THE THEORY OF ROLLING CURVES. 11 



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

 second, we obtain 



Jdx^\(dx 3 Y I /dr.y 



(dyJ LWJ +] W' 



If we know the equation to the rolled curve, we may find |-^| in 



\dffj 



terms of r i} then by substituting for r 2 its value in the second equation, we 



have an equation containing x 3 and -y-?, from which we find the value of ~ 



dy* dy, 



in terms of x 3 ; 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 hyperbolic 

 spiral which rolls on a straight line. 



The equation of the rolled curve is # 3 = , / 



a? 



dy 



dx 3 x 3 

 ' dy 3 ~ 



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

 is the curve traced by the pole of the hyperbolic spiral. 

 By eliminating x 3 in the two equations, we obtain 



dr t _ idx 3 



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

 is given. 



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 



a 



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