280 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
being solenoidal which it presents ; while the formulae of § 76. merely expresses a fact 
with reference to lamellar distributions, and being only applicable to lamellar distribu- 
tions, do not indicate the effect of a deviation from being lamellar, in a distribution 
of a general form. Certain considerations regarding the comparison between common 
magnets and electro-magnets, suggested by Ampere’s theorem that the magnetic 
action of a closed galvanic circuit is the same as that of a “magnetic shell” (as de- 
fined in § 67 .) of any form having its edge coincident with the circuit, led me to a 
synthetical investigation of a distribution of galvanism through the interior and at 
the surface of a magnet magnetized in any arbitrary manner, from which I deduced 
formulae, for the resultant force at any external or internal point, giving the desired 
indication regarding effect of a deviation from being lamellar, on expressions which, 
for lamellar distributions, depend solely on the tangential component of magnetiza- 
tion at points infinitely near the surface. These galvanic elements throughout the 
body, from the action of which the resultant force at any external point is com- 
pounded, produce effects which are not separately expressible by means of a poten- 
tial, and therefore, although of course when the three components X, Y, Z of the total 
resultant force at any point (x, y, z) have been obtained, they will be found to be 
such that 'Kdx-{-Ydy-\-Zdz is a complete differential, the separate infinitely small ele- 
ments of which these forces are compounded by integration with reference to the 
elements of the magnet, do not separately satisfy such a condition. Hence the in- 
vestigation does not lead to an expression for the potential ; but by means of it the 
following expressions for the three components of the force at any external point, 
or on a point within any infinitely small crevass perpendicular to the lines of mag- 
netization, have been obtained*: — 
The investigation by which I originally obtained these expressions is, with reference 
to galvanism, precisely analogous to the investigation in ^ 42. with reference to 
imaginary magnetic matter. It cannot be given without explanations regarding the 
elements of electro-magnetism which would exceed the limits of the present communi- 
cation ; but when I had once discovered the formulee I had no difficulty in working 
out the subjoined analytical demonstration of them for the case of an external point, 
* The expression Xdx + Ydy + Zdz will not be a complete differential for internal points, unless the distribu- 
tion of magnetism be lamellar, since, for any internal point, X, Y, Z differ from the rectangular components of 
the resultant force, as defined in § 48, by the quantities 4ir«, 4ir^, 4Try, respectively, and since (§ 52) the re- 
sultant force, for all points, whether internal or external, is derivable from a potential. 
