MB CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 523 



Also, r^^ = CB2 = CR2 + -RW = (CQ + PB)- + (AP - AQ)^ 



== {Pl + Pif + (?2 - ^l)- 



= Pl^ + 2 P^ Ps + P2' + ^2 - i»2^ - 2 ?i 9, + »-^i - p2, 



r^^ = ri^ + r^ + 1p^ P^ ~ 2 q^ q.. 



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

 Thus «3 = DOB = DCA + ACQ + RGB 



= 6, + tan-i ^ + tan-i 52 

 ' Pl RC 



dr. 



-1 92-^ 1 



6. =6,+ tan-i ^^^ + tan-i ^^-^J 

 rjrftfj i02+i»i 



Thus we have found three independent equations, which, together with the 

 equations of the curves, make up six equations, of which each may be deduced 

 from the others. There is an equation connecting the radii of curvature of the 

 three curves which is sometimes of use. 



The angle through which the rolled curve revolves during the description of 

 the element d s^, is equal to the angle of contact of the fixed curve and the rolling 

 curve, or to the sum of their curvatures, 



d Sg _ d s d Si 



»-2 R, R2 



But the radius of the roUed curve has revolved in the opposite direction 

 through an angle equal to d 6.,, therefore the angle between two successive posi- 



d s 



tions of To is equal to -^ - d 6.,. Now this angle is the angle between two suc- 

 cessive positions of the normal to the traced curve, therefore, if be the centre 

 of curvature of the traced curve, it is the angle which d «, or d s^ subtends at 0. 

 Let OA=T, then 



As an example of the use of this equation, we may examine a property of 

 the logarithmic spiral. 



In this curve, p = m r, and R = — , therefore if the rolled curve be the 

 logarithmic spiral 



VOL. XVI. PART V. 6 T 



