ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 493 
and, neglecting squares of 6,, , 03, we obtain 
P= PLY, + 3° (@? +H’) — o7y,2,9 + 07229] — pp. . . (36), 
where in the small term # we may replace a, y, z by 2, y, 2}. 
If L, M, N denote the couples on the shell about the axes Ox,, Oy,, Oz, 

Py,z Pye CB 
— [| do} ie EL i =@-A@-a ({ Ppy,2z, do, 

ey Poe 
ve (|p doy ~ ma = <5 [{Ppa.e, do, 
N= he ie = Sel Te a 2) it Ppxy, do, 
where do is an element of the surface of the displaced ellipsoid, and the integrals are 
taken over the whole surface. 
Let us now consider separately the parts of these couples introduced by the 
different terms in the expression (36) for p. 
(a) Take p = p, Vj. 
The pressure at every point is the same as if the fluid were at rest under a 
potential V,. 
V, will, in general, consist of three parts due respectively to («) the attraction of the 
shell; (8) the mutual attraction of the fluid particles ; (y) any external attracting 
system. 
If the part («) gave rise to any couple, it would be exactly counterbalanced by the 
couple on the shell due to the attraction of the fluid, since the attractions of the shell 
on the fluid and of the fluid on the shell are equal and opposite. 
The system of forces (8) also form a system in equilibrium, and, therefore, can 
give rise to no resultant couple on the shell. Thus no couple can arise from the 
mutual attractions of the parts of the system. 
The pressure at any point due to the part (y) is the same as if the fluid were at 
rest. Thus the couples due to any external attracting system will be the same as if 
the fluid were supposed to be solidified. If we add to these couples, due to the 
attraction of the external system on the fluid, the parts due to the direct attraction 
on the shell, we see that the total couples due to any external system will be the 
same as if our system were solid throughout. 
(b.) Take 
P= 30'p, (UP + IY") — © py (1, — @9,). 
Integrating over the surface of the ellipsoid 

