96 DIFFERENTIAL AND INTEGRAL CALCULUS. 



therefore 



dy dy dx 3a sin <2 cosfl 



= tan -vjr ; 



therefore O = TT- ^r, or if the tangent at P (fig. 21) meet 

 the axis of a; in Z, O TT Z. PLx = L PL 0. Hence 



OL = ON+NL = ao,oe + a sin 3 cot 



= a cos (cos 2 + sin 2 0) = a cos 0, 



or LM a, the most important property of the curve, that 

 the part of the tangent intercepted between the axes is of 

 constant length. Again x = a cos 3 \|r ; therefore 



dx . dx ds . ds 



=3a Biny cos -\Jr = -= = cosilr -= , 



C?^ C?5 C?\|r r rf^r ' 



or ^- =3a sini/rcos^/; therefore 5 = 3a , 



measuring from the point where "^ = 0; i.e. x a^ ?/ = 0. 

 This curve is generated by any point on a circle of radius 



- which rolls within a circle of radius , giving an equal 



branch in each quadrant, the length of which is ; or the 

 length of the whole curve is 6. 



CURVATURE, RADIUS OF CURVATURE. 



o 



In any circle the radius is equal to if s be any arc 



. 

 whatever, and i// the angle through which the tangent 



turns as the arc 5 is traversed ; hence the curvature of any 



circle, being inversely as the radius, is proportional to , 



s 



and we may take this conveniently as the measure of the 

 curvature ; and the arc s or the angle ^ may be as large 

 or as small as we choose. So, in general, in any curve, 

 5, i// having the same meaning, the average curvature of the 



arc s is , since this would give the same total deflection 

 of the tangent in passing over the arc s, hence the average 



