( 101 ) 



we find from formula [11] 



Q 1 = -f 0.03201 m, in, 



Q, = — 0.03794 m, m 3 

 Q 3 = + 0.00994 in, m s , 

 From the formulas [12], on the other hand, we have: 

 Q, = {-f 03009 — -00460 m, — -01156 m, — -00958 m,| m s m, = 



— _;_ 001815 m. : in, 

 Q, — \— 0-03436 + -00389 ni, + 00933 m s + -00809 ni 3 J m, m 3 = 



= _ 0-02438 m, in, 

 Q s — [-f 0-00794 — 00020 in, — -00016 m 2 — -00042 m 3 \ m, m 2 = 



— _|_ 0-00751 m, m.,. 



The numerical coefficients depend almost exclusively on the ratios 

 of the major axes, i.e. on the mean motions, and they can be laken 

 as correct to the last figure given. 



The corresponding periods, computed by the formula 



t= 2 A 



are, expressed in years: 



from formula [11] .... T= 6-318 



from formula [12] . . . . T= 7985, 



The difference is considerable. 



The question naturally arises: why have these important terms of 

 the second degree l>oen overlooked by Laplace and Souillart? For 

 Laplace, the answer is very simple: be has neglected the pari R t 

 of ihe perturbing function throughout. For Souillart it is different. 

 It is one of Souillart's great merits to have discovered the importance 

 of this same part of the perturbing function, especially for the 

 determination of the quantities B, B'. . . The corrections which have 

 been added by Souillart on this account to these coefficients, amount 

 to a considerable part of the whole. Also Souillart evidently intended 

 to find the expression for the period of the libration as completely 

 as possible. On the pages 46 and 47 (Memoirs of the Royal Astro- 

 nomical Society, Vol. XLV he considers the different parts of the 

 perturbing function, which can in the differential coefficients of the 

 mean longitudes introduce the argument ! l — 34-1-2/5. He, however, 

 rejects them all, as giving negligible coefficients, and retains only 

 the terms which had already been discovered by Laplace. Among 

 the rejected terms are also the new terms treated above, which are 

 discarded by Souillart on the ground that they are of the second 

 degree in the excentricities (page 47, bottom). He here overlooks 



