40 Mr. W. G. Horner on the Properties of the Dzedaleum. 



the circular parts, this gives x 2 = 



_ .r 



a -jr 



which bears a strik- 



ing analogy to the equation to the common cissoid. In fact, 



y 



if we assume x s = „_ ^ as the type of the cissoid family, 

 that of Diocles is of the first order, and this of the second. The 



general polar equation will bew = 



a cos <p^ 

 sin <f 



Hence the immediate relation between the first and se- 



cond cissoids is apparent. For, since u x = 



a cos <p- 

 sin <f> 



and 



a cos <p , I,- 



Uc, = — -. -, we nave «, = w 9 cos <p ; or the radius vector 



simp L T 



(from the vertex) of the cissoid of Diocles is equal to the 



corresponding abscissa of the optical cissoid. (See fig. 2.) 



Fig. 2. 



A relation not very dissimilar exists between the latter and 

 the Apollonian parabola, whose radius vector (always taking 



the vertex for the pole) is U = — : ; whence u 9 = U sin 



r ' sin f 2 2 



<p ; or, the radius vector of the second cissoid is equal to the 

 corresponding ordinate of the parabola. 



If we assume x m *—= 



a cos tp „ 



a " + y n sin tp (sin f m ± cos f m ) « , 



for the more general type of the cissoids, as x m = ■?- 



,m+n 



a* 



.•. U„ as a cos ^" is of the parabolas, it becomes apparent 



sin <f> n + 1 

 that the latter form a genus intermediate to the two genera 

 into which the former resolve themselves. This relation is 

 comparable to that which the common parabola bears to the 

 ellipse and hyperbola, whose vertical equations are 



