Mr H0LD1TCH, ON ROLLING CURVES. 77 



a + b ds ., , 



If r = — = o. or the curve makes a maximum or minimum 



2 dr 



angle with the radius vector at the mean distance; and the reciprocal 



of the radius of curvature 



k. 



The area of a wheel may be found: for the area of a lobe is the 

 integral from = — - to = - of 



•«- {*,+ *. (^)". -»•♦}. '(S±* ♦ SeJ; sin *) .**, 



, . , a + 6 , , a + J / , N2 



which = — - — . * 7r + « . - =-g . (a — o) . 7T ; 

 2 lb 



therefore, the area of a wheel 



a + b , , a + b . .., 



= — — — k.rnr + k . -73- .{a - b) . nir, 

 2 lo 



in which expression, if the value of k, be substituted from the equation 



2k t , (a + bf i . .. 2 

 —=*= + k . - — 7== - k . {a + b) = - , 

 y/ab 2Vab ' * 



the area of a wheel 

 = - .{a + b) .Vab + ~- . {4, .(a + b)Wa~b - 2. (a + bf + (a + b) .(a- b)'\, 



= *- . (a + b) .y/ab + ^ . (a + b) . { (a - by - 2 . (a + V) . (y/a - y/b)*} , 



= ^.(a + b)Val + ^.(a + b).(\/a-v / by.K^ + Vby-2:(a + b)}, 



(y/a - y/b\ 4 



= ^.(a + b).\/ab — Jenir .(a + b) . I j . 



