688 CAV ALLIN, SECULAR PERTURBATIONS 



and hence it follows by (6) and (7) 



t — X,. t — ?., 



m„ Mi «+A,. i — - — „—i — — - 



r = l 



2^■ 



2to^+m, ^, ^ T 



= Cé* 2m ' W sin 



,.=1 'im 



where C is a constant the value of whicli we do not need to 

 determine. 



Thus we get since by (8) 



2mn + i-ii = 2m„ + m, — m« = ?Wj + »w„ , 



r = l 



By logarithmic differentiation of both members of (13) we get 



- ^ + « V = « ^7 - + ^r- / cot -^r — ^ . (14) 

 7\ dt dt 2m Im / j "Im 



As i\ and ö, respectively constitute the niodulus and the 



argument of the right member of (6) they are both real and thus 



, dr, , de, 

 also -^ and -^ . 

 dt dt 



Comparing real and imaginary parts in both members of 



(14) we arrive at the formulas 



-^=^-^y cot ^-^ (15) 



i\ dt 2m / j 2m ^ 



and 



de,^ m +m ^ Ij-^iy ^^^ t-K (1^^ 



dt 2m 2m i / j 2m ^ ' 



where the svnibol R has tlie meaning: »the real part of». 



