8 THE THEORY OF ROLLING CURVES. 



rdr 



~d8 



/^7/^V' 

 V ' + ( 



We come now to consider the three equations of rolling which are involved 

 in the enunciation. Since the second curve rolls upon the first without slipping, 

 the length of the fixed curve at the point of contact is the measure of the 

 length of the rolled curve, therefore we have the following equation to connect 

 the fixed curve and the rolled curve 



s, = s r 



Now, by combining this equation with the two equations 



it is evident that from any of the four quantities Of&r, or x^y l x^ t , we can 

 obtain the other three, therefore we may consider these quantities as known 

 functions of each other. 



Since the curve rolls on the fixed curve, they must have a common tangent. 



Let PA be this tangent, draw BP, CQ perpendicular to PA, produce CQ, 

 and draw BR perpendicular to it, then we have CA=r lt BA=r t , and CB = r t ; 

 CQ=p>, PB=p t , and BN=p t ; AQ = q lt AP = q t , and 



Also r,> 



= Pl * + 2p lPt 

 r,' = r, 1 + r* 



Since the first curve is fixed to the paper, we may find the angle 6 r 

 Thus 



t + tan- JV- + tan 



