738 REPORT—1 890. 
force into two components, one, perpendicular to 8, which we may neglect, as it 
has no influence in connection with the currents we are to consider, the other, 
parallel to S, which we shall eall the effective component and denote by Y. 
Through any point O, of S, draw three rectangular lines OX, O Y, OZ, of which 
OY and OZ are in §, and O X is parallel to the direction of the effective magnetic 
force component Y. Let now the value of Y at time ¢ be 
Qrt 
— 
where M denotes a constant, and T the period of the alternation. The varying 
magnetic force Y, to whatever causeit may be due, implies currents parallel of 
OZ in the conductor, expressed by the formula for y, the current intensity at 
distance X from the plane S, provided T be small enough to fulfil the condition 
stated below :— 
Y=M cos 
_ M282 | (Get Bee 1), 
a Aaya A COS T a NX +47 ? 
where A denotes what we may call the wave-length of the disturbance, and is 
given in terms of T, the period of the disturbance, and p and II the resistivity 
and magnetic permeability of the substance, by the following formula :— 
a= 4/32. 
II 
For copper we have I1=1, and p=1611 square centimetres per second; and thus 
for 80 periods per second \=4°49, or, say, 4$ centimetres. In order that the 
formula for y may be approximately true it is necessary, in the first place, that A 
must be small in comparison with the distance we must travel in any direction in 
the surface of S before finding any deviation of it from the tangent plane through 
© comparable with A. Secondly, for a very good approximation, A must be so small 
that we may be able to travel inwards in any direction from O, through a space 
equal to at least twice A, without coming to any other part of the bounding surface 
of the conductor. If, for example, the surface be a flat plate, this condition 
requires that the thickness be more than twice A. But (because e—7 is less than 
2) the formula gives a very fair approximation requiring for a half the thickness 
of the plate inwards from S no greater correction than about 4 per cent., even if 
the thickness of our plate be no greater than A. When the thickness of the plate 
is less than 2 A, we may consider waves of electric current as travelling inwards 
from its two sides, and being both sensible at the middle of the plate; and a 
complete solution of the problem is readily found by the method of images. But 
a direct analytical investigation, by which the proper conditions of relation to 
varying magnetic force on the two sides of the plate are fulfilled, is the most con- 
venient way of fully solving the problem, and it is thus that the results given 
below have been obtained. 
7. The smallness of the insulating space between the successive turns in each 
layer of our coil A A A A, and the equality of the whole current through them 
all, prevent any surface disturbance from being produced at the contiguous faces, 
and allow the problem to be treated as if, instead of a row of squares or rectangles, 
we had a continuous plate forming each stratum. The smallness of the thickness of 
this plate in comparison with the radius of the cylindric surface to which it is bent 
allows, as said above, the mathematical treatment for an infinite plate bounded by 
two parallel planes to be used without practical error. I have thus found an 
expression for the intensity of the current at any point in the metal of any one 
of the layers of a coil of one, two, three, or more layers; and have deduced from 
it an expression for the quantity of heat generated per unit of time, at any instant, 
per unit breadth in any one of the layers. I need not at present quote the former 
expression ; the latter is as follows :—With gq to denote the dynamical value of the 
time-average of the heat generated per unit of time at different instants of the 
period, per unit breadth and unit length in layer No. ¢, from the outside of the 
a. *=—- += 7) 
