DIFIT.KKNTIAL CALCULUS. 20? 



990. Prove that in any epicycloid or hypocycloid, the radius 

 of curvature is proportional to the perpendicular on the tangent 

 fruin the centre of the fixed circle. 



991. The curvature at any point of a lemniscate varies as 

 the difference of the focal distances. 



992. If the tangent and normal at a curve be taken as the 

 axes of x t y, the co-ordinates of a neighbouring point are, ap- 

 proximately, 



s* 8 s 8 s fdp 



"v f y= 2p~^ 



p being the radius of curvature at the origin, and a the arc 

 measured from the origin. 



993. A loop of a lemniscate rolls in contact with the axis of 

 x, prove that the locus of the node is given by the equation 



>/ 



and if p. f>' he corresponding radii of curvature of this locus and of 

 the lemniscate, 2pp=a*. 



If tin- eurve r m = a m cosmO roll along a straight line, 

 the radius of curvature f the path <>(' the p<>le is 



(*!) 



'". A rectangular liyp.-rli.ila rolls on a straight line : prove 



that the radius of o <>f tin- path of the centre i- half tho 



distance from the centre to tho point of contact; ami th.it tho 



li of any portion of the path <>f the centre i-; e|iial to the 



pondini: an- of the locus of the feet of the |.erj 

 let fall from tho centre on the tangent. 



996. A pi n.iu along , -I line; proro 



the radius of curvature of the path of any point fixed \\ ith i < 



