OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
407 
rP 
§ 20. If we put ci=b then V 3 becomes — (a 2 —c 3 )—, an d the potential of a solid 
ellipsoid of revolution of uniform density becomes 
M/, 3P 2 a 2 —c 2 , 3P 4 /<x 2 —c 2 \ 3 3P e /a 2 -c 2 \3 , 
3-5 r 2 ' 5'7 \ r 2 j 7*9 V r 2 / + * ' 
. (38) 
§ 21. The expansion involving surface spherical harmonics of the potential of a solid 
ellipsoid of uniform density is easily derived from (36). 
In virtue of Laplace’s equation, the operator V 3 becomes 
(a 2 -6 2 )f — (6 3 —c 3 )t 
\ / ( f x \ / rlv 
dz 
which we will abbreviate to 
Now by Theorem i. 
,„d 2 ,d 2 \d 
3; ^ dz 
i 
'?- 2i+ V ^ 
cZ 2 ' 
_P2_ 
^ dx 
dx v dz) r 
Since there is here no operator in regard to y we may put y —0 in the expression 
for Qo iT~‘ before differentiating. In that case y=x, and the expansion will take a 
comparatively simple form. 
We take, according to custom, the axis of z for the axes of the zonal and tesseral- 
sectorial system. The expansion of Q 2z r 2 ‘ then becomes, by Laplace’s Formula 
P oj-(- . . 
2 4o ' 1 2 i\2i\ 
=PL 
2i + 2a\ 2i — 2a ! 
. 2i(2i—l) . 3 „ 
( 2 f, 2 cr) V 3 '( 2 q 2 cr)+ • • • 
The general term of the expansion of the potential of an ellipsoid may therefore be 
expressed thus :— 
where 
(-m 
im(AP / 2*+ • • • +B 3o .(2f, 2<r) , -|" • • . ) 
(2i + 3)(2i+l) r 
a 2 s 
2/ 2^2 I 1 ) 3 1 o;_yj,_ ( ^ e 
B^=(—l)- 
cr! i — cr] 
v2i-3<r/2ff-_I_£^2i-2o--2/’2<r+2 
T 
(* ^X'*' ^ 1) 2cr+3 1 2 i_2<7“4/’2<r+4. \ 
WL2 2T+2 2^ X ^ ' 
(39) 
