152 THE CYCLIDE. 



If the origin be transferred to a conical point, and if the dimensions of 

 the figure be then indefinitely increased, the cyclide becomes ultimately a right 

 cone, having the same conical angle as the original cyclide. If b = c, the cone 

 becomes a plane. 



If b remains finite, while c, r, and x are each increased by the same 

 indefinitely great quantity, the cyclide ultimately becomes a right cylinder, whose 

 radius is r-c. 



Inversion of the Cyclide. 



Since every sphere, when inverted by means of the reciprocals of the radii 

 drawn to a fixed point, becomes another sphere, every cyclide similarly inverted 

 lecomes another cyclide. There is, however, a certain relation between the 

 parameters of the one cyclide and those of the other, namely 



or 



(15). 



If the point of inversion be taken on either of the circles 



c ' = > z = (16), 



or a ? + z I -2^-&'-r'--c 1 = 0, y = (17), 



c 

 crx 



the cyclide will become a surface of revolution in which 6 = 0, and 



r" c'-r 5 



if the point of inversion be on the first circle, or 



if it be on the second. 



When r is less than c, the first circle is real; and when r is greater 

 than 6, the second circle is real In the ring-cyclide r is between b and c, 

 and the cyclide can be transformed into an anchor ring in two different ways. 



