16 



C. V. L. Chailier 



10. Substituting these formulae in (9), we now final!}' get: 



2m„ C / . V . uw 

 n = M p — i ■■ + rj — In 



(18) <=^i ^ I -— = -f -rj — — C « 



= f — 'r\ \/ -\- V — Cw 



and 



^2 = ^^2 H r — ? r + 'n r — '^^ 



^2 ~\ 



Observing that 



w' = w„ A ~ — — ri]/'u^ 4- — liw 



4 = sin %■ cos 4» 

 ■/] = sin sin ^ 

 . C = cos 



we find that trough the formulae (18) and (18*) the velocities of the stars after a 

 passage are explicitly expressed as functions of the velocities (at infinite 'distance) 

 before the passage and the parameters 4, -^j, C or, alternatively, and 



The parametei's |, rj, C appear in the second degree. The angles ■& and 

 occur only in the following combinations 



cos %■ sin %■ cos 

 cos %■ sin ^ sin 

 cos cos ■9'. 



If M^', Mg' etc. are considered as functions of the velocities , etc. before 

 the passage, the relations are more complicated. In the formulae (9), where I, m, n 

 were parameter, we got the velocities after the passage as linear functions of the 

 velocities before the passage. These simple relations subsist no longer. We observe, 

 however, that the right membra of (18) and (18*) are everywhere of the dimension 

 + 1 in w^, «2 etc. as is, indeed, necessary. We shall find, moreover, in the appli- 

 cations that we have to make of these formulae, that they will possess the same 

 proporties as linear relations. The reason is that the functions of u^' etc., 

 which are to be considered, are to be integrated over all values of between 

 ^1^ = 0 and cjj = 2:t, whereby the denominators and the square roots vanish. We 

 observe that if the polarcoordinates (17) are introduced all square roots and (deno- 

 minators) do vanish. We get, indeed, 



