178 
SIR G. H. DARWIN ON THE 
The typical term written down must be deemed to include sine-functions as well as 
cosine-functions, and all those types which I have-denoted by P. C- S in “ Harmonics.” 
On multiplying each side of our equation by any harmonic function, and integrating 
over the surface of the ellipsoid, an element of which surface is denoted by da, we 
find in the usual way 
| / (/*> $) W M (<f>)pda 
e/ = l _.. 
J [%‘M<R-(<f>)Ypda 
Suppose that / (/a, </>) is zero everywhere except over a small area Sa situated at 
the point p !, (f)', and that it is there equal to a constant c; also let p' be the value 
of p at this area So-. 
Then the mass of the inequality is 
| pf(p> <j>)pdcr = cp'pSoc, 
where p is the density of the solid ellipsoid which is deformed. 
Next let us suppose that the mass of the inequality is unity, so that 
Cp'pSa. = 1 . 
Then we have 
/to 4>) W to) «.* (*)pdo- = c®/ (/) «,• W)p ’Sa = I«• (V). 
Hence 
e- = 
w to) c/ m 
P J [V to ffiV (4>)Jp da- 
I now write M for the mass of the ellipsoid, and shall subsequently make it equal 
to iTrpa 3 —, while the mass of the distant particle will be — or #?rpa 3 --- . 
1+K X 3 1 l+X 
Since ^ppda = 3 M, and ip o (y) (£„ (<f>) — 1, we have e 0 = -— . 
3 jjI 
Thus an inequality representing a particle of unit mass at p!, ft on the surface of 
the ellipsoid is expressed m ellipsoidal harmonics by 
P 
3 M 
+ S 
w(p') & n^')w(p) 
p QPi'W (tri'Mfpdo- 
By the formula (51) of “ Harmonics, ’ the external potential at the point v, p, <f> of 
the inequality is 
+ f / to) (f) f(,») Qi‘ (Q w to) e , 1 W _ 
kp r 
[W (l 1 ) (< P)Jpdcr 
