1304 



then the effect of tlie variations on the radius-vector and the longitude 

 is found to be, to the first order 



(fn — I Hi -2" C'" Bq cos (pg a,^ I ^ {Ciq 4- C'iq ) Bq COS (A/ ^ ^) -\- \ 



q i^f q ^ 1 



-h * {<^iQ ~ C JO ) Bq COS («,■ T -f fpq) , I 



>.(29) 



dtVi = JS" c|.' Bq Si7l fpq + ^ { (Cjy 4" c' (q ) Bq sin {Xi ~'^q) + I 



V '^ ? / 



+ {<^iq —Ciq) ^q SlTl {ci X + (pq) \. ' 



As a first approximation we have (fii= — >i, ai j and bij being 

 of the second order. Also with the same approximation, for 



^ = 1 4, ,?^ = x, and consequently from (25) Ciq=zc'iq approxi- 



matel3\ The difference c,^ — c'iq is thus of a higher order, and the 

 last term of" (29) can be omitted. Further also dij and eij are at 

 least of the second order, and consequently by tlie last of (25) 

 c"ij is of a liigher order than c'iq and c"iq. it follows that the first 

 term of rf/v can also be omitted in the first approximation. The 

 equations (29) then have entirely the foiin (6). At the same time we 

 see the reason why the inequalities II and III are so much snjaller 

 in the radius-vector than in the longitude. 



5. The perturbations. 



We must now take into account the part of the peiturbative 



function 



R — [Ril 



which contains terms whose argument D varies with the time, 



thus D=Ex. We will only give the theory in its broad outlines. 



For details we refer to the publication in the Leiden Annals. We 



put for abbreviation 



hi = Xi, k^ = yi, n = x,-f 4 , ioi = y,-|_4 . 



The differential equations then assume the form 



dxi 



— = ^ aj E «^^ Ex -\- 2 2 fi j ]£ sinEx Xj'\-22 gi j p^ cosEx yj, J 



dx E ' j e' " j E^ ''' j 



i ^ (30) 

 ay,- ^ ^ I • 1 



—— = — ^ a'i E cos Ex — 2 S f'ij E cos Ex X; — ^ Sg'i j e «'" Er y ^ , ) 

 dx E ' J e' ' J e' ' ' 



where i and j take the values from 1 to 8. The arguments are of 



the form 



D = Ex = kx -\- k' c^x, 



k and k' being any integers, positive, negative or zero. If we take 



only k^k'=^0, the equations (30) are reduced to (23). Thus we 



have, e.g. 



/i,;,o = 0, ^,-,y,o = a,,;, 9iJ+i,o = ^ij etc. 



