530 MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 



of the other curves is equal to a, and the angles which they make with the radius 

 at the point of contact are equal, 



, -, d6. d6, 



■■■ r. = ± (a ± r,,) and r.-^= r., --' 



d6^ ± (a ± r^) djj^ 



d r„ r„ dt 



p mnv finrl 



dr. 



d S 

 If we know the equation between ^i and )\, we may find — ' in terms of /■,, 



''i 



substitute =t (« ± /•,) for /■,, multiply by "^'-""^H and integrate. 



''2 



Thus, if the equation between 6^ and i\ be 



which is the polar equation of a straight line touching the traced circle whose 

 equation is r = a, 



then 



d 6 a 



NoAv, since the rolling curve is a straight line, and the tracing point is not in 

 its dh-ection, we may apply to this example the observations which have been made 

 upon tractories. 



Let, therefore, the curve r = ^— ^ be denoted by A, its involute by B, 



and the circle traced by C, then B is the tractory of C ; therefore the involute 



of the curve /• = -^ — r is the tractory of the circle, the equation of which is 



1 f I «2 2 a 



6 = cos - - . /-^ — 1. The curve whose equation is r = -, — r seems to be 



a ^ r' 9—1 



among spirals what the catenary is among curves whose equations are between 

 rectangular co-ordinates ; for, if we represent the vertical direction by the radius 



