108 
THE HON. J. W. STRUTT ON THE THEORY OF RESONANCE. 
resistance, and was the subject of some controversy*. It is perfectly correct in the sense 
that when the number of observations is increased without limit it coincides with (53), 
itself, however , only an approximation to the magnitude sought. The extension of our 
second method (for the higher limit) to tube not of revolution would require the general 
solution of the potential problem in two dimensions. It may be inferred that the dif- 
ference between the two limits is of the order of the square of the inclination of the 
tangent plane to the axis, and is therefore very small when the section of the tube alters 
but slowly. 
Tubes not nearly straight. 
In applying (53) to such cases, we are at liberty to take any straight line we please 
as axis ; but if the tube is much bent, even though its cross section remain nearly con- 
stant, the approximation will cease to be good. This is evident, because the planes of 
constant potential must soon become very oblique, and the section a used in the formula 
much greater than the really effective section of the tube. To meet this difficulty a 
modification in the formula is necessary. Instead of taking the artificial planes of 
equal potential all perpendicular to a straight line, we will now take them normal to 
a curve which may have double curvature, and which should run, as it were, along the 
middle of the tube. Consecutive planes intersect in a straight line passing through the 
centre of curvature of the “ axis” and perpendicular to its plane. 
The resistance between two neighbouring equipotential planes is in the limit 
where lb is the angle between the planes, and r is the distance of any element da of the 
* See Sabine’s ‘ Electric Telegraph,’ p. 329. To prove (54), we have 
resistance=- 2 -, and <r\= constant = k, say, 
n a 
.-. 2 1=1 2 \. 
G K 
But Y = volume = - 2(7= - k2 1, 
n n K 
. 1 7 V 1 
and 
■»v X’ 
21=4,2X21 
a n Y A 
The correction for the ends of the tube employed hy Siemens is erroneous, being calculated on the supposition 
that the divergence of the current takes place from the curved surface of a hemisphere of radius equal to that 
of the tube. This is tantamount to assuming a constant potential over the solid hemisphere conceived as of 
infinite conductivity, and gives of course a result too small — R for both ends together. The proper correction, 
which probably is not of much importance, would depend somewhat upon the mode of connexion of the tube 
with the terminal cups, but cannot differ much from "1 R (for both ends), as we have seen. (I have since found 
that Siemens was aware of the small error in this correction.) 
