326 Astronomy and Electrical Theory of Matter. 



Making this substitution, and dividing both sides by dOjdt, 

 we have 



-j a = - r- \ —cos 6 sin (# — '37) 4- sin cos (0—vt) 



dO ■ " )jl M 1 



+ - sin (9 



[dm 



e d6 ~~ 



/i V I 



- |r> j cos # cos (0 — <sr) + sin# sin(0 — -or) 



r • /1 • //1 n \ dm 



(1) 



Instead of (5), we must now use the rigorous value of 

 dm/dO, taking account of the component v e. It is easily 

 found that 



1 dm u Y a u 2 e . , n s 



To obtain the secular terms we must pick out the non- 

 periodic part of (7). It will be seen that the first two 

 terms in the bracket give sin ot dm\d0 and cos ot dm/dd for 

 the two equations — expressions which are purely periodic. 

 We need therefore only consider the third terms. Re- 

 membering that u =fjb/h, the formulae reduce to 



Ya-r 



>2 ae 

 r 



sin 0( V cos + u e sin (0 — ot) ) ? 



Y (S) 



de 



d0 = c 



'dJ^ " ? a(l-e*) sin ° sin (0- OT K V cos ° 



-f w £sin (^—'cj)). J 



We can expand r in powers of £cos(0 — -cr), and with 

 a little trouble the non-periodic part of (8) can be found as 

 a series in powers of e. It will suffice here to give only the 

 first terms of the series. 



de V 2 . 9 

 - = _, S m2^ 



eh 



V 2 



de=s? ecos2 ^ 



Integrating and adding to the results already found, 

 we have finally 



, u o V0 V 2 . 9 



de = ~^r-f cos ^ + -^"2-^ sin Its 



LC OC 



u o Y0 . , u<; 2 , Y 2 9 



(9) 



2c 2 



2c 2 



8c 2 



correct to the first power of e. 



It will be noticed that the terms independent of e (the 



