VOL* XXVI.l PHILOSOPHICAL TRANSACTIONS. 43/ 



is parallel to sp, the angle asp = kal = toa, because tao is the com- 

 plement of each to a right angle ; therefore ka ; al :: SA : sp, hence 



L SA J L X SA 



SP = - X — , and ka = — - — . 



2 KA* 2sp 



Again, since by the equiangular triangles km^, gps and ota, spa, 

 it is KM : K^ ;: gp : gs :: ap : sk 



also K^ : AT :: sk : sa H 



and AT : Ao :: ap : SA, 



L* X SA* 



it will be km : ao :: ap'^ : sa^ ::sa^ — sp^ : sa'^ :: sa'^ JliF' * ^^* ** 



4ak^ — L^ : 4ak^; hence l* : 4ak^ :: (ao — km : ao ::) ak : ar, and 

 therefore ar = -^. In like manner — — = ^ , is found equal to the 



L* t. 2SP* * 



radius of curvature in the hyperbola. 



But in the parabola the calculation will be easier. For, because of the sub- 

 normal being given, it is always K.k = at the fluxion of the axis, fig. 4 ; and 

 the equiangular triangles k^m, ato, spa, akl, hence km : ink :: ap : SA ; also 

 at or ¥ik : AO :: ap : sa ; hence km : ao :: ap^ : sA* :: SA^ — sp'^ : sa^ ; 

 hence it will be sp* : sa^ :: ao — - km : ao :: ak : ab, and therefore ar 



= ^^ ^ ^^ : but AL = half the latus rectum = 4l, and ak : al ;: sa : sp j 



sp* * ' 



therefore sp = 



LXSA J 2 L^XSA* ,, r 4ak' L X SA' , LXSA 



-, and SP'' = — — — ; therefore ar = — — = —, because ak = 



2ak ' 4ak* l* 2spi ' 2sp 



And hence arises a very easy construction, for determining the radius of 

 curvature in any conic section. For let ak be perpendicular to the section, 

 meeting the axis in k, fig. 5, erect kh perpendicular to ak, meeting as pro- 

 duced in H ; erect hr perpendicular to ah ; then will AR be the radius of curva- 

 ture. In the parabola the construction becomes a little simpler. For since, by 

 the nature of the parabola, it is sa = sk, and akh is a right angle, s will be 

 the centre of a circle passing through a, k, h ; whence the radius of curvature 

 is found by producing sa to h, till sh = sa, and erecting the perpendicular 

 HR : then r will be the centre of the circle osculating the parabola at a. 



The centripetal force tending to the focus of the conic section^ in which 

 the body moves, is reciprocally proportional to the square of the distance. 



T^ . LXSA* ., .„ , SA SAXSSP* 2 .1 . • 1 



For smce ar = ^r-^-, it will be , = — = -; that is — ■ 



2SP^ SP^XAR SPJXLXSA* LX8A* SA* 



is as the centripetal force, because l is a given quantity. 



ILet BAD, fig. 6, be an ellipsis, to which ge is a tangent at a ; and to which 

 tangent ak and ps are perpendicular, and s the centre. Then sp X ka will be 

 equal to a 4th part of the figure of the axis, or = the square of the les» 



