OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
41 1 
a di +b %) 
2 2 ' +1 i\i + 1! 
Y 
(47) 
where, after the differentiations are performed, x, y, z are to be put equal to zero. 
If we suppose the ellipse coated with matter of unit density we thus get for the 
potential at any outside point ( f \ g, h), sufficiently distant, 
w(f-%y+((/-yy 2 +h 
j - 27 rab% 
dx 
1 
o 2 2i+1 i! i + 1 ! r 
7 v \ d f^ dg) i 
-‘ 2 2i+ i H 7+1! W/M-'^ + A 2 
= 2mtbS 2 J +1 J i+1 , -nfAP , 2 ;+ . . . + B, a (2i, 2<r)+ . . . ) . . (48) 
Where A, . . . B 2o . . . . are the same as in § 21, provided we write f~ = cr — b' 2 and g 2 =b 2 
in the expansion of that article. We must have r>b and > \Ar—/r, if the above 
expansion is convergent. 
If a—b or/=0 the expansion just found reduces to the wed known expansion for 
a circular plate of uniform thickness, given in Thomson and Tait, p. 406. 
(Added September 23, 1879.) 
§ 25. The series given above for the potential of an ellipse may be thrown into the 
form of an integral. Writing the result for the elliptic ring in the form 
2 il ilil 
we see that this series is the part of the expansion not involving powers of Jc or its 
reciprocal of 
7 SuL+d) L(i*- e ±\\ 
2Trabe 2Kdh d f’ e 2 * v dh d f’~ 
r 
where c 2 =ar — b 3 . That is, the series is equal to 
d 1 
ab\ eJ b 003 *J h ~ csin *7/ -cty. 
3 o 2 
