46 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



Making use of this simplification, we arrive at the following scheme for the 

 contributions to the internal or boundary potentials of the various solutions up to 

 n = 3. Only typical terms are taken ; <j> b represents the value of tf> at the boundary, 

 V 6 represents the contribution from the typical term to the boundary or internal 

 potential. 



n = 0. 

 = !. 



.00 



b = K\ 



Jo 







r-\ x v v ' x y-^ z- 



W +"=^ ~Jo2a 2 Xl B C 



= 3. (i) fc = -^, V t = ' 



v r - - , 



-~ " 



x v ^x ?-x z- , 



" = -" ~~ 3C 



18. In any physical application of this method, and in particular in its appli- 

 cation to the discussion of rotating masses of liquid, it will be important to know 

 what changes are produced by the distortion upon the mass (or density), the 

 position of the centre of gravity, and the moments of inertia of the body. These 

 changes are given at once by a study of the limiting form of the external potential 

 at infinity. 



The potential at infinity of any mass whatever, taken as far as terms of order ^ t 

 has the limiting form 



m + m u . 



r r 2r 



where m is the mass of the whole body, x , y a , z , are the co-ordinates of the centre of 

 gravity, and, L, M, N, P, Q, E, are products of inertia defined by px^dxdydz = L, 



I pyzdxdydz = P, &c. The moment of inertia about the axis of z is \\\ p ( 

 dxdy dz = L + M, and so on. 



