PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 277 
thus obtain, for an external point, 
and for an internal point. 
’(p . cos 9JS . 
^-JJ ■ 
(V.) 
Cor. 1. — The potentials at two points infinitely near one another, even if one be in 
the magnetized substanee and the other be external, differ infinitely little ; for tiie 
value of 
ff 
<p . cos MS 
^2 J 
at a point infinitely near the surface and within it, is found by adding — 4‘r{(p) to the 
value of the same expression at an external point infinitely near the former. 
Cor. 2 . — If the value of 
<p . cos 
be denoted by — Q for any internal point, x,y, z\ and if (a), (j8), (y) denote the com- 
ponents of the intensity of magnetization, and X, Y, Z the components of the resultant 
magnetic force at this point, (that is, according to the definition in the second foot- 
note on § 48, the force at a point in an infinitely small crevass tangential to the lines 
of magnetization at x, y, z) we have 
V— . /ox 1 
dVdQ 
2=— & = 
(VI.) 
The resultant of the partial components, — 4'r(a), — 4 t(| 3), — 4T(y),is a force equal 
to 4 t(/) acting in a direction contrary to that of magnetization, and this, com- 
pounded with the resultant of 
dQ dQ 
dx’ dy’’ dz’ 
which depends solely on the edges of the shells, gives the total resultant force at the 
internal point. We thus see precisely how the statements made at the commence- 
ment of ^ 76. are fulfilled. 
Cor. 3. It is obvious, by the preceding investigation, that 
dOi dOi dOi 
dx^ dy^ dz 
are the components of the force at a point in an infinitely small crevass perpendicular 
to the lines of magnetization at z. 
77- An analytical demonstration of these expressions maybe obtained by a partial 
