148 THE TIDAL PROBLEM. 



V. THE EQUATIONS OF EQUILIBRIUM FOR CONSTANT 



MOMENT OF MOMENTUM. 



Equation (1) is the relation connecting the shape of the Maclaurin 

 spheroid, its density, and its rate of rotation. Let m represent its mass, a 

 its polar radius, and M its moment of momentum. Then we have 



m = |;ra3(l+P)<T (4) 



M = fma2(l+/l> (5) 



Eliminating oj and a between equations (1), (4), and (5), we have 



-2Kfc>^l_,S = ii+^ \ -X- tan-U-3 \ (6) 



Qk^m^ 



[(?+i!)tan-U-3J 



For a homogeneous body of given mass and moment of momentum, this is 

 a relation between the density and oblateness which must always be satis- 



fied so long as the figure is a spheroid. It is easily verified that -tt^ is 



positive for all real values of X, from which it follows that when the left 

 member of the equation is given there is but a single real solution for P. 



The Jacobian ellipsoids are defined by the equations^ 



-/} 





Jo 



(7) 



(8) 



where the axes of the ellipsoid are a, b, and c, and 



Equation (7) defines the relation between X and ^' which must be satisfied 



io' ^ 



by these figures of equilibrium, and equation (8) expresses — in terms of 



X' and A'^ " 



The equations corresponding to (4) and (5) are in this case 



w = |;ra3^1+P-^l+ra M =^i2 + X' + X'')(o (9) 



By means of these relations, equation (8) reduces to 



25(|;r)iM^ _ (2+X^ + n f^ C^d-C^X qO) 



c/ 



Now we may write (7) and (10) respectively 



d(X,X')=0 <p{X,X')=Ka'- (11) 



* Tisserand's M(5canique Celeste, 2, Cliap. VII. 



