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



clode may be said to be an integral algebraical spiral, i. e. one 

 in which the perpendicular on the tangent becomes a rational 

 integral function of the angle of contingence. 



I find in a certain question, presently to be alluded to, the 

 theory of the class so indisputably bound up with that of the 

 genus, as to persuade me of the importance of the theory of the 

 former being gone into by some one who has leisure for the in- 

 vestigation, and of the desirableness of an organic description 

 being discovered or devised for the rational fractional case. The 

 peculiar feature of the cyclode class is the absence of points of 

 inflection, real or imaginary. The cusps of cyclodes are strictly 

 analogous to the asymptotes in algebraical curves, like them 

 entering and disappearing in pairs, creating partial interrup- 

 tions of continuity, and thus separating the curve into distinct 

 branches*. In the same way as the order of an algebraical 

 curve is determined by the number of its intersections with 

 any right line, so that of any such spiral may be characterized 

 by half the number of its intersections with any circle having its 

 centre at the pole. When the rational fraction which expresses 

 the value of the perpendicular is of the degree m in the numerator 

 and n in the denominator, the order will thus become the dominant 

 of the two quantities m + n, 2n. 



Plate III. figs. 1, 2, 4<, 5, exhibit examples of cyclodes of the 

 first, second, and third orders, distinguished respectively, where 

 required, by the number of accompanying dotted lines f. Let 

 us consider more closely those of the second order, which sepa- 

 rate themselves into two classes, the cusped and uncusped. The 

 cusped class are the analogues of the hyperbola, the uncusped. 

 class of the ellipse, and the very remarkable secondary cyclode 

 whose tail (to use the late Dr. WhewelFs expression) is zero, and 

 which may be termed the natural one of the order, is the analogue 

 of the parabola. In the former Note I spoke of the point where 

 it meets the cusp of the parent curve as an abortive loop or a co- 

 incident pair of cusps ; the words marked in italics are calculated 

 to convey a false impression, and are to be considered cancelled. 

 So, too, the passage (p. 301) commencing with the words "at such 

 points/'and ending with the words " points of retrocession/'' and 

 containing the misprint " will change its curve" for " change its 

 course," is erroneous and is also withdrawn. Instead of points 

 of retrocession, I propose to call these iC points of radiation" or 

 " radiant-points ;" the intervention of the cusps prevents the 

 happening of the supposed " retrocession" at such points. This 

 error illustrates the danger of, so to say, fighting in the dark, i. c. 



* Parallelism for cyclodes bears some analogy to projection for algebraic 

 curves, and operates in the way of addition or diminution upon the cusps 

 as the latter process does upon the asymptotes. 



f In figs. 5 and 3, which refer to cyclodes of the second order exclu- 

 sively, it has not been thought necessary to adjoin the dotted lines. 



