200 mOFESSOli G. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
§16. The Integrals cto, cr^, 
Ill accordance with equation (14) of the “Pear-shaped Figure” the third zonal 
hai'inonic is defined by 
wher 
S, = sin e siu^ e - - K- cos 3 cos^ (^), 
•e 
,3 _ ^ 
r = t [1 + — (1 — = 1 — </■ 
The numerical values for the critical Jacobian are 
ic‘2 = -9231276, f = -5746473. 
Writing -- q^, we have 
hg = (yd — /c‘^ cos® ^) \/( 1 ■“ cos® 0) (2 >® + k'® sin® (f)) (k® + /c'® sin® ^). 
Now let 
a =: 
T>\ 
/3 = 2p®K:® + 2 ^, 
4 I ^ A O 
y = /c + 
8 = K^, 
rT = pV®, = 2j;®K®K'® + pV'®, y = k-k'^ + ■lqTK\ 8' = k'\ 
and we have 
(Ng)® = (^a — 13 cos® d + y cos^ 6 — 8 cos'’ 6) {a -{- (3' sin® (f) -}- y' siu^ (/) -h 8' sin’’ (b). 
The numerical values of the logarithms of the coefficients are 
Let 
log a — 9-0843568, 
log/3 = 9-8835606, 
logy= -1748006, 
log 8 = 9-9305236, 
loga = 9-0496186, 
log/3' = 8-7693310, 
logy = 7-9810798, 
log 8' = 6-6573112. 
f{A,„) = aAS, - /3Al, -f yAi, - 8A?, 
for 5Z = 0, 1, 2. 
J -p y'Wi + S 
The definition of cr^ in (8) then shows tliat 
'^- = 1 yWh /(-^»)/(«J-/(A*)/(«o) - ff[/(A„),An.) -r(-\,)/(p.„)]}. 
In order to find and Ci, Ng must be raised to the fourtli power, and we now define, 
for = 1, 2, 3, 
/ pL.q = — 2a/3A.2„ (2ay -j- /3“) Al,j “ (2a8 + 2/Sy) A|i 
+ (2/38 + f) Xl, - '2yh\l + 8®A1® , 
