1303 



A = 



^ „--/?' A 



A. 



C 



1 



A. 



i^"- 



A, 



'A. 



C. 



C. 





B, 



(28) 



Now it can be shown that 



Tj, Ail 4- iu Ai2 i- ■>], Ai^ -\- 11, An -\- Bii 4- Bi2 f As 4 Bu = 0, 



4 5,1 + 2 5,2+^.3=0, j(29) 



Bu =z 0, I 



and the same equations remain true if Aij is replaced by C'ij, and 

 Bij by Dij. It follows that the equation A = 0, which is of the 

 gth degree in ^\ has three roots ^* = Ó, and can therefore be 

 reduced to an equation of the fifth degree. To prove (29) it would 

 be necessarj^ to develop the coefticients Aij, Bij . . . ., which cannot 

 be done here. The proof will be given in a more detailed pnblica- 

 tion that will soon appear in the Annals of the Observatory at 

 Leiden (Vol. XII. Part I). 



There are thus 5 different values of ii\/. To each of these belongs 

 a set of values of c'i,i and c"/,/ , which are found from (27), and 

 of Ct,i and c'"/,/, which are then found from (he fir.^t and last of (25). 



The fust four elements of the diagonal of the determinant A are 

 approximately 



An = X . 



All other elements are at least of the third order. It follows tliat 

 four of the roots /^,/ are very nearly equal to >c, the fifth being 

 much smaller. If we neglect the masses of the satellites and the 

 compression of the planet, then this fifth root becomes zero, and 

 the four others are rigorously equal to x. The motion — x of 

 the perijoves in the intermediary orbit is then exactly cancelled 

 by the variations, and since in that case also ly = 0, and the inter- 

 mediary orbit is a circle, the varied orbit consists of four Keplerian 

 ellipses with the excentricities f/ and the fixed perijoves Tcr , as it 

 evidently must be. 



If we consider the constants of integration f, as quantities of the 



first order, like ?/,'), and if we put 



(pi = /^j T + TxfiO zr: XT + ttT,-, 



1) If follows from (18) that y^.r,i is of the second order, and consequently yh 

 of the first. 



