Page:Scientific Memoirs, Vol. 2 (1841).djvu/461

${\frac {\chi (u_{1}-u)\,dt}{dx}}$

is the quantity of electricity passing over from $M_{1}$ to $M$ when the expression is positive, and from $M$ to $M_{\prime }$ when it is negative. The total change of the quantity of electricity which the disk $M$ undergoes from the motion of the electricity in the interior of the body in the particle of time $dt$, is consequently

${\frac {\chi (u'+u_{1}-2u)\,dt}{dx}},$

and an increase in the quantity of electricity is denoted when this value is positive, and when negative a diminution of the same.

But according to Taylor's theorem

$u'=u+{\frac {du}{dx}}\centerdot dx+{\frac {d^{2}u}{dx^{2}}}\centerdot {\frac {dx^{2}}{2}}+\ldots ,$

and in the same way

$u_{1}=u-{\frac {du}{dx}}\centerdot dx+{\frac {d^{2}u}{dx^{2}}}\centerdot {\frac {dx^{2}}{2}}-\ldots ;$

consequently

$u'+u_{1}=2u+{\frac {d^{2}u}{dx^{2}}}dx^{2}.$

According to this the expression just found for the total change of the quantity of electricity present in the disk $M$ is converted during the time $dt$ into

$\chi \cdot {\frac {d^{2}u}{dx^{2}}}dx\,dt,$

where $\chi$ represents the power of conduction which prevails from one disk to the adjacent one, which we suppose to be invariable throughout the length of the homogeneous body. It must here be observed, that this value $\chi$ is, on account of the infinitely small distance of action, proportional to the section of the cylindric or prismatic body; if therefore we denote the magnitude of this section by $\omega$, and separate this factor from the value $\chi$, always calling the remaining portion $\chi$, the former expression changes into the following:

$\chi \omega {\frac {d^{2}u}{dx^{2}}}dx\,dt,$

in which $\chi$ now represents the conductibility of the body independent of the magnitude of the section, which we will term the absolute conductibilty of the body in opposition to the former, which may be called the relative. Henceforward wherever