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



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

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

 potential C/^ would become 



^1+5|2«^(3-M^)^. 



M i ^ da 

 B 



1 3 , 



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



then, we put 



(1) N= — ^Ma^Ua<i>—ah\ 



we shall have 



M jsr 

 (2) ^=i^ + :R3(i-^^)- 



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 /* 



z 

 will be equal to -p and 



(3) ^ ^-B -^B^- R' ' 



IB= y'x^-^y^-^zK 



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 OK in this plane, namely, OK in the direc- 

 tion of the mean * force of gravity at 0^, and O^ in the 



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

 and the moon are neglected. 



