[164 
164] ON gauss’ method for the attraction of ellipsoids. 
27 
ng a, b, c 
tes of P, 
•esponding 
esponding 
upon the 
he third 
, B+BB, 
die point 
i+8A)£, 
distance 
3 of the 
C=p8N; 
which gives 
Now from the equation 
we find 
da. 8MP = dS cos MQ. 
ABC ^ 
X :=ffgS, 
A 
ABC jJ MP 
A8 
X X 
+ -nwv 8 A 
ABC ' ABC 
But 
and consequent!}’ 
£da8MP 
MP 2 
Sri f [A £ cos MQdS 
ABC JJ Mp- 
X g j Sri f f (a — A g) cos MQdS 
ABC 
ABC. 
MP 2 
. £ X _ a8A i f cos MQdS 
6 'ABC~~XBcJJ ~MP~r ■ 
Hence, by the first theorem: 
In the case of an exterior point, we have 
8 ’AW =0, 
i.e. the attractions, in the directions of the axes, of confocal ellipsoids vary as the masses * 
which is Maclaurin’s theorem for the attractions of ellipsoids upon an exterior point. 
In the case of an interior point, we have 
£ X a8A 
8 ‘ ABC ~~ 47r AMBC’ 
or, taking a, /3, 7 as the semi-axes of an ellipsoid confocal with the ellipsoid (A, B, C), 
but exterior to it, and supposing that (X) refers to the ellipsoid (a, ¡3, 7), we have 
a P7 
•a/37 
Now introducing instead of a the new variable 6, such that a 2 = rl 2 + 0, we have 
89 
8a 
a 2 (A 2 + 6J 
, /3— (B‘ 2 + 0) 2 , 7 = (C 2 + 6)\ and consequently writing d for 8, 
XX) = _ d£ 
a#y ?ra (A* + Of (.B 2 + 0) 2 (O + 0)4 ’ 
4—2