194] 
NOTE ON THE THEORY OF ATTRACTION. 
155 
into normal thickness divided by mass of (to, to + 8m, Jc), and attraction = 47t into 
density, = 4>tt x mass of shell (to, m + dm, h) into normal thickness of shell (to, m + 8m, Jc) 
divided by mass of the last-mentioned shell. 
And, secondly, the attractions of the solids bounded by the two surfaces (n, Id), 
(n, hi) respectively upon the same exterior point are proportional to their masses. 
For the solid (n, li) may be divided into shells (m, m + dm, h) and for this shell 
the equipotential surface is (m, Jc) and the attraction of the shell varies as mass of 
(in, m + dm, h) into normal thickness of the shell (to, m + 8m, Jc). But in like manner 
the solid (n, hd) may be divided into shells (to, m + dm, hd) and the attraction of the 
shell varies as mass of (to, to + dm, hd) into normal thickness of the shell (to, to + 8m, Jc) 
and the attractions are in each case in the direction normal to the shell (to, Jc), and 
therefore in the same direction; that is, the attraction of the shell (to, to + dm, h) 
is in the same direction as that of the shell (to, to + dm, hd) and the two attractions 
are proportional to the masses. Hence integrating from to = 0 (if for this value the 
included space is zero) to m = n, the attractions of the solids (n, h), (n, lid) are composed 
of elements proportional and parallel, the elements of the attraction of (n, h) to the 
elements of the attraction of (n, hd); and consequently the total attractions are in the 
same direction and proportional to the masses. 
Thirdly, the attractions of the two surfaces (to, h), (n, h) upon the same interior 
point are equal. 
A surface having the properties in question is of course the ellipsoidal surface 
¿e 2 y 2 z 2 
1 *2 J = 1 
m (a 2 + h) to (If- + h) m (c 2 + h) 
where if to varies (h being constant) the several surfaces are similar to each other, 
but if h varies (in being constant) the several surfaces are confocal to each other: for 
it is in fact well known that the infinitesimal shell bounded by similar ellipsoidal 
surfaces has the properties assumed with respect to the shell (to, to + dm, h). The 
first theorem in effect reduces the problem of the determination of an ellipsoid upon 
an exterior point to a single integration, and constitutes the foundation of Poisson’s 
method for the attraction of ellipsoids. The second theorem (Maclaurin’s theorem for 
the attraction of ellipsoids on the same external point) shows that the attraction of 
an ellipsoid upon an external point can be found by means of the attraction of the 
confocal ellipsoid through the attracted point; and by the third theorem the attraction 
of an ellipsoid upon an interior point is equal to that of the similar ellipsoid through 
the attracted point; hence the second and third theorems reduce the determination 
of the attraction of an ellipsoid upon an external or internal point to that of an 
ellipsoid upon a point on the surface. 
2, Stone Buildings, W.G., 7th April, 1858.