82 Mr HOLDITCH, ON ROLLING CURVES. 



t 

 = 2b* + bl + -r , omitting the negative powers of b, as b is infinite; 



4 



^.V^.tan-VW,-^ 

 2 a a I - 



!/ 



= («&« + bi + 1) .tan- \Zjhj + { b + i) -^y^?; 



also b . \/ly — y* + b.(a + b). tan -1 Vj-^j 



r 



= b^ly-tf + {2b* + bl) . tan" 1 \/X , 



which quantities being substituted in the above equation, we have the 

 equation to the rack 



x = («, + «j . tan- VjL ♦ * . ^ .v^i 



from which it appears that each lobe of the rack is composed of four 

 similar and equal parts. 



This equation may also be found from the differential equation 



dx 



k ' + k \y-i) 



which is immediately deducible from 



b * 



rde *' + *•('•- H") 



dr ' y/{ a - r).{r-b) 



If & = 0, y = /.sin a -y (fig. 21), which is a rack that will roll with 

 figures (17), (18), (19). 



HAMNETT HOLDITCH. 



