524 MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 



therefore AO in the figure 



Let the locus of 0, or the evolute of the traced curve LYBH, be the curve 



OZY, and let the evolute of the fixed curve KZAS be FEZ, and let us consider 



FEZ as the fixed curve, and OZY as the traced curve. 



OA BP 



Then in the triangles BPA, AOF, we have OAF = PBA, and -jf ~ '^ = r^' 



therefore the triangles are similar, and FOA = APB = ~, 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 = sin ' «;, which is constant, therefore the ciu-ve, 

 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 

 p^ = ni^ t\, and roll another curve having p., = m.-, r., on the curve traced, and so 

 on, it is immaterial in what order we roU these curves. Thus, if we roll a loga- 

 rithmic spiral, in Avhich p = m r, on the Jith involute of a circle whose radius is a, 

 the curve traced is the n + 1th involute of a circle whose radius is V 1 — m^ 



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

 n + jwth involute of a circle, whose radius is 



a V 1 — m^ V 1 — m^ J etc. 



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 perpendicular 

 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 .r , and y,. 



r ^ 



• -'•= ■ —-—. . . (1., 



J-^m' 



