302 Mr. J. J. Sylvester's Note on Successive 



direction from point to point in a curve. Accordingly we ought not 

 to say, as is usually done, that at a cusp the tangent is suddenly 

 reversed in direction, but, rather, that the increment of the arc on 

 passing through a cusp changes sign, as it ought to do according 

 to first principles ; for the flow of the incremental arc, from being 

 concurrent with, becomes opposite to that of the rotating tangent 

 line which carries it, or vice versa. Thus in the common cycloid (a 

 curve of infinite length to the eye and with an infinite number of 

 cusps) we have s=c cos <£, which, subject to this interpretation, is 

 perfectly true and self-consistent for the whole extent of the curve 

 from infinity to infinity. In that case we have a visible repre- 

 sentation of quantity undergoing an infinite number of periodic 

 changes, although the subject matter of the quantity is continually 

 changing and never recurs. In the case of the involutes of the 

 circle, the number of those periodic changes is of course finite and 

 equal to the number of the cusps. If A, B, C, D, . . . , L be the 

 cusps in natural order on the curve whose involute is to be found, 

 and if we call % the radius of curvature of the point of the involute 

 corresponding to A (x being taken positive when this radius 

 is in the position into which it would be brought by unwind- 

 ing a string from the infinite branch adjacent to A), and if we 

 form the series a?- AB + BC -CD . . . ± KL, where AB, BC, . . . 

 are the arithmetical lengths of the branches, it is clear that at 

 each term of this series in which a change of sign in the sum 

 takes place the involute will have a cusp; if the number of 

 banches is odd and x is negative, the sum may remain negative 

 at whatever term we stop, and then there will be no cusp in the 

 involute so engendered ; but when the number of points A, B, 

 C, . . . , L is even, then it is easy to see that one of the infinite 

 branches must contain a cusp of the involute and the other be 

 vacant. The second involute, whether cusped or not, manifestly 

 consists of two parts symmetrically arranged about its apse. If 

 we form a third involute by unwinding from this apse as origin, 

 the figures so formed will again be symmetrical, and the cusps 

 will lie at the vertices of an isosceles triangle; and now every in- 

 volute of this symmetrical third involute will again be symme- 

 trical, and so on continually, the number of conditions imposed 

 on the parameters in order to ensure symmetry in the involute 



i — I 

 of the 2th order being thus the integer part of — — • When this 



symmetry obtains, the algebraical equation requisite for deter- 

 mining the polar equation depends on the solution of an equation 

 of only the ith. instead of the 2ith degree ; for it is clear that in 

 this case the functions G 2 and G' 2 may be made to contain only 

 powers of </> 2 . Thus we may very easily write down the general 

 polar equation to the absolutely general second involute, and 



