THE THEOEY OF . EOLLING CUEVES. 21 



We may likewise prove by the same method as before, that the result of 

 performing this inverse operation an infinite number of times is the logarithmic 

 spiral 



As an example of the inverse method, let the traced line be straight, let 

 its equation be 



r. = 2a sec ft, 



, , 



then P=i=& = to = 



therefore suppressing the suffix, 



r 4 



-ar, 



\de) a. 



1 



dr 



r I 



.. /2a _ \ 

 .'. ft = COS M 1 ), 



2a 

 ~l-cos0' 



the polar equation of the parabola whose parameter is 4a. 



The last case which we shall here consider affords the means of constructing 

 two wheels whose centres are fixed, and which shall roll on each other, so that 

 the angle described by the first shall be a given function of the angle described 

 by the second. 



T A a ^ i d&s I", 



-Let P, = <pc/i, then r l + r, = a, and -jy = > 



do, r. 



_ 



d& l ~ a *i ' 



Let us take as an example, the pair of wheels which will represent the 

 angular motion of a comet in a parabola. 



