532 MR CLEKK MAXWELL ON THE THEOKY OF ROLLING CURVES. 



vector, and cos~' -- - cos ' ^' is the variation of tliis angle when 6„ varies 

 by an angle equal to «. Let this variation = </) ; then if ^o~ ^o^- f^ 



a /3 



<p = - + - 



n n 



Now, if w increases, (p will diminish ; and if n become infinite, 



a 3 

 a = 1- _ = when a and 8 are finite. 



'^ 00 CO 



Therefore, when ti 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 ciu-ve 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 ; there- 

 fore the cm-ve required is the logarithmic spiral. As an example of a curve 

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



cos IJ 

 Here-J^: = 2".(.ini») (cos 1«) -' 



(^ \ 2 n 

 COS J ) 



^/22« a- (cos ^A -'» + 2-' ■' n2 (gin ^A ^ (cos ^A ' 



cos 1) 

 , _ . J ^' I d \ n+1 



' ~ . , = 2" + i a(co.sru) 



Now S^ — Sf, = — COS 



-iZ? 



n + 1 



substituting this value of f„ in the expression for i\ 



T, =2. a \ COS ^-- \ 



\ " + 1/ 



