PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 273 
74 . Prop. — A distribution of magnetism expressed by {(05, |8, y) at {x, y,z))* is so le- 
noidal if, and is not solenoidal unless 
The condition that a given distribution of magnetism, in a substance of finite 
dimensions, may be solenoidal, is readily deduced from the investigations of § 42, 
by means of the propositions of §§ 71 and 72. For, if the distribution of magnetism 
be solenoidal, the imaginary magnetic matter by which the polarity of the whole 
magnet may be represented will be situated at the ends of the solenoids, according to 
§ 7lj and therefore (§ 68.) will be spread over the bounding surface. On the other 
hand, if the distribution be not solenoidal, that is, if the magnet be divisible into 
solenoids, of which some, if not all, are complex ; there will, according to § 72, be an 
internal distribution of imaginary magnetic matter in the representation of the pola- 
rity of the whole magnet. Hence it follows from § 42, that if a, j3, y denote the 
components of the intensity of magnetization at any internal point {x, y, z), the 
equation 
dx'dy' 
(I.) 
expresses that the distribution of magnetism is solenoidal-f-. 
75 . Prop. — A distribution of magnetism {(a, /3, y) at (x, y, z)} is lamellar if, and is 
* Where a, /3, y. which may be called the components, parallel to the axes of coordinates, of the magnetiza- 
tion at (a?, y, z), denote respectively the products of the intensity into the direction cosines of the magnetization. 
t The analogy between the circumstances of this expression and those of the cinematical condition ex- 
pressed by “the equation of continuity” to which the motion of a homogeneous incompressible fluid is subject, 
is so obvious that it is scarcely necessary to point it out. When an incompressible fluid flows through a tube 
of variable infinitely small section, the velocity (or in reality the mean velocity) in any part is inversely pro- 
portional to the area of the section. Hence the intensity and direction of magnetization, in a solenoid, accord- 
ing to the definition, are subject to the same law as the mean fluid velocity in a tube with an incompressible 
fluid flowing through it. Again, if any finite portion of a mass of incompressible fluid in motion be at any 
instant divided into an infinite number of solenoids (that is, tube-like parts), by following the lines of motion 
the velocity in any one of these parts will at different points of it be inversely proportional to the area of its 
section. Hence the intensity and direction of magnetization in a solenoidal distribution of magnetism, accord- 
ing to the definition, are subject to the same condition as the fluid-velocity and its direction, at any point in 
an incompressible fluid in motion. It may be remarked, that by making an investigation on the plan of § 42. 
to express merely the condition that there may be no internal distribution of imaginary magnetic matter, the 
equation is obtained in a manner precisely similar to a mode of investigating the equation of 
dx ■ dy dz 
continuity for an incompressible fluid, now well known, which is given in Duhamel’s Cours de Mecanique, 
and in the Cambridge and Dublin Mathematical Journal, vol. ii. p. 282. The following very remarkable pro- 
position is an immediate consequence of the proposition that “ a closed solenoid exerts no action on any other 
magnet” (§ 71, Cor. 2 above), in virtue of the analogy here indicated. 
“ If a closed vessel of any internal shape, be completely filled with an incompressible fluid, the fluid set into 
any possible state of motion, and the vessel held at rest ; and if a solid mass of steel of the same shape as the 
space within the vessel be magnetized at each point with an intensity proportional and in a direction corre- 
sponding to the velocity and direction of the motion at the corresponding point of the fluid at any instant ; the 
magnet thus formed will exercise no force on any external magnet.” 
MDCCCLI. 2 N 
