a. ee 
TRANSACTIONS OF THE SECTIONS. 9 
From this it follows, that, as far as regards the directive action of terrestrial mag- 
netism, the ellipsoid with three unequal axes may be taken as the general type for a 
body of any form whatever. 
Besides other special cases of interest, to which it is unnecessary for me at present 
to call attention, on account of the close analogy which is presented by the well- 
known theory of principal axes in dynamics, there is one to which I shall allude, on 
account of its importance with reference to the general principles on which the di- 
rective agency depends. Ifthe body considered be a cube, the three principal mag- 
netic elements will be equal, and therefore the corresponding ellipsoid must have 
its three axes equal; that is, it must be a sphere. Hence a cube supported by its 
centre of gravity cannot experience any directive tendency, and will therefore be 
astatic. 
Now a mass of any form may be divided into an infinite number of small cubes, 
and the resultant of the actual directive couples on all of these cubes will determine 
the directive tendency of the whole mass. Hence if each small cube were acted 
upon only by the terrestrial magnetic force, there would be no directive agency on 
the body; and it is to the modification of the circumstances introduced by the 
mutual action of the different parts of the body that we must ascribe the directive 
tendency which is actually experienced, in general by irregular, but especially by 
elongated masses. This modification is distinctly alluded to by Mr. Faraday in his 
memoir on the General Magnetic Condition of Matter (Experimental Researches, 
§ 2264), and the directive tendency which he has observed in needles of diamag- 
netic substances is shown to depend on essentially different physical circumstances 
(§§ 2269, 2418) connected with the variation of the total intensity of the resultant 
magnetic force in the neighbourhood of the poles of a magnet, and quite independent 
of the actual.directions of the lines of force. A mathematical investigation of the 
circumstances on which these phenomena depend will be found in the Cambridge 
and Dublin Mathematical Journal (May 1847). 
From the principles alluded to above, we may draw the following general conclu- 
sions with reference to the action experienced by a body subjected to magnetic in- 
fluence when the intensity of the magnetizing force is constant in its neighbourhood. 
1. The directive tendency on a diamagnetic substance of any form must be ex- 
tremely small, probably quite insensible in any actual experiment that can be made; 
depending as it does upon the mutual action of the parts of the body which are 
primarily influenced to but a very feeble extent in the case of every known diamag- 
netic. 
2. An elongated body, whether of a magnetic or of a diamagnetic substance, will 
tend to place itself in the direction of the lines of force; so that, for instance, either 
a bar of soft iron,.or a diamagnetic bar, supported by its centre of gravity, would, 
. if perfectly free, assume the position of the dipping-needle. 
On the Theory of Electro-magnetic Induction. By Prof. W.'THomson. 
The theory of electro-magnetic induction, founded on the elementary experiments 
of Faraday and Lenz, has been subjected to mathematical analysis by Neumann, 
who has recently laid some very valuable researches on this subject before the Ber- 
lin Academy of Sciences. The case of a closed linear conductor (a bent metallic 
wire with its ends joined) under the influence of a magnet in a state of relative 
motion is considered in Neumann’s first memoir*, and a very beautiful theorem is 
demonstrated, completely expressing the circumstances which determine the inten- 
sity of the induced current. It has appeared to me that a very simple @ priori de- 
monstration of this theorem may be founded on the axiom that the amount of work 
expended in producing the relative motion on which the electro-magnetic induction 
oo must be equivalent to the mechanical effect lost by the current induced in 
e wire. 
In the first place, it may be proved that the amount of the mechanical effect con- 
tinually Jost or spent in some physical agency (according to Joule the generation 
of heat) during the existence of a galvanic current in a given closed wire is, for a 
_* A translation of this memoir into French is published in the last April number of Liou- 
ville’s Journal des Mathématiques. 
