78 Mr HOLDITCH, ON ROLLING CURVES. 



Those systems of curves where k = 0, have no tangential points ; for 



dd = i 



dr ~ ^/{ a - r).(r - b) ' 



and therefore cannot vanish. 



If k t = 0, there is always a tangential point in the middle of each 

 half-lobe. 



The former deserve a more particular consideration, as being in 

 general more simple in form, and admitting of easy and elegant con- 

 struction: if a„, b n be the major and minor apsidal distances of a wheel 

 of n lobes, the equations of condition (2) are reduced to «„ — b x = con- 

 stant = I, 



k, 1 



and 



\/^J B »' 



and therefore, a n = - + \/n 2 k i l + - , 



2 V - ( T|: 



I / P 



and b n = - - + V n*k? + -, 



and the equation to a curve of n lobes will then be 



9 = - . tan " 



r~ p i 



V n*k? + - + - 



nk, ' V 



2n*k 

 or, r = 



2 V»^ J + 7 + /.cos0 

 ' 4 



Describe therefore a circle whose diameter is /, and draw (fig. 16) a 

 tangent at any point A, in which take AC = k t and AE = nk t and 

 draw EG through the centre: then the apsidal distances for a wheel 

 of n lobes are EG and EF; 



