Successive Involutes to Circles. 465 



so ; it is true that only cyclodes of even orders are reducible, i. e. 

 capable of giving rasa rational integral function of $ ; but after 

 the sixth order, i. e. beginning with the eighth, non-symmetrical 

 reducible cyclodes come into existence, and, as the order rises, 

 become infinitely more numerous than those of the symmetri- 

 cal kind. 



Calling 2m the order, every distinct mode of making the par- 

 titions of numbers expressed by the two simultaneous equations, 

 f 2 , 1 +# 2 + ... -\-Xi=m\ 

 ^2/i+2/2+ •>• +yi=mf ' 

 where i takes all possible values, gives rise to a system of equa- 

 tions yielding in general many solutions ; and it is only when 

 x \ = Vv x 2~Vv • • • y x i — y% tnat the solutions are of the symme- 

 trical kind. Moreover, even in that case, in general, and subject 

 only to rare cases of exception, the reducing system of equations 

 gives two distinct groups of solutions, one corresponding to 

 symmetrical and the other to non-symmetrical cyclodes*. This 



cyclode of the order 2m defined by the equation 



r ir{2m) K ^ J 



corresponding to which 

 Thus, when 771=2, 



when <£=0, we have 



r =^y^ 2 -™ 2 )"'" 1 ^+" ,! )- 



r =a ( * 4_l6) ' 



whence we may derive the following construction : — Draw an uncusped se- 

 condary cyclode with a tail equal to one-third of the radius ; unwind from 

 this a ternary cyclode beginning from the apse, which will become a cusp 

 in the cyclode so engendered ; and from this last cyclode, beginning at its 

 cusp, again unwind a new cyclode, which will possess a triangular point at 

 the apse of its atavian secondary cyclode. This will be a quartic reducible 

 cyclode, and, as regards form (irrespective of position and magnitude), the 

 only one that exists. By the way, it may be noticed that a system of coor- 

 dinates consisting of the vectorial angle and angle of contingence furnishes 

 what may be termed a. form equation, i. e. one in which actual magnitude 

 is ignored. Thus, ex. gr,, tan 0-=zk tan <fi is the form equation to a conic. 



* It is to be understood that every x and y must be an actual integer, 

 zero being for this purpose to be regarded, not as a number, but as a ne- 

 gation of number. Furthermore, if the x and y numbers are not only re- 

 spectively equal each to each, but have all the same value (as ex. gr. unity), 

 the corresponding system of equations become incompatible ; or, to speak more 

 philosophically, the order of the system becomes zero, which here per contra 



Phil. Mag. S. 4. Vol. 36. No. 245. Dec. 1868. 2 H 



