FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
10. We shall now limit ourselves to the distortion which was found, in the earlier 
paper, to lead to the pear-shaped figure of equilibrium. For this particular distortion 
we found (cf. equations (93) and (94) of the earlier paper), 
P — £ ( a f" + /3>; 2 + y<,"“ + K),.(50) 
Q - |[L^+M, 7 4 + Nr + 2^ 2 + 2m^ 2 + 2^V + 2 (pf + qj + r^ + s]. . (51) 
We shall now assume for R the value 
R = i^[iL ^ 4 +W+^ 4 + 2 h 7 2 f + 2m£ 2 f+2nfr, 2 +2 (^ 2 +q v 2 +rf)+s], (52) 
this being found adequate to satisfy all the conditions for a figure of equilibrium of 
a rotating liquid mass. 
11 . The values of u 1 +fv 1 and of u 2 +fv 2 were calculated in the former paper (see 
equations ( 121 ), ( 122 ) and (132)). In the same way the value of u 3 +fv 3 as given by 
equation (47) is found to be 
u 3 + fv 3 = f [15|A V + 2fABa 2 /3 +2fA(V y + p 2 a/3 2 + pCa/3y + |C 2 ay 2 ] 
+ fV [131AV/3 +7UBT/1 + 2iACa/3y+lp 2 /3 3 +pC/3 2 y + |C 2 /3y 2 ] 
+ [l3|AVy +2|-ABa/3y + 7|-ACay 2 + |B^ 2 y + fBC/3y 2 + l|C 3 y 8 ] 
+ G; 4 [ ifA 2 a/3 2 -(- if AB/3 3 + fAC/Ty] 
+ ^ 4 [ lfA 2 ay 2 + fAB,dy 2 +lfACy 3 ] 
+ i,T [ 3fA 2 a/3y + 2fAB/3 2 y + 2fAC/3y 2 ] 
+ ?k [l3fAV + 2fAB a/3 + 2-gACay + |B 2 /3 2 + fBC/3y + f€ 2 y 2 ] 
+ fr 2 K [ 3fA 2 a/3 + 2yAB/3 2 + fAC/Sy] 
+ #c [ 3fA 2 ay + |AB/3y + 2fACy 2 ] 
+ ^ 2 [ l|A 2 a + fAB/3 + fACy] 
-£ 5 [ 2§ALa + p(L/3 + 2na) -l- fC(Ly + 2ma)] 
— f "rf [ if A (L/3 + 2 ymx) + fB (Ma + 2n/3') + fC (^a+ tyi(3 + ny)~\ 
— g 3 £ 2 [ lfA(Ly + 2ma) + fB (la + m/3 + ?ly) + fC(Na + 2fty)] 
-£/ 4 [ fA(Ma + 2 np) + lfBM, (3 + fC(My + 2//3)] 
-^ 4 [ |A(Na + 2ny) + fB(N/3 + 2 ly) + lfCNy] 
-£>iT[ fA(Za+m/3 + ny)+ p(M y +2Z/3) + fC(N/3 + 2?y)] 
— f 3 [ lfA(L/c + 2ap) + fB (n/c + ag + /3p) + fC (mic + yp + an)] 
[ 4 A (?nc + + /3p) T fB (M/c + 2/3g) + fC (Z/c + yc^ + /3n)] 
" [ 7A (nnc + yp + an) + fB (//c + yg' + /3r) + fC (N/c+2yn)] 
£ [ is A (2p/c + sa) + fB (2qic + s/3) + fC (2r/c + sy)J 
+i£ \%t +iH>7 4 +Wf 4 +2l ^ 2 + 2 nR 2 f+ 2 n£V +2 (pf+% 2 +tf 2 )+s] 
