70 Mr HOLDITCH, ON ROLLING CURVES. 



If k be positive, there are no tangential points unless k is equal to, 

 or greater than, 



4s\/ab 



n(a + b). (\/a - \Zbf ' 



they begin at r m — - — , and as k increases, one moves nearer to, and 



the other farther from the centre; and when k is infinite, 



a + b y/ a + b . r- /Ts • 

 ■g- ± — (y/a - s/b). 



r = 



If k be negative, the values of r in equation (5) must be within the 



limits of the curve, and therefore there are no tangent points unless 



2 

 k > — jr— , and if k be infinite their distances are the same as 



n yv a — y/of 



when k is positive : comparing this condition with equation (4) it will 

 be seen that when k is negative and the curve is retrograde at the apses, 

 there are always tangent points. 



Other forms of self-rolling curves may be found, as 



and 9 = A .hyp log r + Bar + (C« 2 - B) . - - ^'. r 1 + — , 



2 3 4 



the latter including the logarithmic spiral. 



Fig. 22 is a self-rolling curve, where the minor apsidal distance 

 vanishes, and rolls round the point C in its circumference. 



We will now proceed to the consideration of rolling curves when 

 they are not necessarily similar and equal to each other. If c be the 



distance of their centres, and -=— =f(r) be the differential equation of 



one of the curves, and r t and 6 / belong to a curve that will roll with 

 the former, then, since 



rdO r t d6, .^. „ , 



^ = i^' and ^>=^ c - r) ' 



