74 Mr HOLDITCH, ON ROLLING CURVES. 



and therefore, the equations will be the same as for those that roll 

 externally, and the equations of condition are also the same, and con- 

 sequently all curves (whose equations are of the forms that have been 

 considered) that roll externally, are also capable of rolling on each other 

 internally; in the latter case, the major axes come into contact; 



a + b a+ b, 

 for if r, = a,, r = r i ±c = a,±c = a i + — '—t 



a — b a + b a - b a + b ,.„ , , 



= 3 t + = — (- — - — = a : and if r = b , r = b. 



2 2 2 2 u 



The curves in fig. (13) will therefore roll in the positions, figs. (14) 

 and (15), which are two different attitudes of the curves. 



It will be necessary hereafter to know the radii of curvature at 

 different points of the curve; 



Let 



rd9 



tet-ksnSl 



dr " */(u-r).{r-b) \7Y \Zl-s*' 



where s is the sine of the angle between the curve and radius vector, 

 then p m sr, p being the perpendicular upon the tangent ; 



dp _ s ds 

 rdr r dr ' 



1 X 



? .** X* ; 



therefore, to find the radius of curvature at the tangent points where 

 * = 0, and X = 0, 



ds Y dX 



s*dr X 3 ' dr ' 



ds_ *_ YdX 



dr ~ X s ' dr ' 



x * - x*+r- r- ■ ds dX 



