464 Mr. J. J. Sylvester's Second Note on 



vector r, reckoned from the centre of the circle. Now it is easy 

 to sec that whilst ( d(pp represents the arc of any curve, \ d<pr will 

 represent the corresponding arc of its first pedal ; so that the 

 spiral in question possesses the remarkable property (capable, one 

 would think, of some practical kinematic application) that these 

 two arcs always remain equal to each other. More general ty, 

 if ]j 2 +p' 2 *, where jo is a rational integral function of <£, and p' 

 its first derivative in respect to <f>, is a perfect square, the arc of 

 the curve and of its pedal will always remain algebraically related. 

 Here, then, we are led to consider the possibility of satisfying this 

 diaphantine condition for cyclodes beyond the second order. At 

 a first glance the problem might seem to be impossible. For if 

 the condition is satisfied by p = F<£, a rational integral quantic in 

 <£> of the order n, it obviously will be satisfied also by F((£-t-X), 

 X being an arbitrary constant ; and consequently we have only 

 {n — 1) and not n disposable constants (or ratios) wherewith to 

 satisfy the n conditions involved in a function of <f) of order 2n 

 being a perfect square. 



This objection, however, is only apparent, and may at once be 

 seen so to be, at all events as regards cyclodes of an even order — 

 say, of order 2m. For we may suppose p = F(<£ + X) (f{(j> + X) ) 2 , a 

 quantic of the order m in (0 +X) 2 ,then F 2 + F' 3 =f 2 + 4(</> -f *,)»/« 

 is a quantic of the order 2m in ((/> + X) 2 , and the m disposable con- 

 stants in /are sufficient to make this a perfect square. Thus, 

 then, the n conditions are not absolutely incompatible. Still the 

 disproof of the incompatibility might seem to involve the necessity 

 of F being a function of ((/> + X) 2 , i. e. of the cyclode being of the 

 symmetrical kind. Moreover, if the problem be attacked by a di- 

 rect exoscopic method for cyclodes of the second, fourth, and sixth 

 orders, it will be found that the only cyclodes which possess the 

 required property are of the symmetrical kind, viz., for the second 



order, p= - (<£ 2 — 1), for the fourth, p= -(0 2 ~4) 2 , and for the 

 sixth p^ % (0 2 -9) 3 f , 0179=4 (<£ 2 ~ 9) (</> 2 - 3G) 2 . The infer- 



ell <V 



ence, then, might appear to be almost irresistible as to the 

 necessity of the symmetrical form holding good. But it is not 



* It will be remembered that r 2 =j9 2 -f-.p' 3 . I may remark incidentally 

 that this equation enables us to extend the well-known one, p 2 = r a — a' 2 , ap- 

 plicable to the first cyclode : the general theorem which includes this as 

 a particular case is obviously 



^ 2 =r 2 -r' 2 +y /2 + ... ±a\ 

 p being the perpendicular on the tangent of a cyclode 'of any order, and 

 r, r', ?•", .. . the distances of the corresponding points in the cyclode and 

 its successive evolutes from the centre of the originating circle. 



t It is very easy to see that there is always one reducible symmetrica! 



