1294 



the exception of j = /, and further over the indices j which refer 

 to the sun, Saturn, etc. 



If the plane of (xy) is chosen near the mean position of the equator, 

 then a and (3 are very small, and y is very nearly equal to unity. 

 The latitudes si of the satellites are very small (the largest does 

 not exceed 0°.7), and also for bodies outside the system of Jupiter 

 Sj is small (e.g. for the sun it never exceeds 3°.l). 



Regarding the value (7) of i2, we may remark, that the terms 

 multiplied with J and K in the complementary part of the pertur- 

 bative function (second line of the formula) are here given for the 

 first time. These terms are neglected by Laplace, and all subsequent 

 investigators adopt Laplace's perturbative function without any 

 criticism. Laplace was perfectly right, for these terms are beyond 

 the limit of accuracy which he had set himself; but Souillaht, who 

 includes other terms of the same, and higher orders, ought also to 

 have included these terms. 



If now we put 



, _ dxi , _ dt/i , dzi 



dt dt dt 



F, = T, - Hi, 

 then the equations become 



d.vi dFi dxi' dFi 



dt dxi' dt dxi 



and similarly for the other two coordinates. 



We now introduce the canonical elements of Delannay 



^/i </m »*^i» ^'t ^it ^/» 



where 



Li = ^i \/ai, Gi ■= LiV\— et^, &i = Gi COS ii, 

 I now put ^) 



s: = si, — ~ .... (8) 



r. 



Then we have 



Fi = --^-Si' 

 2U 



and the equations become 



1) See : On Canonical elements^ these Proceedings Sept. 1913, Vol. XVI 

 pages 285 and 287. 



