736 REPORT—1890. 
or frictional resistance to change of magnetism nothing for the iron in the actual 
circumstances. This last supposition is probably true approximately with a 
permeability of 4; for iron or steel, according to Lord Rayleigh, if the current is 
so small that the greatest magnetising force acting on the iron is less than 
10C.-G.-S. 
VI. The dependence of the total quantity carried on extent of surface, accord- 
ing to the electrostatic problem described in II., justifies Snow Harris, and proves 
that those who condemned him out of Ohm’s law were wrong, in respect to his 
advising tubes or broad plates for lightning conductors; but does not justify him 
in bringing them down in the interior of a ship (even through the powder maga- 
zine) instead of across the deck and down its sides, or from the masts along the 
rigging and down the sides to the water. The non-dependence of the total quan- 
tities of current on the material, whether iron or non-magnetic metals, seems 
quite in accordance with Dr. Oliver Lodge's experiments and doctrines regarding 
‘alternative path’ and lightning conductors. ‘The case of alternate currents is, of 
course, not exactly that of lightning discharges ; but from it, by Fourier’s methods, 
we infer the main conclusions of II. and V., whether the discharges be oscillatory 
or non-oscillatory, provided only that it be as sudden as we have reason to believe 
lightning discharges are. 
4. On Anti-EHffective Copper in Parallel Conductors or in Coiled Conductors 
for Alternate Currents. By Sir Witu1am TuHomson, D.O.L., LL.D., F.B.S. 
1. It is known that by making the conductors of ‘a circuit too thick we do not 
get the advantage of the whole conductivity of the metal—copper, let us say—for 
alternate currents. When the conductor is too thick, we have in part of it com- 
paratively ineffective copper present ; but, so far as I know, it has generally been 
supposed that the thicker the conductor the greater will be its whole effective con- 
ductance, and that thickening it too much can never do worse than add compara- 
tively ineffective copper to that which is most effective in conveying the current. 
It might, however, be expected that we could get a positive augmentation of the 
effective ohmic resistance, because we know. that the presence of copper in the 
neighbourhood of a circuit carrying alternate currents causes a virtual increase of 
the apparent ohmic resistance of the circuit in virtue of the heat generated by the 
currents induced in it. May it not be that anti-effective influence such as is thus 
produced by copper not forming part of the circuit can be produced by copper 
actually in the circuit, if the conductor be too thick? Examining the question 
mathematically, I find that it must be answered in the affirmative, and that great 
augmentation of the effective ohmic resistance is actually produced if the con- 
ductor be too thick; especially in coils consisting of several layers of wire laid over 
one another in series around a cylindric or flat core, as in various forms of trans- 
former. 
2. Fig. 1 may be imagined to represent the secondary coil of a transformer con- 
sisting of solid square copper wire in three layers. For simplicity we suppose the 
axial length to be infinitely great, and straight; but the uniformity which this 
involves, and a close practical application to its simplicity, is realised in that ex- 
cellent form of transformer which consists of a toroidal iron core completely 
covered by primary and secondary wires laid on toroidal surfaces. To simplify the 
mathematical work, I suppose the whole thickness of the three layers to be small 
in comparison with the greatest radius of curvature of the circular or flat cylindric 
surface on which the wire is wound, but if it is not so the solution is easily 
obtained, for the case of circular cylinders, in terms of the Fourier-Bessel functions. 
It is of no consequence for our present question what there be inside of coil No. 3, 
and, if we please, we may imagine there to be nothing but air; the drawing, how- 
ever, indicates an iron core and aspace which might be occupied by the primary coil, 
if a transformer is the subject ; or our coil A A A A may be the primary coil of 
a transformer with secondary coil and core inside it, and the alternate current 
maintained in it by an external electro-motive agent acting in an arc between its 
ends outside. Our present results are applicable to all these varieties of cases 
