PEOFESSOK G. H DAEWIN ON ELLIPSOIDAL HAEMONIC ANALYSIS. 
507 
and 
/ pdmdf 
( 
l<?dfid^ 1 
5 / ^ 
1 + /3 
1-/3 
( 1 -/ 3 ) 
fZ fZ 
fZ?i k~ v^{Lvq 
n - 8 cos 2.;. o\2 
(-T^ri- 
(1 - 
8 cos2,#,)(l 
2\ ’ 
/X-) 
(50). 
Two fiinctio2is, written in alternative form, 
are solutions of Laplace’s equation, and together form a function V continuous at 
the surface of the ellipsoid v = vq. Reading the upper line we have a function 
always finite inside the ellipsoid, and reading the lower line one always finite 
outside. Hence V is the potential of a layer of surface density on the ellipsoid i^q, 
and bv Poisson’s equation that density is equal to — — 
Let the surface density, which it is our object to find, be 
13/ (/x) (IT/ (</>). p , 
a surface harmonic multiplied by the pei'pendicular on to the tangent })lane and by 
a quantity p. 
% (outside) 
dY 
dll 
(inside) 
d p 
d 
'dn ~ k- 
V^flvQ 
8 = - 
X 
Airk~v, 
0 L 
i9/(j^o)J^ €/(^u) - ©/(^o) ib PiU^o) 
dvn 
But 
©'(!<„) = ef.‘(*'o) (' 
j P 
dv 
[f/(-)P(^--i)M-^-i:-y 
Differentiating this logarithmically we find 
P = 
mi 
^ 70 , .T i\jZ o 1 + ) 9. constant. 
47r/.-7/o(ro“ - IV (r,,- - 
Hence surface density 13/(g)C/(</)). p, where p is constant, gives rise to potential 
f inside 
47r/dt 
.... (51). 
The same investigation holds good with SV(^), with P, Q, C, S in place of 
the corresponding letters above. 
•a." ) 23/ M w 
3 1’ 2 
