 2. 
Hence for the bicircular and its evolute we get 
Bicircular, ^ = 4, v=5, (5=2, r= 2, x=l, i = 4. 
Evolute, [Jj= 4, =5, & = 2, r=2, «=1, ; = 4. 
This bicircular is the inverse of a parabola; and we see that it has the same characteristics 
as its evolute (see foot-note, art. 70, ‘ Bicircular Quartics’). 
229. By considering special sections of A and % we get, as in the preceding articles, 
the singularities of bicircular quartics, Cartesian ovals, circular cubics, and the evolutes 
of these respective species of curves. The reader who has followed out the method of 
reasoning in recent articles can easily account for the result in each case. In order to 
save space, I shall give only the) particular cone whose intersection with the inverse of 
the/sphere U gives the bicircular and the evolute. The numbers for the several cones 
are taken from Salmon’s 4 Geometry of Three Dimensions,’ art. 324 (see also Cambridge 
and Dublin Mathematical Journal, vol. v. p. (23)-(4G)). 
I. When U and V do not touch. 
1°. Cone whose vertex is a point of the system A : 
(M = 3, z = 0, v 
i=6, r=0, V A circular cubic of the sixth class. 
i=9, a=o. J 
