1886.] On Systems of Circles and Spheres. 245 



exception ; there are eight pairs of spheres which touch four given 

 spheres, but except in very special cases no spheres exist analogous 

 to Dr. Hart's circles. 



The discussion of the equation of the first degree in power-coordi- 

 nates is much the same as that in Part I. The reduction, however, 

 of the general equation of the second degree is more complicated ; 

 there are four distinct forms to which the equation may be reduced. 



axt + bf + czZ + dw^ + ev^O U) 



the equation of the absolute being 



a 2 + 2/ 2 + z 2 + w 2 + ^=0; 



this is the general cyclide, of either the fourth or third order ; if d = e, 

 it has two cnic-nodes. and if b = c, d = <?, it has four cnic-nodes ; but 

 in this case the sphere x = must be imaginary. 



ax 2 + by 2 + cz 2 + dw 2 =0 (/3) 



the equation of the absolute being 



x 2 + y 2 + z 2 —4>wv=0. 



This is the general case of a cyclide having one cnic-node, if b = c it 

 has three nodes; the former case is the inverse of a central quadric, 

 the latter the inverse of a central quadric of revolution : the spheres 

 x, y, z are real in this case. 



ax 2 + by 2 + 2hzw=0 (7) 



the equation of the absolute being 



x 2 + y 2 + z 2 —4<wv=Q. 



This represents a cyclide having two principal spheres and a binode ; 

 if a or b = the node is a unode. 



ax 2 + 2hyz + dw 2 =0 (t) 



the equation of the absolute being 



x 2 j-y 2 + z 2 — 4<wv=0. 



This represents a cyclide having only one principal sphere ; and 

 a cnic-node, which becomes a binode when a = 0, and a unode when 



n = 0. 



The different species of cyclides are then briefly discussed in 

 detail. 



VOL. XL. 



s 



