734 REPORT—1890. 
in Figs, 1 and 2, all the parts of A are in mutual metallic connection, and all the 
parts of A’ are in mutual metallic connection, this would entail uniformity of 
potential through A, and uniformity of potential through A’, even without the 
limitation of our subject laid down in section 3. 
7. The following are some of the most noteworthy results of the full mathe- 
matical treatment of the subject :— 
I. When the period of alternation is large in comparison with 400 times the 
square of the greatest thickness or diameter of any of the conductors, multiplied by 
its magnetic permeability, and divided by its electric resistivity, the current 
intensity is distributed through each conductor inversely as the electric resistivity ; 
the phase of alternation of the current is the same as the phase of the electro- 
motive force; and the current across every infinitesimal area of the cross-section — 
is calculated, according to the electro-motive force at each instant, by simple 
application of Ohm’s law. 
II. When the period is very small in comparison with 400 times the square of 
the smallest thickness, or diameter of any of the conductors, multiplied by its 
magnetic permeability and divided by its electric resistivity, the current is confined 
to an exceedingly thin surface-stratum of the conductors. The thickness of this 
stratum is directly as the square root of the quotient of resistivity, divided by 
magnetic permeability, of the substance in different parts of the surface. The total 
quantity of the current per unit breadth of the surface independent of the material, 
and, except in such cases as those referred to at the end of II. below, varies in each 
cross-section in simple proportion to the electric surface density of the’ static 
electrification induced by the electro-motive force applied between the extremities 
for maintaining the current. The distribution of this electric density is similar in 
all cross-sections, but its absolute magnitude at corresponding points of the cross- 
section varies along the length of the conductor in simple proportion to the 
difference of electric potential between A and A’, and is zero at one end, in the 
particular case in which the conductors are connected through zero resistance at 
one end, while the electro-motive force is applied by an alternate current dynamo 
at the other end. On the other hand, the surface distribution of electric current is 
uniform throughout the whole length of the conductors, and it is only its distribu- 
tion in different parts of the cross-section that varies as the electric density. 
The proportionality of surface intensity of the current to electric density, 
asserted above, fails clearly in any case in which the circumstances are such that 
the distance we must travel along the surface to find a sensible difference in 
electric density is not very great in comparison with the thickness of the current- 
