44 Mr. I. Todhunter on Jacobi's Theorem. 



The attractions which the ellipsoid exerts on a point y, z) parallel to 

 the axes of coordinates are known to be respectively A-r, By, Cz, where 



A __3MfWit B _3Mr i u*du C = '^C u9du 



k 3 J H' ^Jo(H-AV)H' FJo(l+^)H' 

 Here M denotes the mass of the ellipsoid, and H stands for 

 ^(1+W)(1+/x 2 m 2 ): 



see the ( Mecanique Celeste/ Livre III. No. 7. 



Take the axis of x for that of revolution, and let w be the angular 

 velocity. Then the necessary and sufficient condition for relative equi- 

 librium is that the equation 



kxdoc + (B - io~)ydy + (0 — w 2 )z<fe = 



should coincide with the differential equation to the surface of the ellipsoid, 

 namely, 



Hence we have the conditions 



B — w 2 1 C — w 2 1 



If we eliminate w 2 between these we obtain 



(l+\ 2 )(l +M 2 )(B-C)=A(^-\ 2 ) (2) 



But from the values of B and C we have 



Thus (2) becomes 



{(1+X«) (l+^j^-J^j^O. 



This may be satisfied by putting ju 2 = \ 2 , which gives us an ellipsoid of re- 

 volution. Or it may be satisfied by making 



/ 1 L x»w, , 2 ^C 1 u i du C l u 2 du 

 (l+X)(l+^)J o 1T r=J o -H-; 



this may be reduced to 



^ u 2 (\-u 2 )(\-X 2 f x 2 u 2 )du _ Q ^ ^ 



o ia 



We have then to show that for suitable values of X and fi this relation will 

 hold. This will be shown in the sequel. We must also show that the 

 value of is positive. From (1) we have 



B-M 2 __l+/i 2 

 C-w 2 1.4- \ 2 ' 



