402 
and the above equations become 
dVdL daVdaL YS EES ND 
See aes = aa eae 
dy dx dx dy dy 
aval _aval_,,, al 
dz dx dx ds di 
Tf, in conformity with General Schubert’s* recent determinations, we 
assume the earth’s surface to be that of an ellipsoid, with three unequal 
axes, we should substitute for L 
7 yp? 3 
eo Cae 
or 
dL _ 22 dZ_2y dL 2a. 
de @’ dy Bb’ de o° 
whence we have 
dV V dV av 
Spey ee eA AF ON Nepean ae eee ey ae 
bx F ay a, wey (a? — b*), cx de Gg OO 
Each of these partial differential equations can be easily integrated, and 
the value of V, finally obtained, is equivalent to the equation of fiuid 
equilibrium ; or 
7+ tsy a6 
Let @ represent the complement of the latitude, and ¢ the longitude, 
counted from the meridian of the greatest axis, then s =r cos 0, =r 
sin 6 cos $, y =r sin 6 sin ¢, and 
Zp ps 
V+ sin = C. 
In the case of an ellipsoid having the ellipticity e, we have, neglecting 
small terms, 
r = a (1—e cos*@). 
From these equations, and from the properties of Laplace’s functions 
into which V can be expanded, an expression can be obtained of the same 
kind as that deduced by Professor Stokes from his own and Gauss’ the- 
orems relative to attractions. 
* Mémoires de l’Académie Imperiale des Sciences de St. Petersbourg, v11° serie, tome i. 
