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



Foremost amongst these stands the algebraical form of equa- 

 tion (and that a quantic), which connects not only the arc, but 

 also the squared radius vector with the angle of contingence, and 

 consequently the two former with one another. In a marvel- 

 lous and, so to say, transcendental fashion, these curves partici- 

 pate in the nature of algebraical curves — their apses, cusps, and 

 points of retrocession being counted by the order of the involu- 

 tion, and becoming imaginary in pairs. 



I need hardly say that by a second involute I mean an in- 

 volute of the first by a third, an involute of a second, and so 

 on in continual progression. To any given curve all its first 

 involutes form a system of parallel curves, so that in general 

 the number of /wm-parameters to a curve will be augmented 

 by i when we pass to its general involute of the zth order. 

 In the case of the circle, however, owing to its homogeneity, the 

 first involute, like the curve itself, contains only one /om-para- 

 meter (it being, in other words, a property of the first involute, 

 that when rotated round a certain point, the curves so generated 

 continue always parallel to each other) ; and so the number of 

 /orm-parameters in the general ith involute will contain i, and 

 not t + 1 parameters, as the general formula would require. 



I shall use (/>, s, r, 6 to denote the angle of contingence, arc, 

 radius vector, and vectorial angle of the curves under conside- 

 ration. 



Starting from the circle s = acj) ) a set of corresponding succes- 

 sive involutes will, as is well known, be represented by 



and so on, according to the obvious law 



Now in general for any curve whatever, if we call p the per- 

 pendicular on the tangent from an arbitrary pole, q the projec- 

 tion of the radius on the tangent, we have 



*=-$; n 



also 



rf course not new*; they are given by 

 * We have only to take P, P', two consecutive points, and on PP', P'T, 



These equations are of course not new*; they are given by 



