of Edinburgh, Session 1871-72. 785 
where by (c) and (a) we see that the right hand member may be 
written 
= fff (S . (rV) er + crS . Vr — VS . err) ds 
= V(V<r)Ttfe. (d). 
This, and similar formula, are applied in the paper to find the 
potential and vector-force due to various distributions of magnetism. 
To show how this is introduced, I briefly sketch the mode of ex¬ 
pressing the potential of a distribution. 
Let <r be the vector expressing the direction and intensity of 
magnetisation, per unit of volume, at the element d<$. Then if the 
magnet be placed in a field of magnetic force whose potential is u , 
we have for its potential energy 
E = - f/j${erV)uds 
— Jjfu$(ycr)ds - uS . Uvc~ds . 
This shows at once that the magnetism may be resolved into a 
volume-density S(Vcr), and a surface-density -S.Uvcr-. Hence, 
for a solenoidal distribution, 
S . V<r- = 0 . 
What Thomson has called a lamellar distribution {Phil. Trans . 
1852), obviously requires that 
S. crdp 
be integrable without a factor; i.e ., that 
V.V<^ = 0. 
A complex lamellar distribution requires that the same expression 
be integrable by the aid of a factor. If this be u, we have at once 
Y . Y(^cr) = 0, 
or 
S . a-Vo*" — 0, 
