506 Mr. O'BRIEN'S CONTRIBUTIONS TOWARDS A SYSTEM 



and therefore 



du u. , 



— = — j3 + constant (3), 



dt h 



and therefore putting for — its value, observing that rw = -, and assuming e to be the direction 



unit of the constant, and e its numerical magnitude, we have 



dr h - IX ^ , ^ 



— a + -fi = j(i + ee (4), 



dt r h 



performing on this the operation A/B, observing that A/3, a =0, il/3./3= 1, we find 



— = ^ + eA/3.6 (5), 



r n 



which is the polar equation of a conic section, the origin being focus, e being the eccentricity, 



and e perpendicular to the axis major ; for AjS . e is the cosine of the angle which /3 makes with c. 



i. e. the cosine of the angle which the radius vector makes with a perpendicular to e. 



If we perform the operation A a upon (4), we obtain 



dr 



— = eAa . e ; 



dt 



Aa.e denoting the cosine of the angle which the radius vector makes with a perpendicular to the 



axis major. 



38. To determine the motion of the particle when it is acted upon by any disturbing force 

 U in addition to the central force. 



In this case instead of the equation (1), we have 



S=-^+f/ (fi). 



dr r 



Treating this equation as we did (1), we find 



Du.—— = Du. U; 



dt' 



and .-. ^^^^ = Du.U (7), 



dt 



for Du. -— =r^aiy = hy, using h to denote r'cy. 



By integrating equation (7), we find h and y, and thus by integrating one equation we d 

 termine three elements of the orbit, for -y, being perpendicular to the plane of the orbit, determim 

 both the inclination and the position of the node. 



If we integrate (7), after having performed the operation A 7 on each side, we find 



h = fAy.(Du. U)dt. 

 Now A 7 . {Da . f/) = AjS . f/*, 

 hence, since ti = ra, we have 



h = fr/Sli.Udt (8), 



le- 

 nes 



• For V = a{^a.U) + |3^l^|3.U) + y(^y. U), 

 and therefore perfonning successively the operations Da and Ay, we find 



