STABILITY OF THE PEAR-SHAPED FIGURE OF EQUILIBRIUM. 
17 
A similar notation enables us to evaluate </>/, which is given by 
so that 
ft = - | j (<r cos' (9+ k' 2 sin 3 4) AT' dOcty, 
4 = - {« 3 /(V)/(n;)+ K y(A,»)/(W)}. 
7r 
appropriate forms being attributed to /(A), /’(H). 
The result of the procedure sketched is the following series of values, in which 
only the cases i — 6, s — 2, 4, are derived from approximate forms. For the sake of 
comparison I add the approximate values of <f>/ as computed from the formulae in 
“Harmonics.” In the cases i= 2, s = 0, 2, the approximate results, derived from 
that paper, are multiplied by such factors as to make the approximate formulae for 
|A (p) and C 2 (<£) agree with the exact one when p = 1, <f> = 45°; and for |S 2 2 (pd and 
(£, 2 (<f>) to make the coefficients of p 2 and cos 2 </> agree with the exact formulae. 
Table of Logarithms of <uf, pf, <£/. 
i. s. 
log (of +10. 
log Pi + 10 . 
log &*. 
Approximate 
from formula 
in “ Harmonics.” 
0 0 
7-6310567 
2 0 
7-6714241 
7-0286816 
9-0051748- 10 
9-00518-10 
2 2 
(-) 5-6818162 
(-) 5-0264000 
7-0397377 - 10 
7-03981 - 10 
4 0 
8-0332932 
7 • 3558076 
9-6886735- 10 
9-68861 - 10 
4 2 
(-) 8-25158 
(-) 7-32157 
1-72739 
1-72729 
4 4 
8-30779 
7-37092 
3-81610 
3-81612 
6 0 
7-96786 
7-32449 
9-69177 - 10 
9-69303- 10 
6 2 
(-) 8-72778 
(-) 7-94094 
— 
2-20562 
6 4 
9-10094 
8-13161 
— 
5-29999 
8 0 
7-78437 
6-96857 
9-75611 - 10 
9-76872- 10 
10 0 
(-) 7-9838 
6•6024 
9-8473 - 10 
9-87800- 10 
Note that w 10 is negative while <Lo remains positive. 
The calculation of the integrals for i = 8 and i = 10 was very laborious, and as the 
results tend to present themselves as the differences between large numbers, it is 
difficult to obtain accuracy with logarithms of only seven places of decimals. The 
integrals (f> are much the most troublesome ; indeed I do not claim close accuracy for 
<f> 8 ; and as it appeared to be impossible to compute (f> w to nearer than 10 per cent, 
from the formula, I computed the several constituent integrals for the tenth harmonic 
by quadratures and combined them to find (f> 10 . The results derived from the 
approximate formulae of “ Harmonics ” are given for the sake of comparison. They 
clearly give somewhat too large a value for the higher harmonics. I believe w 10 and 
/>io to be nearly correct. 
VOL. CCVIII.—A. 
D 
