FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
15 
The values of the four coefficients ti 1} ti 3 , are most conveniently expressed in 
the form :— 
n I dx g 
!o A A 3 
tii — <5i—J . yv (apA+IqB + ^r 1 
tio = S; — 
A- 
1 
o AAB 
1 
(IpA+iqB+irC) dx + o q 
dx 
Jo AAB-' Jo AAB 
A A 2 
G 
0 AAC 
1 
q (fpA + iqB + 1 rC) dX + lx | 
dX 
^ ~ * 4+ I -^x-(tP A +iqB+irC)^x+ls j o j Q 
AAC 2 
dX 
o AA 
-2 
G 
0 A AG ’ 
G 
AA 
dx, 
dX, 
dX, 
dX, 
(62) 
(63) 
(64) 
(65) 
where § 2 , S 3 , d 4 are quantities which do not depend on p, q, r, S, being given by the 
following equations :— 
<h = Pv-Tl (13|*A 2 a 2 + 2yABa/3 + 2iACay + §B 2 /3 2 + 5-BC/3y + fC 2 y 2 ) 
Jo AA 
— lfA (L/c + 2ap)— X B (iik + aq + fy ))—fC (niK+py + ra) 
+ 2 Gl-A'^ + ifB 3 /3 3 + {\.,cy + 6ffA 2 Ba 2 /3 + 6 &A 2 Ca 2 y 
+ lfAB 2 a/3 2 + l|AC 2 ay 2 + lyABCa/Sy + t\-B 2 C/3 2 y + ff BC 2 /3y 2 ] dX 
[6ffAV + lfA 2 Ba/3 + if A 2 Cay + fQAB 2 ^ + x %AC 2 y 2 + fABO/Sy] dX 
K. 
AA 2 
_ I A A 3 ^ ^A"La+-jfB 2 (Ma + 2n/3) + X yC 2 (Na + 2wy) 
+ 
+ fAB (L/3 + 2na) + fAC (Ly + 2?na) + fBC (1 a + m/3 + ny)~\ dX 
yy DitlA" (L/c + 2 a p) + ffB' (M/c + 2 / 3 g) + ffC' (N k + 2 y r) 
+ g AB {yik + ag + ftp) F fAC (tyik F yp F ar) F | BC (Ik F y q F /3r)J dX 
+j [liAi+iBn+JOm] ./a.(66) 
K = [ [* (3tAV/3+2}AB/3 2 +fAC^y) 
— fA (tik + aq +/3p)— fB (M/cF2/3g) — fC (L + y g + /3r) 
+ 6ff AV/3 f 5|A 2 Ba/3 2 + if A 2 Ca/3y 
+ 2||AB 2 A 3 + T f AC 2 /3y 2 + f ABC/3 2 y] dx 
[6ffA 3 a 2 F lfA 2 Ba/3F llA^a^ + t^AB^F AAC 2 y 2 + |ABC/3y] dX 
AAB 
3 1 
AAB 
F 
1 
0 AAB 
5 [xtA 2 (L/3 + 2na ) + 2ffB 2 M/3 + ff C 2 (N/3 + 2/y) 
+ 1-gAB (Ma + 2n / 8) + fAC (Za + m/3 + ny) + fBC (My + 2//3)] dX 
[ifA 2 (L/c + 2ap) + ifB 2 (M/c + 2(3q) + ffC 2 (N* F 2y?■) 
F §AB (n/c f ag F /3p) + fAC (m/c + yp F ar) + fBC (Ik + yg + /3r)] dx 
+j oX jBs[Un + fBJM + iCl]d\.(67) 
