FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
13 
Let us put 
f 
't; v, _ 
dX — X (CjjX 1 + Cgo?/ 4 T C332 4 + C^ST?/ 2 + t$iZ X + £■>:$ 2 2 + tlj-'C 2 + tL?/ 2 ~t +tJ 4 ) . ( 57 ) 
The values of the various coefficients in this expression are found to be as follows :— 
Li = 
0 AA 5 
[ 15f A 2 a 3 + 2f AB« 2 /3 + 2| A(V y + fB= a /3 2 + £BC aj 8y 
+ §C 2 ay 2 —2f ALa —g-B (L/3 + 2na) — g-C (Ly + 2ma) + ylL] dX 
A A 
-[26|AV + AB' ! A + ACV + 6 A'A 2 Ba 2 /3 + 6^A 2 €a 2 y 
+ l£AB 2 a/3 2 + lgAC 2 ay 2 + I j ABUa/3y + A B 2 0/3 2 y + A BC 2 /8y 2 ] dX 
+ ] [6 T g-A3La + AB“ (M a + 2w/3) + A ^ 2 (Na + 2 ny) 
+ f- AB (L/3 + 2na) + -f AC (Ly + 2ma) + -gBC (/a + m/3 + ny)] c/X 
~ [ -^g[l|AlL+|Bn+yCm] + | dx. 
(58) 
Cl 2 = 
1 
A Add 
[ 13iAV/3 + 7gABa/3 2 + 2iACa/3y + lp 2 /3 3 + |BC/3 2 y 
+ fC 2 /3y 2 — lyA (L/3+2na)—fB (Ma+ 2nf3) — yC (/a + W^ + Wy) + in] dX 
£ WV+A^ 3 +AC V+6 AA 2 Ba 2 /3 + 6 AA 2 C« 2 y 
+ lgAB 2 «/3 2 + l | AC 2 ay 2 + llABCay+ x 3 6 B 2 C/3 2 y +ABC 2 /3y 2 ] dX 
[6 A AVjS + 5f A 2 Ba/3 2 + lfA 2 Ca/3y + 2b|AB 2 /3 :j 
1 
0 AA 2 B 2 
+ f , ( . 1 AC 2 /3y 2 + 1 gABC/3 2 y] dX 
+ 
1 
0 AA 2 B 2 
[IfA 2 (L/3 + 2na) + 2AP 2 M/3 + AC 2 (N (3 + 2 ly) 
+ IgAB (Ma + 2n/3)+fAC (lot + m(3 + ny) + §BC (My + 2//3)] d\ 
+ | 2 yy>[ 6 A A 2 La + AB- (Mot + 2nj3) + AC 2 (Na + 2ny) 
f 
1 
AA 2 B : 
+ y AB ( L/3 + 2 net ) + g-AC (Ly + 2mct) + gBC ( la. + mf3 + n~y )J dX 
;[|An+fBm+iCi] dx 
~~ Jo ^^[IplL+pn+pttt] dx + 2 j 
G 
AA 2 B 
dX, 
(59) 
