PKOFESSUU (4. II. l)Ak\VlK OX KLLII’SOTDAL HAILMOXKJ ANALYSIS. 
51 
r> 
• ) 
]>y means of (51), we now at once write down the external potential of tlie ellipsoid. 
It is 
1 + n (a.ip) 
■in (S., 
l>/3 
in Iq? 
In this expression Mq denotes the mass of the ellipsoid, and tlie (Jr’s ai'e merely 
coeliicients determined a})})roximately in § to. 
In order to tind the potential internally, let 
)■■- = + (/'- + r -; 
and, as suggested by the form of (61), let 
I’ly' _ I + n n + 1 — ‘6/3 
/.- 
iUJ 
A WA), 
_ * + ^ n — \ + s/3 ^ 
12JJ ' 1 — /•? la 2/,,^ 1^0 {n)^-2 ( 9 ) "h ^0 
3(1 - /3) 
Then i\j' i'’ a solution of Laplace’s equation throughout the interior of tlie ellipsoid, 
and at the surface, where a = it is ecpial to x' + _y- + 
Now consider the function 
= 
y) + 
'L±^ 
in 
3jS @y(i^o) 
in 
The whole of it, excepting the term in r'\ is a solution of Laplace’s equation for 
space inside the ellipsoid, and the term in }'~ gives V’ V = —47rp. Also at the 
siuTace, where r= zy, this expression agrees with (63). Hence we have found the 
potenticd of the ellipsoid internally. 
The potential at an internal point does not lend itself to expression in elliptic 
co-ordinates, hut it may he given another form which is perhaps more csinvenient. 
In our present notation the well-known formula is 
“ 2 /7 
1 — 
7 -2 
dv 
Since I;3o(^) = L ~ 
A±Jy 
A. 
1 
— "')a Pi(^) = 
le 
integrals may be expressed in terms of the Q-functions, and we have (omitting the 
divisors (J? and E for brevity) 
VOL. CXCVII.—A. 
'■- Qd(a,j) _ _ s @1 (a,,) 
' /*■ /d Pd(zv) A'"^3ih’(.) 
3 L' 
V — 3^io/^o(^( 
> - o' , 
. (U5). 
