1882.] Liquid Ellipsoid about an Axis. 213 



y2 (c 2 -a 2 ) f X [(b 2 -4a 2 )\ 2 -{Sa 4 +(c 2 + a 2 )(a 2 -b 2 )}\] a ^ 



2-rrp * (a* + b 2 + p) (6 J - c 2 ) [3a 4 - (c 2 - a') {a 1 - ¥) \ 



Q - o . c 2 + a 2 + ^ 



n y a 2 + b*+ fi 



therefore 



<w 2 2 a 2 — 6 2 c 2 + a 2 + /a 



27T/3 b 2 - c 2 ' 3a 4 - (c 2 - a 2 ) (a 2 - b 2 ) 



abcdX 



p3 ' 



( [(c 2 - 4a 2 ) \ 2 - {3a 4 - (c 2 - a 2 ) (a 2 + 6 2 )} X] 

 Jo 



co 2 c 2 — a 2 a 2 + b' 2 + /j. 



2irp ¥ - c 2 ' 3a 4 - (c 2 - a 2 ) (a 2 - 6 2 ) 



f*[(P- 4a 2 )X 2 - (3a 4 - (c 2 + a 2 ) (a 2 -6 2 )}X]^ 



Then the pressure is given by 



fa? 

 * + %? 



y 2 z 2 \ 

 ^- fi + "2 ) = constant, 



/, af y* z'\ 

 p = „ + (r [l---^^ } 



where -57 is the pressure at the surface, and 



a = \pa 2 A = ±pb 2 B' = ipc 2 ^', 

 and therefore 



a 



(6 2 - c 2 M - (4a 2 - 3fr 2 - c 2 ) B + (4a 2 - fr 2 - 3c 2 ) G 



a*bYd\ 



= ^- J (SX 2 + 6a 2 \ + Z>) — p 



= ^f o W {3(a 2 + X) 2 -( C 2 -a 2 )(a 2 -6 2 )} a ^^, 

 putting 3a 4 - (c 2 - a 2 ) (a 2 - b 2 ) = D. 



In order that the surface may be a free surface, and therefore 

 that ot may vanish, it is necessary that <r should be positive fur p 

 to be positive in the interior of the ellipsoid. 



