( 102 ) 



thai in these terms, for the same reason as in those of 1 lie tirst 

 degree, the excentricities must be replaced by their perturbations 

 with the arguments u and u' , in order to find the terms determining 

 (lie libration. These terms thus are of the second degree, not in 

 t he excentricities, but in the quantities Ji, B'. . . and of these the 

 squares are not negligible, as we have seen. 



The question further arises : do not the terms of the third degree 

 in the excentricities, i. e. those of the types 



Pe' cos (21' — l —ÖÏ), Q c- e cos (2 I' — I — 2 (5 + <»), 



I! e' cos (6 /' — 3 / — 3 u>), S e 3 cos (4 1" — / — 3 to), etc. 



also contribute appreciably towards the coefficients Q, ■? To find the 

 answer to this question I have computed all the terms of this kind 

 in t^. These terms of the third degree, which are of the fourth order 

 in I he masses, are: 

 ,f Q, = {+ -00012 m," + -00079 m a s + 00034 m 8 5 + -00061 m, ra 2 4- 

 4- -00(150 m, m 8 + .00124 m, m,J ra 3 m 3 — + .00071 m a m,. 

 They are thus not wholly negligible. I have, however, not carried 

 out the computation — which is rather complicated — for Q s and 

 Q t , nor have I computed the terms of the fourth degree (i.e. of the 

 fifth order in the masses). The development of the period Tin powers 

 of the masses evidently converges very slowly, and the period com- 

 puted by the formulas [12] may very well be erroneous by a few 

 tenths of a year. 



2. The equations of the centre. The large inequalities, which in 

 the integration by the method of variation of elements appear as 

 perturbations of the excentricities and perijoves (formula [7] above), 

 are in practice added to the longitudes and radii-vectores, and the 

 excentricities and perijoves are conceived to be affected by their 

 secular, bul not by their periodic perturbations. I now return to the 

 notations used in all my other work on the satellites, and I denote 

 the excentricities and perijoves, defined in this way, by E t and <2,-. 

 We have then ') 



hi — 2 Ei sin S2i = 2 Sj n, e j sin <5j | 

 hi = 2 Ei ens £2; = 2 2, tijej cos t3, \ 



The sums extend over the values of j from 1 to 4; e, and to,- are 

 the "own" excentricities and perijoves of Laplace, the values of 

 e; are constant and w, are linear functions of the time. Further 



[13| 



') Those It, and A', are thus not the same quantities as those denoted by h, k, It', 



by TlSSEKAfiD. 



