PHYSICAL ASTRONOMY. 333 



Therefore, if the velocity of a body attracted to a 

 centre, be as the square root of the distance it has 

 fallen through, divided by the square root of the 

 distance between it and the centre, the centripe- 

 tal force by which it is impelled is inversely as the 

 square of its distance from that centre. 



Hence also, if the centripetal force at G be = F, and 



Afr r* 



if CG = , F = e X TJ s= 1, and *F = c\ 





39. If to a tangent of an ellipsis, a perpendi- 

 cular be drawn from either focus, the distance 

 of that focu3 from the point of contact will be 

 to the distance, of the, other focus from the same 

 point, as the square of the perpendicular drawn 

 to the tangent, to the square of the seini-con- 

 jugate axis. 



. Let P (fig. 23.) be a point in the ellipsis ADBE, 

 AB the transverse, DE tfie conjugate axis, C the 

 centre, 8 and F the foci ; GPH a tangent to the 

 ellipsis in P, SG the perpendicular on it from S. 

 Draw SP, PF, and make FH perpendicular to 

 GP. Because the angles SPG, FPH,are equal, 

 from the nature of the ellipsis, the triangles SPG 

 FPH are equiangular, and therefore 



SP : PF : : SG : FH, and 

 SP : PJF : SG* : FH x SG. 



But 



