Involutes to a Circle. 303 



might, if it were worth while, do as much for the symmetrical 

 class of third and fourth involutes, of which the former will con- 

 tain two and the latter three arbitrary parameters, by solving a 

 cubic and biquadratic equation in these two cases respectively. 



It is easy to see also that the arco-radial equation to the symme- 

 trical involute of an odd order is of only half the degrees in s and 

 r 2 that it is of in the general case, and for the symmetrical invo- 

 lute of an even order, although of the same degrees in r 2 and s as 

 in the general case, involves only the even powers of s. 



A few words upon the second involute, and I have done ; for 

 it is difficult to deal with theory in any detail so as to be intel- 

 ligible, or even safe, without the suggestive and regulative aid of 

 drawn figures, which I have not yet been able to obtain in a form 

 fit for use. 



The two principal classes to distinguish in the second invo- 

 lute are the cusped and uncusped species. The cusped second 

 involute winds round the parent curve upon which the extremities 

 of its finite branch rests. The uncusped species crosses itself, 

 and intersects each branch of the first involute of which it en- 

 closes the cusp, its node being on one side of it and its apse on 

 the other. The transition case is when the unwinding begins 

 from the cusp of the first involute; the second involute so ob- 

 tained has a very singular point at that cusp, which may be re- 

 garded either as an abortive loop, or as a coincident pair of cusps*. 



The general connecting equations for this involute may be 

 put under the form 



0= s in-^+(A, 

 r T 



where a is the radius of the circle ; and there will be a loop or 

 cusps according as a -\-b is positive or negative; when b= — a, 



* Mr. Crofton has noticed, in an ingenious paper published in the 

 ' Mathematical Messenger,' that this involute is the locus of the centres of 

 all the circles cutting orthogonally the originating circle and the parent 

 first involute. This is seen very easily as follows. 



P= a (^ ~l)> s'=p+p" = a¥, 



r 2 =p 2 -\-p' 2 —a 2 ^~+a, or r 2 — a 2 = s' 2 , 



showing that the tangents to the circle and first involute from any point in 

 the second are equal to one another. 



