a Rotating Ellipsoidal Shell containing Liquid. 547 



If we now suppose as usual ©i, « 2 , ^u &h> £> V to vary as 

 e ikt equations (4) are equivalent to 



iMlx — (1 + e) nCl 2 = ikcoi 



(6) 



from which we obtain 



+ €)nQ l i = ika>2 ) 

 __ Po) 2 + /A;(1 + €)wg) 1 "| 



i2 2- #»_(l+ e )V 



... • (7) 

 n _ ft 2 ^ — ik(l + e)nco 2 I 



Substituting these values in (2), the latter become 

 + {o+V-A-Af- £$& } -»-«*'. 



•, r»,», ,1/ ,^ a -(i+e)» 2 \ 



>- («) 



-{a+«-*-*-£ffifa}*«- 



Me ikt , 



Denoting the coefficients of ikco^ n&> 2 in the former of these 

 by A", C"— A" respectively, we have 



co^ico^-m^KniO'-A^-kA"). . . (9) 



We will now connect our moving axes of x, y, z, denoted 

 respectively by OA, OB, OC, with axes fixed in space 

 OX, OY, OZ. Let denote the arc CZ, i/r the angle which 

 the arc ZC makes with the arc ZX measured from the latter in 

 a direction similar to that of the rotation of the shell, cf> the 

 angle the arc CA makes with the arc ZO produced measured 

 from the latter in the same direction as that of rotation*. 



Euler's geometrical equations are 



<£>+T|r cos = ra, (10) 



= 0)! sin + o) 2 cos (j>, (11) 



— i|rsin = 0)! cos (f>— ct> 2 sin<£ (12) 



Since &> 1? w 2 are small quantities 0, ^r are small also, and if 

 * For diagram see Routh, l Rigid Dynamics,' vol. i. 



