BY SUEBACES OF A HYPOTHETICAL CLASS INCLUDING THE ELLIPS(JlD. 7 
This result may be verified by actual difierentiation, as will be sliown afterwards 
(art. 12). 
7. Kesuming the equation (3.), art. 2, and supposing that 0 is the function /’(A) 
determined in the last article, so that Wf{h)—Q, let us put u—\ ; then the equation 
becomes 
if this be multiplied by dJc, it expresses the following proposition : — 
If the homogeneous infinitesimal shell ^h, the density f'(h), tJte mass of 
the shell is independent of h. 
It follows that the potential of such a shell, on a given interior mass, vanishes when 
h has the value which makes the surface [h, Jc) extend to infinity in all directions ; for 
the mass of the shell is finite, but every part of it is infinitely distant from the 
attracted mass. 
8. Instead of putting u=\, as in the last article, let us now take for u the potential 
of a given mass M, placed anywhere. Then if ^ be the density, at the point {x, y, z), of 
the matter composing M, we shall have 
D%= — 4t^. 
Let the surface (Aj, k) be within the surface (Aa, A:), and let M, be all that part of M 
which is within the former surface, and Mg all that part which is within the latter (so 
that Mg includes MJ. 
Also \etf{h) be still taken for 6, in equation (3.), art. 2; then D^^ = 0, and the last 
term on the right of that equation becomes 
^'^\S&fih)dxdydzJ^, 
Now the whole mass included between the two surfaces is Mg — M, ; hence the above 
integral is equal to 
in which h is put for the parameter of some surface (A, k), which lies between (/q, k) and 
(Ag, A;), and cuts the mass Mg— Mj. If the mass M be concentrated at a point between 
the two surfaces, then h is the parameter of the surface (A, k) which passes through that 
point. 
In the general case, however, equation (3.) becomes 
»/'('*.) ]*■ -«/(?>.) [ J'= - 4xM,/(A,)+4:.M.y(A,)+4^(M,-M,)/(A). 
Let Vi, Vg be put for the potentials on M of the two infinitesimal shells 
(/la, with densities /'(A,), ^'(^3) respectively; then this equation, multiplied by 
dk, gives 
Va-V, = ^ (-Ma/(Aa)+M,/(A,)+(M,-M,)/(/0)* (9.) 
