( 466 ) 



The computation of clM directly from the formulae is rather 

 lengthy ^) ; by considering the geometrical meaning of M, cIM is 

 found at once. Evidently M is purely a function of Jt and q'. 

 If a increases by the amount Lx, the increment of M is a strip 

 2x(pAic; if q' increases by A^', the increment of Af is negative and 

 equal to a strip (crescent) 2 sin (f' . A^' = 2x sin (p . A^'. Therefore 



dM = <fd{ii') — 2 sin (p' . dq' 



2 sin (p' 2 o sin (f 



— <fd{v.'') ■- — d^ -\ — — . df 



If in this expression we substitute the value, given above, of df, 

 we get dM expressed as a function of diy.''), d^, d^. 



5. Calculation of d/i and d^ in function of the variations of the 

 elements of the orbit and of the epoch. 



If sin if ^ e, then {vide Bauschinger, die Bahnbestimmung der 

 Himmelskörper, n". 197) : 



c?ü 1= I — ) cos (fi Ut — T) dpi — ndT\ -\ cos ip sin v\\ -\ ) dtp. 



\r J r \ pj 



— z=r sin (P — (f^) cos {P — ^ sin i tg i . d(o — siti^ {P — ^) tg i . di 



Q a 



-[- I — I [« sin E — sin (P — ,fQ cos (P— j^) sin i tg i cos tp [{t — T) d[i — ndT\ 



— I I {cos E-e) cos (p-\-sin {P- Sh)^<^^{P~ (fb) ^"^ i ^9 i ^^^^ -^ I — h cos^<p J | d<p. 



According to the detinition of to adopted above, we have to put; 

 Q sin (P — ^) ■=: r cos i cos (a> -\- v) = r cos i cos ji 

 Q cos (P — (f^) =: — r sin (to -\- v) ^= — r sin ^ 



We now pass to the following particular case : 



a. the original orbit is circular; 



b. if i = 90" — t', then i' is so small that 3"^ and higher powers 

 may be neglected ; the same is true for sin tp . 



Furthermore let dii = 0; c/w =: ; 2e cos to = jj , 2e sin to ^ y; 



2^ ^ 

 n = — {17= period = 12.91 days), t^ = time counted from "superior 



conjunction" ; dM, = — (.idT. 



Then with sufficient approximation 



1) See : Untersuchungen über den Lichtwechsel des Sternes /3 Persei, von J. Harting. 

 (München 1889) p. 41. 



