OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
405 
If P' be any point inside the ellipsoid we then have ultimately when P' coincides 
with the origin 
e v — 1 
v QP' 
But, if f g, h be the coordinates of Q, 
QP'*=(f-xy+( g - y y+(h-zy 
Hence 
A 1 — d 1 
dx QF df QF 
cP 1 _d 3 1 
dx QP' df QP' 
&c. 
We may therefore take V 2 to stand for 
/ 2 l7 W 3 , od 3 
a j/ +b % +e dk 
and in that case we may put x, y, z zero before differentiating, 
denoted by r, we have 
—e _v 1 
OL — ... 
V r 
or, in series, 
2 ( 1 +Iv=+... 
2i+l! T Jr 
If then OQ be 
• • • • ( 32 ) 
. . . . (33) 
If we multiply a by 277 pcibcSd we have the potential of an infinitely thin shell 
bounded by two similar and similarly situated ellipsoids. When a shell is hereafter 
referred to, this kind of shell will be meant, unless otherwise specified. 
§ 18. Since the whole ellipsoid may be divided up into shells, the potential of the 
ellipsoid at Q is 
f«.<“> 
where p is the density at any point in the shell whose semi-axes are ad, bd, cd. It 
will be obs( 
be written 
will be observed that, since pbd—d&p, — is an element of volume, so that (34) may 
pdxdydz 
QP 
If we take the series (33) we find for the potential of the ellipsoid at Q the series 
jlTrpabcd 2 (l-\-—' V'-p . . . +y—ypj V - '-\-&c}j-dd .... (35) 
