•2^)2 PROFESSOR G. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
can be seen, without actually writing down the product, how the coelEcients will 
occur; I write therefore those coefficients as follows :— 
Iq = eta, — ((/3 ha, ~ (ly -f" h/S -j“ Ca, /g ~ uS ffi hy -|~ CfS ffi dci^ &c. 
= a'a, m .2 = a'ji' + h'a\ = ay' -f- + c'a', — a'K h'y' + c'/8' d'a', &c. 
Next let 
f (-Y,.) ^0^2,> ^2^2)1 "T ~ /fiA®;, . . . 
for n = 0, 1, 2. 
/ m^9X„ + mM.„ + 111^91,, + m^9l„ ... 
Then it follows from the defnitions of oj/, pf in (8) that 
-f(\)f{n,) - (?[/(A„)r(n,) -/(.v,)/(n„)]j. 
p: = 
TT snr 7 
6 cos- (3 COS” 7 
TT sin- 7 siu (3 
{/(Ao),/'(n^)-/(Aj)/(!io)}. 
It is, of course, necessary to reduce the two factors of Nf to the required forms. 
The harmonics of the second order are 
S.f = (k“ siir 0 — q,“) {qy — k'- cos'- (p), {s = 0, 2), 
and I find = ‘SlOfSIO, qd — ‘9623311 ; whence we may find n, 6, a', b' for 
these harmonics. 
For the harmonics of the fourth and sixth orders I take the formulm of “ Harmonics,” 
and attributing- to the parameter /3 its value ‘0399726 (or more shortly ‘04 in the 
terms of the sixth order), I reduce (p), ((/>) to the required forms and determine 
a, 0 , c, &c., a', h', c', &c. The numerical values of these coefficients are given in the 
tables of ^ 20 hereafter. 
It may be well to remark that is needed (l^ut not &>g), and in this case Sq= 1 , so 
that a — a' = 1. 
It seems useless to go in detail through the tedious operations involved in carrying 
out this process in the several cases. 
Ap23roximate formulm are given for tlie integrals in 5 22 of “ Harmonics.” The 
12 >o do- of that paper is the same as ^rrJd ~ [| T dO d(f) of the present one, and 
ros oo.s 
the factor there ^vritten M is Id — . i, . Hence it lidlows that 
sin-^ /3 
cj)" = I do- of “ Harmonics.” 
In order to ap])ly this to the harmonics of the second degree, it must be borne in 
mind that a different definition of = 0, 2) is being used here. If [^o], [<^ 2 ~] be 
