THE HON. J. W. STRUTT ON THE THEORY OF RESONANCE. 
109 
section from the line of intersection of the planes. Now if ds be the inter- 
cept on the axis between the normal planes, and p the radius of curvature at the point 
in question. The lower limit to the resistance is thus expressed by 
( 
ds 
(55) 
In the particular case of a tube of revolution (such as an anchor-ring) ~ i 
is a con- 
stant, and the limit which now coincides with the true resistance varies as the length of 
the axis, and is evidently independent of its position. In general the value of the inte- 
gral will depend on the axis used, but it is in every case less than the true value of the 
resistance. In choosing the axis, the object is to make the artificial planes of constant 
potential agree as nearly as possible with the true equipotential surfaces. 
A still further generalization is possible by taking for the artificial equipotential 
surfaces those represented by the equation 
F = const. 
For all systems of surfaces, with one exception, the resistance found on this assumption 
will be too small. The exception is of course when the surfaces F= const, coincide 
with the undisturbed equipotential surfaces. The element of resistance between the 
surfaces F and F -j- <7F is 
1 
da 
■ da 
where dn is the distance between the surfaces at the element da, and the integration 
goes over the surface F as far as the edge of the tube. Now 
(Sr 
dn- 
+ 
" 1,V +(f); 
d 'J 
limit to resistance 
dV 
+ 
ds 
• (56> 
an expression whose form remains unchanged when /(F) is written for F. If F — r 
so that the surfaces are spheres, 
ftV? V=i 
d x 
dr ) 
limit ={’A = ! V -- 
J jjf/cT J 
This form would be suitable for approximately conical tubes, the vertex of the cone 
being taken as origin of r. 
MDCCCLXXI. q 
