528 MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 



aij for ;•„ and multiply by ,*\, equate the result to — , and integrate. 

 Thus, if the equation of the rolled curve be 



^ = A /•-" + etc. + K r-- + L r"' + M log ?• + N >• + etc. + Z r" 



— = - « A i-("+" - etc.-2 Kr-' -L r-' + M ;■-' + N + etc. + nZ r"~' 

 dr 



-^ = - n A z-"- - etc. - 2 K r-^ - L a;-i + M + N »• + etc. + n Z x" 

 dx 



y = -^ A z"-" + etc. + 2 K x-' - L log x + M a; + i N a;^ + etc. + -^ Z x"*' 



which is the equation of the fixed curve. 



If the equation of the fixed curve be given, find ^ in terms of a-, substitute 



d 6 



r for X, and divide by r, equate the result to -r , and integrate. 



Thus, if the fixed curve be the orthogonal tractory of the straight line, whose 

 equation is 



+ Va^ 



dr 





this is the equation to the orthogonal tractory of a circle whose diameter is equal 

 to the constant tangent of the fixed curve, and its constant tangent equal to half 

 that of the fixed curve. 



This property of the tractory of the circle may be proved geometrically, thus — 

 Let P be the centre of a circle whose radius is PD, and let CD be a line constantly 

 equal to the radius. Let BCP be the ciu^e described by the point C when the 

 point D is moved along the circumference of the circle, then if tangents equal 

 to CD be drawn to the curve, then- extremities wiU be in the circle. Let ACH 

 be the curve on which BCP roUs, and let OPE be the straight Une traced by the 

 pole, let CDE be the common tangent, let it cut the circle in D, and the straight 

 line in E. 



