4 Trans. Acad. Sci. of St. Louis. 



and «', in function of ^, ^, 8, 0, \, is derived from the third 

 one of the formulas (4). 



9U 9U 9U 

 5. The derivatives tFj nT » ^ will now be expressed in 



function of ikf, iV", e, X, 5, |, ?;, ^. All terms whose ratios to 



M 



M~ are small quantities of order higher than the first power of 



the oblateness of the earth will be neglected. It should be 

 observed that e and gj- are small quantities of the order of the 

 oblateness. In regard to e this fact is well known. To be 

 convinced of it in regard to ^ one only needs to divide both 

 sides of (1) by S*. Then 



JSr 1 M/aY /I . j\ 



M 



and smce ^ represents the approximate value of the accel- 



M 



eration of the force of gravity at 0, that is, -^ = 9'", 78 and 



■^ differs little from unity, it is clear that _ is a small 



quantity of the order of ~ a<f> — ah, that is, of the oblate- 

 ness of the earth. If, then, we put 



(9), a=-— ^sme 

 + ~gi~ [sin 2\ — 3 ( 1 + sin 2 \) sin € + 9 sin X cos X sin ^e] , 



„, , 3M . X'i.N 



(9)j o = — -gg-smeH p— sm 2X, 



