ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 489 
but p*, pw, v° are the three roots of the equation in « 

therefore, 
(2 — p*) (4 — p*) (a —-v?) =a(a— b*) (a — c?) — a? (a — b*) (2 — c?) 
—ya(e—c) —#a(a — b*); 
putting 2 = — @ in this identity, we obtain 
RMN = (8 + p*) (B+ ) (8 +) =B(B + L*)(B +c’) 
+ «(8 + 0°) (B+ &) + ¥B(B+ &) + 2B (B + 0b’), 
and, therefore, at the surface 
RMN = (X? + Y? + 22) B(B + 8°) (8 + &) + p?X?.(B +B) (B+) 
+ (p? =P) VY. B(B +c) + (p? — &) 2.B(B +B) 
=(pP + B)[(B+ (B+ e)X +8 (B+e)V +88 +2) 2), 
or, since by (23) (8 +- 8) (8+ ¢*) = — B(B +’) — B(B + 0b’), 
RMN = (p’ + B)[B (8 + ©) (Y° — X*) + B(B + B*) (2 — X*)], 
therefore, when R’ = p? + f’, 
MN’ =6(8' +.)(V—X)+ 6 (6 +0)(2—%), 
and if p, = R/M’N’ 
OY cos a! ae ! cos B’ $+ Mh cosy 
== AE ES Se G4) (VO) ee (Sse OCA 2) (2) 
Cit gee ON Ny a p b2 
7 8 ‘ cos B = EG ) 2XY {28 (6 qr @ ap (8 (8 ap Oks 
or since f’ satisfies an equation similar to (23), 
Oy Si Ca ator OY p> + B’ 2 1\ 7.2 fe 
7 OnE = 7 cos B = Ea oy eee + B’)b?} . . (28). 
Let us denote by f’,, B’z, the two values of @’, and assume that 
MDCCCXCV.—A. 3 R 
