Successive Involutes to Circles. 4G8 



As the tail goes on still to decrease, the double tangents be- 

 come imaginary, the infinite branches intersect and cut out a 

 lune, one extremity of which, the two cusps of the cyclode under 

 consideration, and the cusp of the parent cyclode, together form 

 a quadrangle, which continually contracts its dimension until 

 finally it vanishes with the tail and the central arc, and the four 

 points merge into the remarkable round point indicated in fig. 1, 

 corresponding to the parabolic or transition case between the 

 cusped and uncusped species. This paradoxical point is a mere 

 creature of the reason, and can by no effort be made sensible to 

 the understanding. Observe that, in this point, the curve dips 

 its beak, so to say, into the cusp of the parent first involute, and 

 yet touches the original circle. Professor Cayley informs me he 

 has met with the same kind of point in an investigation into the 

 form of the parallels to an ellipse, and proposes to call it a trian- 

 gular point, as consisting of the union of a node and two cusps. 

 At this point, in the case before us, we have 



cl s ds 



so that, it will be observed, tj^, as well as -yr, vanishes when (f> is 



made zero. v< ^ r 



This gives me occasion to make a remark which I do not re- 

 member having seen in the text-books, viz. that for any curve, 



ds 

 while in general -=-r indicates the existence of a cusp, this law is 



subject to the exception that if a succession of such derivatives 

 ds d 2 s d 3 s 



3j? df 2 ' dfi'" 



all vanish simultaneously, there will not be a cusp in fact unless 

 the last of the flush is of an odd order. 



Fig. 5 exhibits the critical cases (1) of the double tangents 

 in opposition, (2) on the point of quitting the central branch, 

 (3) in coincidence. Mr. Spottiswoode informs me that this 

 figure has not been drawn with the same attention to mechanical 

 exactitude as the other figures of the Plate. 



In fig. 5 are seen examples of the uncusped species. The 

 Norwich spiral (of which a word or two more presently) belongs 

 to this species, but is not drawn ; its apse lies midway between 

 the centre of the circle and the cusp of the first cyclode. In 

 fig. 2 is seen an example of a symmetrical tricuspidal cyclode of 

 the third order; in fig. 4, of a unicuspid alcyclodc of the same 

 order, where a loop replaces the missing cusps. 



To return to the Norwich spiral ; its radius of curvature p has 

 been shown in the preceding rule to be always equal to its radius 



