276 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
surround it, if there are any, and (§ 73, Cor. 2.) together with the product of into 
the sum of the strengths of any open shells which have it between them and their 
superficial substitutes, will be the potential due to the whole of the given magnet at 
this point. 
Now if d(p denote the difference between the values of <p at two consecutive sur- 
faces of the series, by which we may conceive the whole magnet to be divided 
into shells, it follows, from the investigation of ^ 73, that the magnetic strength of 
the shell is equal to d(p. Hence, if A denote the least value of <p at any part of the 
bounding surface, and <p be supposed to correspond to a point in the surface, the 
strength of the complex magnetic shell, found by adding the strengths of all the 
shells of the imaginary series superimposed at this point, will be ®— A ; and if P be 
an internal point, and the value of (p at it be denoted by {p), the sum of the strengths 
of all the shells between that which passes through P and that which corresponds to 
A, will be (<p) — A, from which it maybe demonstrated*, that, whether (p) be > or <A, 
and whatever be the nature of the shells, whether all open or some open and some 
closed, the quantity to be added to the potential due to the imaginary complex shell 
coinciding with the surface of the magnet to find the actual potential at P, is 
4 t{(<p) — A}. Now, from what we have seen above, it follows that the potential at any 
point P, due to an element, dS, of this complex shell if 9 denote the angle 
which an external normal, or a normal through the north polar side of dS, makes 
with a line drawn from dS to P, and A the length of this line. Hence the total poten- 
tial at P, due to the whole complex shell, is equal to 
//^ 
4- A} cos 
A2 
in which the integration includes the whole bounding surface of the magnet. Hence, 
if V denote the potential at P, we have the following expression, according as P is 
external or internal, — 
y_^^ {<p-A}cOSe^S ^ 
or 
v=jXii=^pi^+4^m-A}. 
These expressions may be simplified if we remark that, for any external point, 
/*/*cos 
Jj A2 
and that, for any internal point. 
ff 
cos MS 
•4^ 
(since 9 is the angle between the line A and the external normal through c?S). We 
* See second foot-note on § 48 above, and Cors. 2, 3, § 76, below. 
