2 Trans. Acad. Sci. of St. Louis. 



been fluid." In the case of a homogeneous ellipsoid of 



5 

 revolution, ah would be equal to j a^ and the expression of the 



potential U would become 



Mi 3 a2 



B > ^-^b^^^ 



(a-^)l 



1 3 



In general, ah — ^ ®^ will differ but little from v ah. If, 



then, we put 



( 1 ) -^ = ~ 3 ^^^ (2 "*^ — «^^ ) ' 

 we shall have 



^^ M jsr 



(2) Cr=:-^+-^3(l_3/.2). 



2. Let us take a rectangular system of axes, as follows: 



Z along the axis of rotation of the earth in the northward 



direction, X and Y through the center (C) of the earth in 



the plane of the equator and invariably fixed in it. Then /ti 



z 

 will be equal to -5 and 



M N SNz-' 

 (3) { ^''B^ B'~ R' ' 



B= i/xH^FT'^^. 



3. We will now transform our system of coordinate axes. 

 The new origin ( O) will be taken at an external point on or 

 near the surface of the earth in the meridian plane ZCX; 

 the axes 0| and 0^ in this plane, namely, O^ in the direc- 

 tion of the mean * force of gravity at Ot* and 0| in the 



* that is, on the assumption that the forces of attraction by the sun 

 and the moon are neclected. 



