ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 497 
§ 9. Reduction of Period Equation. 
Expanding out this determinant, we obtain 
[{AW — (C — B+ rv) o*} {BY — (C —A +2) 0} — od’ (A + B— C)’] 
9 19 2 by SO 22 
x | (2p — 09) (2p? — BF — 0) — FE = 07] 
+ per? {Ad — (C — B + v) a} (2p? — 0? — c”) 
+ v(e —b?) V {BM — (C — A + p) w*} (2p? — &”) 
— 2w°d? (p> — ce”) (A + B — C) fu (e — b?) + v0} 
SP UUCH Cha ON Oe Rh Mi Sue. 55 Sy Sul me een (ee 

Now since 

=) 

= p(p? — 1) + (p+ p=) (6? — 02) + (p?— oF (1 — 
1 9.9 9 9 9 9 9 9 f9 2\9 
repel Pe (2k oes) tes Pees) (Ore Caste (Pa) 



=. * (2p — c*) (2p? — b? diay 4ex*p* (p* — b*)5 (43). 
itt 9 9 
BOC aerccn i (2 fae etl, eae) 
if ) ) 2 2% 9 9 ONG 
= 272 {? (2p? == Ge) 4a” (p° a b’)§ 
and 
9 19 1 9.9 9 9 1 9 9 9 2.9 
2p? — c% = —.{rp® + (p® = ©) 5 = ag. M (2p? — &) — daip?} 

> YS 
Hence substituting in (42) we obtain the following cubic for )? : 
[ {Ad — (C — B+ v) 07} {BY — (C — A+ pz) 0} — wd? (A + B—C)’] 
 {d? (2p? — b? — c®) (2p? — c2) — 4e%p? (p? — 2)} 
+ porn’. fAXW — (C — B + v) w} {d? (2p? — b? — c) — 40° (p? — b*)} 
+ v(c? — b?) {BNW — (C — A + p) w} fd? (2p? — c) — 4a*p§ 
— 2u?d4(p? — 2) (A + B —C) {u (c? — 8) + ve%} 
+ py (V — 407) 7 (e — F)= 
MDCCCXCV.—A. 38 
