1070 



CHARLIER, ON PERIODIC ORBITS. 



If we now divide A^ and A^ in their real and imaginary 

 parts putting 



A^ = a^ + «oV — 1 



A^ = «1 «oV 1 7 



we may put also 



B, = ß, + ß.V — 1 



B. = ß, — /^oV'"^=n: , 



and we have then 



A^e^'^' + A^e^'^' = 2a^ cos i'J, + 2a, sin v.^t . 



If the arbitrary constants are so determined that B^=.B^=0^ 

 we have now a periodic Solution of the form 



^ = 2aj cos v^t + '2a^ sin v^t 



ri = '2ß-^ cos vj, + 2/^2 sin ^^^i^ , 



(22) 



and between the coefficients a and ß there are now the rela- 

 tions: ' 



(23) 





and the equation (13) becomes 



(13*) 



d-Q\ / 





An-v- + 



dadh 



From (23) in connection with (13*) we obtain 



^'2 + ^) («)/^2 — «2/?]) = — 2^'X^2 (/^J + /^o) 

 2 r-i2W o os/2 ^'-ßW^- ,^-\ 



The equation of the orbit may be obtained through eliinin- 

 ating the time between the expressions for t, and i^. We 

 thus aet: 



