Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/399

the integration being extended over the whole length, $x,$ of the conductor, then the resistance of the tube $d\psi \,d\phi$ is

${\frac {2}{d\psi \,d\phi }}(A+\psi B),$

and its conductivity is

${\frac {d\psi \,d\phi }{2(A+\psi B)}}.$

To find the conductivity of the whole conductor, which is the sum of the conductivities of the separate tubes, we must integrate this expression between $\phi =0$ and $\phi =2\pi ,$ and between $\psi =0$

and $\psi =1.$ The result is

${\frac {1}{R'}}={\frac {\pi }{B}}\log \left(1+{\frac {B}{A}}\right),$

(14)

which may be less, but cannot be greater, than the true conductivity of the conductor.

When ${\frac {db}{dx}}$ is always a small quantity ${\frac {B}{A}}$ will also be small, and we may expand the expression for the conductivity, thus

${\frac {1}{R'}}={\frac {\pi }{A}}\left(1-{\frac {1}{2}}{\frac {B}{A}}+{\frac {1}{3}}{\frac {B^{2}}{A^{2}}}-{\frac {1}{4}}{\frac {B^{3}}{A^{3}}}+\mathrm {\&c.} \right).$

(15)

The first term of this expression, ${\frac {\pi }{A}},$ is that which we should have found by the former method as the superior limit of the conductivity. Hence the true conductivity is less than the first term but greater than the whole series. The superior value of the resistance is the reciprocal of this, or

$R'={\frac {A}{\pi }}\left(1+{\frac {1}{2}}{\frac {B}{A}}-{\frac {1}{12}}{\frac {B^{2}}{A^{2}}}+{\frac {1}{24}}{\frac {B^{3}}{A^{3}}}-\mathrm {\&c.} \right).$

(16)

If, besides supposing the flow to be guided by the surfaces $\phi$ and $\psi ,$ we had assumed that the flow through each tube is proportional to $d\psi \,d\phi ,$ we should have obtained as the value of the resistance under this additional constraint

$R''={\frac {1}{\pi }}(A+{\frac {1}{2}}B),$

(17)

which is evidently greater than the former value, as it ought to be, on account of the additional constraint. In Mr. Strutt's paper this is the supposition made, and the superior limit of the resistance there given has the value (17), which is a little greater than that which we have obtained in (16).