72 Mr HOLDITCH, ON ROLLING CURVES. 



is the equation to a curve which will roll with the former, the equa- 

 tions of condition being « — b t = a — b, 



,111c, k (a, + 6,)* , . ,. 2 



Hence, if h, k t and a — b be any assumed constant quantities, the 

 values of a and b may be found for n = 1, 2, 3, &c. from the equation 



2£, h (a + b)°- t . -,,2 



Vah r 2- ^ ' r "»■ 



by the solution of a cubic equation, as will be easily seen, and the 

 curves constructed from the equation 



9 - (• + *.<« '+ 4] • tan- V?. V^H? 

 [n ') b a —r 



- k s/(a -r).(r - b) - k . (a + b) . tan" 1 \J r -^-, 



ct — r 



and a system of wheels or curves thus found will roll together in pairs 

 or in any combinations. 



When there are tangential points in one wheel, there will be corre- 

 sponding ones in all of the same system, and in rolling they will come 

 into contact with each other; for those of one wheel are found by mak- 

 ing f{r) = 0, and if a be a root of this equation, c — a, or a will be 

 a root of the equation f{c - r) — 0, or of f{r) = 0, and .*. a + ft/ = c. 



Forms of wheels are readily found from assumed values of & and k t : 

 or if the dimensions of a pair of wheels be assumed, k and k t may be 

 found from equation (2) ; 



Thus, if n = 1, b = 11 __ . ,,, _,. ., „. 



n = 3, b = 5) a = 10 > and ■■", = 1 ^ *+m 



rf 1 2 », \Z l) a = 10 ' and •• a ' = 13 ^- Fi ^- ( 12 )- 



" " ■ i' ; J I J}« = 10, and ... a 4 - 13}. Fig. (13); 

 in all which cases each curve is also a self-rolling curve. 



