Central Forces. 341 



fdistance will approach the centre of force, and after describing an 



P 



•an2;le=— will be at a distance = R from the centre of force, and that 



° m 



the portion of the curve described in passing from the least to the 

 greatest distance is equal and similar to the portion described in pass- 

 ing from the greatest to the least distance, and. that these portions are 

 similarly situated on opposite sides of the greatest distance ; also 



P . 



after having arrived at the distance R, the angle — will be repeated 



and the particle will arrive at the greatest distance R+~5 when it 



will again approach the centre as before, and so on perpetually. Put 



e' e' 



R+-^=R', and -—^,=e, then (17) becomes r=R'(l - e cos. mv), 



(19), which is the equation of the curve described by the particle. 

 Put o7=^3 \/ -"^ =w, then by (19) neglecting quantities of the 



R'-^' V ^7 



c 



order e-, r^=R'-(l — 2ecos. mt')= -(1 — 2e cos. m?;), hence and by 



(14) there results ndt=^dv — 2e cos. mvdv whose integral gives n^=. 



2e sin. mv , 

 V , which needs no correction, since t and v commence 



at the origin ; or mv=mnt-\-2e sin. mv, and by neglecting quantities 



2e sin. mnt 

 of the order e-, v^=nt-{- — , (20), and 7-=R'(l — ecos. m??^), 



(21); (20) and (21) are sufficient to find the place of the particle 

 at any time {t). Let v^ denote the degrees in v as given by 



(18), then since —=\/ -I =\/ — ~ — ; (18) becomes t;° — 



180ov/|^ (22),or^° = 180^^v/'^^,(23); (22) evidently 



agrees with the method given by Newton in the ninth section of the 

 Principia, for determining the angle between the apsides in orbits, 

 which differ very little from circles; but it appears to me that (23) 

 will generally be more convenient in practice ; and it may be observ- 

 ed that by neglecting quantities of the order eR, in comparison with 



R, we may write r instead of R in (22), (23). Let — = — —, A=. 



