10 THE THEORY OF ROLLING CURVES. 



Then in the triangles BPA, AOF, we have OAF=PBA, and ^p= m = ^j r 



therefore the triangles are similar, and FOA = APE = - , therefore OF is perpen- 



dicular to OA, the tangent to the curve OZY, therefore OF is the radius of 

 the curve which when rolled on FEZ traces OZY, and the angle which the 

 curve makes with this radius is OFA = PAB sva.~ 1 m, which is constant, there- 

 fore the curve, which, when rolled on FEZ, traces OZY, is the logarithmic 

 spiral. Thus we have proved the following proposition : " The involute of the 

 curve traced by the pole of a logarithmic spiral which rolls upon any curve, 

 is the curve traced by the pole of the same logarithmic spiral when rolled on 

 the involute of the primary curve." 



It follows from this, that if we roll on any curve a curve having the 

 property 2 ) \ s * m i r i> an ^ ro ^ another curve having p t = m t r t on the curve traced, 

 and so on, it is immaterial in what order we roll these curves. Thus, if we 

 roll a logarithmic spiral, in which p = mr, on the nth involute of a circle whose 

 radius is a, the curve traced is the w + lth involute of a circle whose radius 



is >/l m*. 



Or, if we roll successively m logarithmic spirals, the resulting curve is the 

 n + mth involute of a circle, whose radius is 



We now proceed to the cases in which the solution of the problem may 

 be simplified. This simplification is generally effected by the consideration that 

 the radius vector of the rolled curve is the normal drawn from the traced 

 curve to the fixed curve. 



In the case in which the curve is rolled on a straight line, the perpen- 

 dicular on the tangent of the rolled curve is the distance of the tracing point 

 from the straight line; therefore, if the traced curve be defined by an equation 

 in x t and y u 



