ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. AT5 
Suppose 
Merve Bay 
=— €1, 
wf Y 


= €9. 
From (3) we have 
vp (a? — y*) =n (8° — y’); 
therefore, if we neglect squares of €,, &, 
B/e, = v/e, = gC say. 
In the case where the thickness of the shell is finite compared with its linear 
dimensions g will be a finite quantity; when the shell is thin g will be large, and 
when the fluid nucleus is small compared with the dimensions of the whole system q¢ 
will be small, the densities of the fluid and of the crust being supposed comparable 
with one another. In all cases qg will be positive. 
If e, « are each equal to zero, the equation (5) becomes 
ABM —[(C — A)(C — B) + AB] No? + (C — A\(C — B) ot = 0, 
or 
(2 — w) [AB — (C — A)(C — B) o?] = 0. 
Thus we obtain as a first approximation to the roots 
SOLAN Es 
Ye es c He B) w. 

Next let us retain first powers of €,, €, in (5); this equation then becomes 
MAB(1 +e, + «) 
= a2[(C — A) (C —B) (1+ 4 +4) + AB(1+ 24 + 26) + g(ACe + BCe)] 
+ ow (1 + 2e, + 2e,)(C — A + qe) (C — B+ gCe) = 0, 
or 
02 =a?) (AB — OC —AC— Ba?) 
+ «, [ABA — wd? {(C — A)(C — B) 4+ 2AB 4 qAC} 
+ w*(C — B) {2(C — A) + gC} } 
+ «, [AB — wd? {(C — A) (C — B) + 2AB + qBC} 
+ of(C—A)(2(C—B)+gC3]=0 . (8); 
dividing by ABh* — (C — A)(C — B)o”’, and putting \*? = o? in the terms which 
contain €, or €, as a factor, we obtain as a closer approximation to the root \* = o’, 
3) Pw j 
