iy 
nt 
433 
of the internal axis of the cone whose vertex is S, and which 
cireumscribes the shell. 
To find the whole attraction of the shell, the expression 
mrw cos @ (2) 
must be integrated. Let ¢ be the angle which a plane, pass- 
ing through SP and the axis of £, makes with the plane £n; 
then 
w = sin 0d0dq, 
1. cos’ _sin?@.cos? sin?@ sin? l 
a= V ( k hier negra 
When these values are substituted in (7), that expression may 
be readily integrated, first with respect to 0, and then with 
respect to @. 
It is evident that, by the same substitutions, the expres- 
sion (g) may be twice integrated. 
An investigation similar to the preceding has been given 
by M. Chasles, for the case in which the force varies inversely 
as the square of the distance (Mémoires des Savants Etran- 
gers, tom. ix.) He uses a theorem equivalent to the formula 
(f), but deduces it in a different way. 
From what has been proved it follows that, if v be the 
sum of the quotients found by dividing every element of the 
shell by its distance from an external point S, the value of v 
will be the same wherever that point is taken on the surface 5 
of an ellipsoid confocal with the surface A of the shell. 
Let >’ be another ellipsoid confocal with A, and indefi- 
nitely near the surface 2. The normal interval between the 
two surfaces = and >’, at any point S on the former, will be 
inversely as the perpendicular dropped from the common cen- 
tre of the ellipsoids on the plane which touches ¥ at S. Hence, 
supposing the point S to move over the surface &, that per- 
pendicular will vary as the attraction exerted by the shell on 
the point S, when the force is inversely as the square of the 
