508 
PROFESSOR C4. H. I)ARWI>i OX ELLIPSOIDAL HARMONIC ANALYSIS. 
Imagine that the surface of a homogeneous ellipsoid of density p, defined by 
receives a normal displacement Sn, such that 
8« =lJ.£y, 
Then the erpiivalent surface density is ^9. (p-)Cf (<^), and we can at once ’sw'ite 
down the exjoressions for the internal and external potentials by means of (51). 
If cCq, yQ, 2 q be the co-ordinates of a point on the surface, it is clear that the 
co-ordinates of the corresponding point on the deformed surface are 
i.2 _ 141) y • ?/ 2/oO + - 1) )' 
Hence the equation to the deformed surface is 
— -0 1 + 
Irv- 
} 
70/ o rr?i + 7 0/ i y, + — 1 + (p) (JT* ('/>) 
(52), 
a;- 
or since 
fi¬ 
x'' 
7/2 ( j 72 _ ^ ^2 ( j ,2 _ 1 ) ^ 
-fi = 1, it may be written 
(w - i/Q^)^^ = 2ep/ (p) C ((^). 
7fi . 
If we substitute for its value from (49), this may be written in the form 
- 7 /,f ) (p^ - pf) - V ,?) 
ry(V-l)(v-lfi-?) 
= 243/(p) C/(<^) 
(52). 
This is the equation in elliptic co-ordinates to the deformed surface, but in actual 
comjiutation the form involving rectangular co-ordinates might perhaps be more 
convenient. 
12. The Potential of a homogeneous solid Ellipsoid. 
It is well known that the potential of a solid ellipsoid externally is equal to that 
of a “ focaloid ” shell of the same mass coincident with its external surface. 
If p' be the density of the shell defined by and fi- Sn, we have 
^n¥p' 
~ -f '2vfv - ) (i^u^ fi- '2vfv — iy{vQ~ fi- 
1 + /3 
r^/3,, 
(ry - 1)^/7^ 
4 
iTrPpivh^ ~ (V - 
