MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 525 



r, = ^J{li^) +1 .... (2.; 



By substituting for r., in the first equation, its value, as derived from the 

 second, we obtain 



''(a)'[©'-]^(^)' 



(d r \^ 

 ^) in terms 



of r „ then by substituting for »\, its value in the second equation, we have an 



equation containing x^ and j^, from which we find the value of ^— in terms 



of «,, the integration of this gives the equation of the traced curve. 



As an example, we may find the curve traced by the pole of a hyperboUc 

 spiral which roUs on a straight Mne. 



The equation of the rolled curve is ^^ = — 



''2 



d sc. 



Ay^ ^a?-x:^ 



This is the dififerential equation of the tractory of the straight line, which is 

 the curve traced by the pole of the hyperbolic spiral. 

 By eMminating x.^ in the two equations, we obtain 



d 6,_ 2 \d}/J 



This equation serves to determine the rolled curve when the traced curve is 



As an example we shall find the curve, which being rolled on a straight line, 

 traces a common catenary. 



Let the equation to the catenary be 



-|( 



e' + e 



Then p= 1^4-1 



