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



which he has taken them ; but in themselves, easily as they 

 can be obtained, they contain the whole theory of the remark- 

 able curves to which this note refers. In the case before, 



ds 



— = ~F6, where F is (a quantic in, i. e.) a rational integral func- 



dcp 



tion of cf). Hence we have for the solution of (1), 



p=Y -F" + F""... + Acos<£ + Bsin</>, 



j9=F"-F'" + -Asin<£ + Bcos0; 



wherefore r 2 is known in terms of F and the arbitrary constants 

 A and B, whose values depend on the position of the origin 

 from which r is reckoned, by a due choice of which they may be 

 made to vanish. One will readily suppose that this eligible po- 

 sition of the pole must be the centre of the generating circle ; and 

 the proof is as follows : — 



If r 13 r 2 ,... n be any radii vectores corresponding tosj, s 2 , . . :s if 

 it is well known, and follows from the definition of the involute, 

 that 



^x^-^g+a? W 



Now, suppose that for any number i the origin has been so 

 chosen that 



then 



T\— {Si_ x — Si_ 3 + . . . ) 2 + (Si- 2 — ^_ 4 + • • • )% 



U dSi *_, \ + ( Si _ 2 - Si - 4 . . .)(*-8 ~5i_5 • . • ) J 



== Si— 2 Si— 4 "r • • • 



Hence 



r? +1 = {s-Si_ 2 + *_ 4 . . . ) 2 + («_, -^_ 3 + . . . ) 2 ; 



and the supposed relation, if true for any value i 3 is true for all 

 superior values; but when the origin is at the centre, s l = a(f> ) 

 r l 2 = a?; and consequently the equation (4) is true universally*. 



* The above result might have been deduced more directly from the 



equation _-1=joj_ 1 , which is true for any curve and its evolute. In fact 

 dcp 



this paper need never have been written (for all that it contains is a straight- 

 forward inference from four equations which may be found scattered up 

 and down in elementary treatises), had it been the custom to regard those 

 equations as forming collectively a connected apparatus. I mean the four 

 following, where the unaccented and accented s and p refer to any curve and 

 its evolute respectively, being the angle of contingence common in mag- 



