Make now, to distinguish between tlic length and direction 

 of the vector, 



a = n, r= v^(-a^), t^ = - 1 ; (6) 



we shall have 



da = r.dt + dr.t, (7) 



and because r and dr are scalar (or real) quantities, 



a. da = r^i.di — r .dr, da. a =. 7-'^di.i — r.dr; (8) 

 therefore 



(i.dt = |(a . da - da . a) = ^ . d£ - dt . = r'^i.di, (9) 



observing that the equation 



t^ = — 1 gives t.di + di.i = 0. (10) 



The fundamental equation (1) of the problem becomes, by 

 (6) and (9), 



d da _ ^ _ dtM . 



d?d7~^~d^^' ^ ^ 



(in which last member the order of the factors is not indiffe- 

 rent), and therefore gives, by integration, since j3 as well as m 

 is constant, 



J -.^' = const.; (12) 



or, as we may also write it, 



.-^^ = e, d. = 0. (13) 



dt M 



We have, consequently, by (6) and (4), 



da 3 , da B ,, .^ 



a£ = — r — a -J--, £a = — /• 4- -T- a ^ ; (H) 



dtM dt M 



and finally, by (3), 



— & 

 at + m + 2r = 2p, (15) it p zz ^, (16) 



