FIC4URE OF EQITILIBPJUM OF A ROTATING MASS OF LIQUID. 
2 G 8 
The equations to be satisfied by the ellipsoid afibrd expressions for and 
in terms of difierentials of T. If these expressions are added together, may be 
eliminated, and the expression becomes 
-9. ira 
2 
■E + Cty 
d'^ 
da 
iJ 
In reverting to the notation ado|)ted here, I remark that p/, will be used to 
denote those functions when the variable is Vq, and the variable will only be inserted 
explicitly when it has any other value. 
In the present case My, the mass of the Jacobian ellipsoid, is M and it was 
shown in “ the Pear-shaped Figure ” that 
Hence 
It was shown in the same paper that the internal potential of the Jacobian 
inclusive of rotation, is 
f Jfj 4 T' + «! 
d^ /sJ 
?r 
dc/y \ay~ dy 
3 i 7, 2 1 _ 2 
Therefore in the present case 
^7 + {l/ + Z^) = 
M ( h 
^'0 \ 
— 7.2 siir 78 (. 1 - sec- y + sec" ^ k 
But the equation to an inequality on the ellipsoid defined by t is in our new notation 
sin'^ {x^ sec" y + y' sec^ f3 z^) = P (1 — 2t) ; 
M I h \2 
therefore 
I,/ + {if + — f 
h \ ]. I i 
'' 0 \ '■ n / 
(59„©o- Pi’Qi')+2tPi'Q,‘5. 
Let us divide this potential into two parts, say U', U", of which the first is 
constant and the second a constant multiplied by r. Also let (k/i)', {JB)" l^e the 
two corresponding portions of the energy JR. 
In order to find {JR)' we have simply to multiply U' by the mass of R considered 
as consisting of positive density. The volume of R is the excess of the volume of J 
above that of the pear; hence the mass of R is M 
kV 
L'A), 
(JB)' = 
IP 
0 LV'^'O, 
kj 
L 
kV' 
(^ 0^0 - 
Therefore 
