FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
301 
be integrated for 0 = 0, 15°, 30°, 45°, 60°, 75°, 90°, drew curves on squared paper, 
and counted the squares on the positive and negative sides of the axis. 
In this way I find logwg = 6‘765, log pg = 5'653. 
The integral (/>g is found at once from §22 of “Harmonics” with 13=0. This 
gives (/)s = or log (^g = 9-247. 
If Pg fp) be expressed in terms of cosines of 0 we have 
Pg (fji) = a ~ h cos“ d + c cos^ d — c/ cos® 0 e cos® 0, 
where a = 1, h = 18, c = 74'25, d = 107'25, e = 50-273, 
Then we may, as in § 18, put 
w,, = Pg {y) = a b cot' X ^ X + X'^ ^ X* 
As was done in that section, I then computed and tq^, and so found the integral 
of the empirical function. The result gave 
log ^g =: 9-191 ; whence log’ Hs = ’370, 
It may be admitted that the determination of ^Ig, Bs is not wholly consistent with 
that of the previous integrals, since I only assume k to be unity in as far as the values 
of a, h, c, d, e are affected. 
Applying these values as before, I find ^ 3 —^ 3 =-197, log C3=8-540, p 3 =-000092, 
11 
~ = - 0027 , and 
[8, 0] = -00000051, = -00000025. 
Hence that part of e (the uncomputed residue of the series) which depends on the 
eight zonal harmonic is only about -0000008. The contribution is so insignificant 
compared with the critical total -00014, that I have not thought it worth while to 
make estimates for the tenth and twelfth harmonics. 
It may then be confidently asserted that the pear is stable. 
I> 
In the course of this estimate we have also found /g = = ’0027e^. 
§ 20. Second Aiyproximation to the Form of the Pear. 
Extracting the numerical values ot the /’s from our results, we find that the 
inequality of the critical Jacobian ellipsoid is 
eS.^ + c” [-15068 S.^ + -50839 + -07705 - ’000506 + -00000019 
H- -01852 aS'g - -000278 + -00000034 + ’0027 - . . .]. 
