52 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



Equation (79) was obtained by the elimination of K from two equations (65) and 

 (74), each of which expressed in effect the condition that the centre of gravity 

 of the mass should be at the origin ; in fact, equation (65) was only a short way 

 of arriving at the value of K, which would in any case have been given by equa- 

 tion (74). We therefore expect that equation (79), derived from (65) and (74), 

 will prove only to be an identity of which the truth is involved in the three other 

 equations (76) to (78). And, as a matter of procedure which is entirely at our 

 choice, we shall elect first to solve equations (76) to (78), and then to verify the 

 truth of (79). 



The elimination of a', /3', y from equations (76), (77), and (78) gives a determinant 

 which on expansion reduces to 



3 _! JL 



-(- b 2 c: 



2 ) +c 3 (3a 8 +6 a )} + = 0, (80) 



Ct 



and this is accordingly the equation giving points of bifurcation on the Jacobian 

 series of ellipsoids. 



22. Two identities of importance are the following : 



--f. . (so 



abc 



From equation (70 ) 



JA = W+| S , ........... (83) 



JB = +p, ........... (84) 



Jo=A ............ (85) 



O 



so that on addition, by the use of (81), 



(86) 



, 



abc \a 2 V <?) 



giving 0, and hence J A , J B , and J c , in terms of a, b, c, arid n. We further have 



o AAB J l(a a -6 8 )AA ( 2 -6 2 )ABj dX = a'-b* 

 T _ a 2 J A -c 2 J c 



IAC - -^-r-' -. (88) 



