AND THE FIGURE OF THE EARTH. 91 



16. To find the equation of equilibrium of a heterogeneous spheroidal mass of fluid, 

 revolving about its axis, with an angular velocity w. 



By the principles of hydrostatics the general equation \sj — f = /(X dx -\-\ dy 



-{- Z dz), §' being the density at the point {^,1/, 2), p the pressure, and X, Y, Z the 



dY d\ dV 

 sums of the resolved parts of the forces ; which are — ^, -- -j-, — -j-, and the 



centrifugal force. Let the axis of z be that of rotation ; then the centrifugal force is 

 ar^ X along X, and aP' y along y. Let us express u in terras of the ratio of the centri- 

 fugal force at the equator to the equatorial gravity. Call this ratio m, which is small 

 in the case of the earth, being of the same order as g. Then 



m = = 1 i — , or a;^ = — ^—5 . 



mass 4 IT g rj; a' a^ 



Therefore 



„ dY '\:T:qm'^a.x „ dY 4'rrgm^a.y ^ rfV 



and 



/dp ^. 2irgm^a , „. „ 



Now / -^ is a constant for a level surface. Hence for any stratum we have 



c = y + '-^£p^O-i^')r'. 



At the surface this is 



where 



Ma=:y^ <^d ^-j^, ^^rfa',andNa = y^ <^d'-^dd. 



For r write a (1 ^ z\i?), then 



Equate the coefficients of \j?, then 



Ma/ w\ 3Na .^i. 



-^ V - W = y^ • • • • (^)- 



-r, we obtain the amount of 



gravity which acts towards the centre ; which, to the order we are now considering, 

 is the same as the whole force of gravity ; since the cosine of the angle of the verti- 

 cal differs from unity only by terms of a higher order. 



n2 



