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



and z^i in fuaction of |, ?, 3, 6, \, is derived from the third 

 one of the formulas ( 4 ) . 



9U 9U 9U .„ , 



5. The derivatives 7^v-, ^, ^ will now be expressed m 



function of iHf, iV, e, \, S, |, r), ^. All terms whose ratios to 



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 ^ 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 rt ^^^ ^^^Y needs to divide both 

 sides of (1) by S*. Then 





M 



and since ^-- represents the approximate value of the accel- 



M 



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



TIT" 



K^ differs little from unity, it is clear that -^ is a small 



o o* 



quantity of the order of — a^ — ah^ that is, of the oblate- 

 ness of the earth. If, then, we put 



M 

 (9), a = — -gysine 



+ -g4~ [sin 2X — 3 ( 1 + sin 2 X) sin e + 9 sin X cos \ sin 'e] , 



(yjj 6 = _ -gj- sin e H — gg— sin 2X, 



