372 Proceedings of the Royal Irish Academy. 



if this pass through the centre of inversion, 



y 



These values satisfy equation (ii.). 



Therefore, the middle points of the 16 double normals of a bicircular 

 quartic lie on the rectangular hyperbola which passes through the four 

 centres of inversion and the centre of the focal conies. 



This curve remains the same if we suppose the radius of the circle 

 to vary : we thus obtain a system of bicircular quartics, having four 

 common concurrent double normals which have the same points of 

 bisection; the locus of the middle points of the 12 other double 

 normals is the rectangular hyperbola under consideration. This curve 

 may also be regarded as the locus of the vertices of the common self- 

 conjugate triangles of a fixed conic and a system of concentric circles. 



The 12 double normals of a circular cubic have exactly similar 

 properties. 



CrCLIDES. 



Let /S = a;^ + y2 + z^ + 2> + 2gy + 2h% + (Z = 0, 



i^=— + ^+--1 = 0. 

 a c 



The coordinates of the centres of inversion are 



, , af ig ch 



•^ ^ « + A' 5 + A' c + A' 



where A is given by the equation 



«+A 6+A c+A 

 The equation of the cyclide is 



[a;^ + y-' + z''-dJ = A \_a {x + ff + h (w + yf + c {z + A)*]. 



Then, by reasoning exactly similar to that employed for plane bicircular 

 quartics, we obtain the following theorems : — 



A. quartic cyclide has thirty double normals passing by sixes through 

 the five centres of inversion. 



The middle points of these thirty double normals lie on a twisted 

 cubic, the intersection of three rectangular hyperbolic cylinders, which 

 passes also through the five centres of inversion and the centre of the 

 focal quadrics. 



