ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 499 
Mt. [A (B? + 7”) + 7 (8 — 7°) I[B (e’ + 7’) + pw (? — y’)] 
— oe + 9) (B+ ¥?)(B—C— (A —C~ p) 
Hi 2((4 + BC) (ae) n(B C=) (BC) cae 
elie (o 98) (BEG aye ipa (7) (es =) i 
+o. (402° (B — C—v)(A—C—yp)} =0, 
or 
NS LAS (Soe) (8 77) BGs.) (ae, 
— oe + PB + 7)(B- C=) (A—C~ p) + He (A+) (BH HY) 
Sone Op ie a ie) | 
+ off 40°8?(B—C—r)(A—C—p)}=0... . . (45). 
In order that the system may satisfy the criteria for “ ordinary ” stability the roots 
of this equation in )* must be real and positive. 
The period equation (45) agrees with the equation (5) (§1), and the solution of it, 
in the case in which the ellipsoid differs but slightly from a sphere is given in § 3. 
§ 10. Nature of the Oscillations. 
From equations (40), (41), (48) we see that the equations giving the ratios of the 
quantities @’,, 65, B,/r, B,/7 are 
0 {AN +(B —C)a"} + 005(A-+ BC) —» (0%, +P) = 0 | 

0’. {BN + (A — C) 0%} — ho (A+ B—C) — 4 (0°, 4+) =0 
f(a? +y°) 2 — 407a?} — 2wihy’? — (Me 4w*)(a? —y*) #',= 0 
3 9 9 ) * 13}. 9 / 9 9 9 7 
SUB by?) BM 40°B?} + 2widy? + d2(N — 40”) (B= y?) 6 = 0 
(a). We have seen that in every case )? = w is one root of the period equation ; 
when \ = o the equations (46) reduce to 
Ba 6 
Otery 2 

GEN BPO} Eh (NBC) =p 
TO 
,{A+B-C—p}—i6, (A+ B-OC)+p-2=0, 
B 
2 
9 
TO" 
B, 9 2 07.9 Be 2 }/ 
2 (Y? — 3B") + Qi? — 3 (8? — 7’) 0 = 0, 
TO 
382 
B, 5 ON 70 
5 1 38 (e — y) F,—= 0, 

(9? — S22) — Bin? 
TO 

