THE THEORY OP ROLLING CURVES. 19 



Now, cos" 1 ^ is the complement of the angle at which the curve cuts the 



_. _ np fY) * 



radius vector, and cos' 1 cos" 1 is the variation of this angle when O n varies 

 by an angle equal to a. Let this variation = <f> ; then if - l = /3, 



n n' 

 Now, if n increases, <j> will diminish ; and if n becomes infinite, 



<j> = ^5- + ^5- = when a and /8 are finite. 



Therefore, when n is infinite, $ vanishes ; therefore the curve cuts the radius 

 vector at a constant angle ; therefore the curve is the logarithmic spiral. 



Therefore, if any curve be rolled on itself, and the operation repeated an 

 infinite number of times, the resulting curve is the logarithmic spiral. 



Hence we may find, analytically, the curve which, being rolled on itself, 

 traces itself. 



For the curve which has this property, if rolled on itself, and the operation 

 repeated an infinite number of times, will still trace itself. 



But, by this proposition, the resulting curve is the logarithmic spiral ; 

 therefore the curve required is the logarithmic spiral. As an example of a curve 

 rolling on itself, we will take the curve whose equation is 



cos 



TT dr I v\ / 0Y 1 ' 1 



Here -~=2"a sin- cos- ; 



a# V n \ n l 



\ i \ f 



m 

 cos 



.'. r. = 2 n =2 i= = , 



' i\-- / 6\ / 0\^-" 



1 I cos 



nl \ n] \ n 



\n+i 



2"a 



\ nl I 



COS 



/ 



32 



