FIGUKE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
255 
condensation as — C. I do not suppose the condensation actually efiected, but 
imagine the surface of J to be coated Avith etpial and opposite condensations + G 
and — C. 
The system of masses forming the pear may then be considered as being as 
follows;— 
Density + p throughout J, say + J. 
Negath'e condensation on J, say — C. 
Positive condensation -f C on J and negative volume density — p throughout i?. 
This last forms a double system of zero mass, say D, and D = C — E. 
Let Vj, Vr be the potentials of + J and + R, and Vj_,. the potential of the j^ear. 
An element of volume being Avritten dv, let [ dv, [ dv, [ dv denote integrations 
throughout •/, R and the pear respectiAmly. 
Let d be the distance along the s axis from the centre of the ellipsoid as origin to 
the centre of inertia of the pear; let w be the angular velocity of the critical 
Jacobian about the axis x, so that or/'lrrp = '14200 ; and let ex' + Sw' be the square 
of the augular A’elocity of the pear. Lastly, let J/ be the mass of the pear. 
Then the lost energy E is gi\^eu by 
E — Vj_fpdv + [ur + hex)-') j [y-' + (z — d)^] pdv. 
Noav 
j zpdv = J/d, so that (— ‘2zd. -j- d~) pdv — — J/db 
]j-r )j-> 
Again, since 
Ave have 
Also 
f, = f. - f . f. ^rpdv = f Vjpdv, 
J j—r J j J /• y} Jr 
^ [ Vj_,pdv = i I — [ V;pdv + i f V^pdv. 
J j^r J j J ?• Jr 
i {w“ H- [ \jj~ + (a — d)~] pdv = j (y* + pdv — [ (y® + z-^) pdv 
+ bSw" j" (y" -|- z") pdv — h {(t)' + hat") ME. 
J j-r 
Hence 
= 2 I [ {y^ + ^~)] — [ [ ^5 + + 2^)] P^^ + i f Vrpdv 
•j Jr 
+ [ (y^ + z^) pdv — ^ + 8fo»”) Jddb 
As the several terms Avill be considered separately, it Avill be convenient t(j have an 
