BY SUEEACES OF A HYPOTHETICAL CLASS INCLUHING THE ELL^J^SOiO. 
4. Now let the general equation (2.), art. 2, be applied to the case of the surface 
considered in the last article, Jc being taken for the parameter so that is JJ% and 
is =(p(h) ; also let the arbitrary function u be put =1, so that D'%=0. Then, observing 
the second of equations (6.), we obtain from (2.), 
Let the volume enclosed by the surface (A, k) be represented by V. Then, since the 
normal thickness at any point of the shell is the above equation is eqiii- 
valent to 
n§^f(k).r, 
from which we obtain by integration, putting for shortness = 
r=F(/c).^(h), ( 7 .) 
where F(A) is an unknown function of A, independent of A. 
But if, instead of putting m— 1, we suppose ti to be the potential, at the point (cc, ^,z), 
of a given mass M exterior to the surface (A, A), then we have (since is again =0) 
n 
rp^T- 
L 
* Q/i 
(p{h). ^^^udxdydz 
and if V be put for the potential on M of a (homogeneous) solid bounded by the surface 
(A, A), this equation is equivalent to 
n^ = oih}.Y, 
and therefore, as before, 
V=F(A).^^(A), (8.) 
where ■^'(A) is the same as before, but F(A) is a new unknown function of A, which will 
also involve the given quantities which define M. 
From (7.) and (8.) we have 
Y-^.r 
F(*) 
which equation expresses 
Theoeem I. The potential, on a given external mass, of a homogeneous solid hounded 
by the surface (h, k), varies as the mass of the solid, if h vary while k remains constant. 
5. If we put F(A, A) for the volume, and V(A., A) for the potential, denoted above 
simply by V and V, we obtain from equations (7.) and (8.) the following: — 
Y{K, h) V{h^, k^) - V(k^ k,) 
\{h„ k^)-\{h„ k,) - F(A, k^)-V[h„ kf 
which expresses 
Theoeem II. The potentials, on a given external mass, of the homogeneous shells 
^ha, ^h,, are proportional to the masses of the shells. 
