of Edinburgh, Session 1871-72. 785 
where by (c) and (a) we see that the right hand member may be 
written 
= #( S • ( tV ) . Vr - VS . cnr) ds 
= -ffV- V(v<r)rds (d). 
This, and similar formulas, 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 cr 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 = - fj] ${cr^uds 
= JJf u$(ycr)d<s - Jf u$. Vivcrds . 
This shows at once that the magnetism may be resolved into a 
volume-density S(V<r), and a surface-density -S.Uvo~. Hence, 
for a solenoidal distribution, 
S. = 0. 
What Thomson has called a lamellar distribution {Phil. Trans. 
1852), obviously requires that 
S . erdp 
be integrable without a factor; Le., that 
Y . V<r~ = 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 . V{ucr ) = 0, 
or 
S . <rVc t - 0. 
