272 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
cos 0 
Now — ^^ 2 — is the solid angle subtended at P by the element dS, and therefore the 
potential due to any infinitely small element is equal to the product of its magnetic 
strength, into the solid angle which its area subtends at P. But the potential due to 
the whole is equal to the sum of the potentials due to the parts, and the strength is 
the same for all the parts. Hence the potential due to the whole shell is equal to 
the product of its strength into the sum of the solid angles which all its parts, or 
the solid angle which the whole, subtends at P. 
Cor. 1. — The expression which occurred in the preceding demonstration, 
being positive or negative according as 0 is acute or obtuse, it appears that the solid 
angles subtended by different parts of the shell at P must be considered as positive 
or negative according as their north polar or their south polar sides are towards this 
point. 
Cor. 2. — The potential at any point due to a magnetic shell is independent of 
the form of the shell itself, and depends solely on its bounding line or edge, sub- 
ject to an ambiguity, the nature of which is made clear by the following state- 
ment : — 
If two shells of equal magnetic strength, X, have a common boundary, and if the 
north polar side of one, and the south polar side of the other be towards the enclosed 
space, the potentials due to them at any external point will be equal ; and the poten- 
tial at any point in the enclosed space, due to that one of which the northern polarity 
is on the inside, will exceed the potential due to the other by the constant 4rX. 
Cor. 3. — Of two points infinitely near one another on the two sides of a magnetic- 
shell, but not infinitely near its edge, the potential at that one which is on the north 
polar side exceeds the potential at the other by the constant At'K. 
Cor. 4. — The potential of a closed magnetic shell of strength X, with its northern 
polarity on the inside, is 4 tX, for all points in the enclosed space, and 0 for all ex- 
ternal points ; and for points in the magnetized substance it varies continuously from 
the inside, where it is 4‘rX to the outside, where it is 0. 
Cor. 5. — A closed magnetic shell exerts no force on any other magnet. 
Cor. 6. — The “resultant force” (§ 49.) at any point in the substance of a closed 
magnetic shell is equal to if r be the thickness, or to 47r«, if i be the intensity of 
magnetization of the shell in the neighbourhood of the point, and is in the direction 
of a normal drawn from the point through the south polar side of the shell. 
Cor. 7- — If the intensity of magnetization of an open shell be finite, the resultant 
force at any external point not infinitely near the edge is infinitely small ; but the 
force at any point in the substance not infinitely near the edge is finite, and is equal 
to 4^1, if i be the intensity of the magnetization in the neighbourhood of the point, 
and is in the direction of a normal through the south polar side. 
