Central Forces. 67 



a^p' ( a^ sm.'^ V -{-p"' COS.- v\ 



cor. 1. and scholium.) Put — ;r=^' X ; ; 



^ p^ \ ap / ' 



then (2) is easily changed to r^= ( _ '\ r~' {^)-> ^"^ (^) ^^ 



r^ =7 — ; — 77 ' (6) ; and it is evident that v= the angle made 



(a+p')cos.^ ?;- a ^ ^ ° 



by r and a. 



r^dv^ r^dv 



By substituting in (4), for &^ its equal .,„ ? I have dt=^—r==^, 

 ^ & ^ ^' ^ d^ v/Aay 



(7); rf^ = the element of the time {t), and dv = the element of the 

 angle v, supposing v to increase with t: I shall also suppose that 

 t and V commence when the particle is at the extremity of a. 

 Substituting in (7), for r^ its value as given by (5), there results 



a-p'dv , . , 1 • /p' 



dt=—j==^ — , which by putting tan. ?;=: \/ t- 



V Aa^j?' X (a — (a— p'jcos.-?;) a 



dv' . v' 



X tan. zj', (8), becomes dt = -y=, or by integration t = -y=, (9), 



which needs no correction, for when ^ = 0, ?; = 0, and by (8) when 

 u = 0, v'=-0. By (9) v'=t'^ k, which substituted in (8) gives tan. ?7= 



\/ ^ X tan. ^ -v/a, (10); hence, and by (5), r''=a^- cos.- 1"/ K-{- 



op' sin. 2 t\^A, (11) ; by knowing when the particle is at the extremi- 

 ty of a, its position is easily found at any other time (t), for v and r 

 are easily found at that time by (10) and (1 1), and hence the place 

 of the particle becomes known at the same time. Let P= the semi- 

 circumference of a circle rad. = l, and T=the time of revolution of 

 the particle. Now when the particle has made a fourth part of a 



. P T ^ P /?/ 



revolution, ^=7:, and t = -7, and (10) becomes tan.-r=\/ - X 



T _ P T — 



tan.^VA, but tan. -^ = Infinity, -■. tan. 7'^A, is also infinite, hence 



2P 



T— -7=-, (12); it is evident by (12) that the time of revolution (T) 



will always be the same, whatever the axes of the ellipse may be, 

 (provided A is invariable;) because T, as given by (12) is indepen- 

 dent of the axes. (Prin. B. I, prop. 10, cor. 3.) If the semiparam- 

 eter j?', is supposed to be indefinitely diminished, (a remaining inva- 

 riable,) the ellipse will coincide with its axis very nearly, and the par- 



