note * page 158 The distance between the hollows is 15½ cms., the bar is about a foot long, and its diameter is 14 mm.

note * page 159 This caution is introduced to avoid leading any reader into an error into which I myself fell in the text, and corrected in footnotes, of an article in Philosophical Magazine for March 1890, “On the Time-Integral of a Transient Electro-Magaetieally Induced Current.”

note * page 160 The thermal analogue for a varying or constant electromotive force applied by a voltaic battery or dynamo, substituted for one of the end-bridges, is positive and negative sources of heat applied at the interfaces between the thermal analogues of electric conductor and electric insulator. The quantity of heat generated per unit of area per unit of time, at any point of either interface in the thermal analogue, is equal to the rate of variation per unit of length along the electric conductor, of the electrostatic force in the insulator in contact with it in the electric analogue. Remark that, in the electric system the potential is uniform over each normal section of either conductor, and that the variation of potential within each conductor per unit of distance along its length—that is to say, the component electrostatic force in the direction of the length is exceedingly small in comparison with the component electrostatic force perpendicular to the length at any place in the insulator, except close to the ends metallically connected by a bridge. The equations in the text are unchanged, except the second interfacial condition, it becomes

where σ denotes the quantity of heat generated per unit of time in the aource.

note * page 162 This collocation of words illustrates the exceeding inconvenience of Maxwell's use of “magnetic induction “to designate the magnetic force in on aircrevasse perpendicular, to the lines of magnetisation in magnetised steel or softiron.

note * page 163 Rowland for one specimen of iron found the magnetic susceptibility as high as 3595 for magnetising force 1·317 (see Phil. Mag., Aug. 1873).Google Scholar

note † page 163 This seems to me a much better word than specific resistance to denote the resistance per centimetre of length of a bar of a square centimetre of crosssection of any substance. The resistivity of Lord Armstrong's steel bar I have found by measurement to he 14,000 c.g.s.

note ‡ page 163 See British Association Report, Bath, 1888, p. 571; or Vol. III. of my collected Papers, to be published in May.Google Scholar

note * page 165 Papers, Vol. II. Arts. 72–77 and 80–83.

note * page 166 This suggests an interesting and, happily, an easy problem regarding electro-magnetic induction in rectilinear electric current through a conductor surrounded by an insulator. Let the electromotive action, whatever its kind, be so regulated that the integral amount of current crossing the normal section of the conductor is kept constant. The mathematical statement of this condition, according to the notation of the text above, is,

where denotes surface integration over the cross-section of the conductor. From this, by the first of the equations of the text,

Now, as is well known, and very easily proved, we have in every case,

where denotes integration all round the border of the cross-section. Hence the condition for constant total amount of current is simply

For the case of circular cross-section with uniform electric conductivity in all parts of it; and with the circuit-completing conductor either a coaxial, cylindric sheath, or a conductor of any form whatever, provided only that no part of it is near enough to the considered part of the given conductor to sensibly disturb the distribution, if the current, through its circular crosssection, from being of equal current-density at equal distances from the axis, the condition for constancy of total amount becomes simply

at the boundary of the conductor, where r denotes distances from the axis. The full numerical solution of this problem, from the instantaneous commencement of a current of given total strength (which must necessarily be in the very outer skin, and must require an infinite current for the first instant) through the whole time until the current becomes as nearly as may be uniform throughout the cross-section, is particularly easy, but must be reserved for a future communication. It is identical with the following particular case of Fourier's thermal problem:—Let a given quantity of heat be initially distributed uniformly through an infinitely thin surface-layer of a solid cylinder coated with an impermeable varnish; it is required to find, for any subsequent time, the temperature at any distance inwards from the surface of the cylinder.