162 
Proceedings of Royal Society of Edinburgh. [sess. 
as that regarding a heated cylinder left to cool, which was pre- 
sented and fully solved by Fourier in the sixth chapter of his great 
work ( Theorie analytique de la Chaleur). Instead of the bodily 
thermal conductivity divided by surface emissivity of Fourier’s 
problem, essentially a line which we shall denote by X, we have in 
the electro-magnetic problem, 
x ns' b 
when us and us' are the magnetic permeabilities of the conducting 
rod and insulating medium around it, and a and b the radii of the 
cross-sections of the rod, and of the inner surface of the enclosing 
conductor. 
Not considering for the present the interesting case suggested by 
Heaviside of an insulating medium composed of soft iron filings 
imbedded in wax or other ordinary insulating solid, we have prac- 
tically us' = l, wdiether the insulator be air or any ordinary insulat- 
ing solid or liquid. 
Consider now two cases — a copper rod and a steel or iron rod, 
each of the same diameter, 1*4 cm. as Lord Armstrong’s steel rod, 
and suppose, for example, b equal to ten times a, we have 
X = 2*3a=l*6 cm. for copper; 
X = sr 1 .2‘3a = sr' 1 .l*6 cm. for steel or iron. 
] 
If b, instead of being 7 cm. were 70 cm. or 700 cm., X would be only 
doubled or tripled ; on the other hand, if b were very small in 
comparison with a, \ = b-a. Excluding this case, we see that for 
copper X is greater than a , or not incomparably less than a. We 
thus have a very fine and a very easy example for working out 
numerical results by Fourier’s solution. 
On the other hand, for an iron or steel rod we have for us some 
large number, possibly about 300 ; or if the currents and therefore 
the magnetic forces concerned are very small, we may, according to 
Lord Rayleigh’s important experimental investigation on the subject 
of magnetic induction * by very small magnetic forces, have us as 
* This collocation of words illustrates the exceeding inconvenience of Max- 
well’s use of “magnetic induction ” to designate the magnetic force in an air- 
crevasse perpendicular to the lines of magnetisation in magnetised steel or soft 
iron. 
