30 



(I) 



Investigation of the differential equations of the orbit described 

 by a body moving in a fixed plane about a fixed centre, tchen 

 acted on by two forces, one directed to that centre, and the 

 other in a direction perpendicular to the radius vector. 



Let APQ be the orbit described 

 about the centre E ; let PE =:a; ; AEP 

 = ► ; velocity at P=:V ; the force at P 

 direction PN pei-pendicular to PE=P ; the 

 centripetal force at P=:R ; area AEP = 3 

 and the time of describing PQ = a f ; the 

 sign A expressing a difference. 



Then if PT be drawn perpendicular to 

 EQ we may thus proceed to investigate 

 the relation of v and ^ 



g = limit ^|^:.lim.g= lim- 



\ FT* 



v PQ 

 now V = Ism. — = 



PQ 



Therefore hm.^p^ = hm 



dAP 



or dAP = vdt 



dAP 



dt 

 PQxQE _ PE 



■■J. PUE 

 ■ dt 



PE vdt .as 



^ dAPE ~ 2 ^ dz 



To investigate the value of ~ we may consider z a function of t, 



then by Taylor's Theorem 



A« = ^^ Ai4-i:4At^u-^A«3+&c. (3) 



