Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/61

$\iint R\cos \epsilon dS=\iint {\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}dx\,dy\,dz+\iint (X'-X)dy\,dz$

$+\iint (Y'-Y)dx\,dz+\iint (Z'-Z)dx\,dy;$

(7)

or, if $l',m',n'$ are the direction-cosines of the normal to the surface of discontinuity, and $dS'$ an element of that surface,

$\iint R\cos \epsilon dS=\iiint ({\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}})dx\,dy\,dz$

$+\iint \{l'(X'-X)+m'(Y'-Y)+n'(Z'-Z)\}dS'$,

(8)

where the integration of the last term is to be extended over the surface of discontinuity.

If at every point where $X,Y,Z$ are continuous

${\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}=0$

(9)

and at every surface where they are discontinuous

$l'X'+m'Y'+n'Z'=l'X+m'Y+n'Z$,

(10)

then the surface-integral over every closed surface is zero, and the distribution of the vector quantity is said to be Solenoidal.

We shall refer to equation (9) as the General solenoidal condition, and to equation (10) as the Superficial solenoidal condition.

22.] Let us now consider the case in which at every point within the surface $S$ the equation

${\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}=0$

(11)

is fulfilled. We have as a consequence of this the surface-integral over the closed surface equal to zero.

Now let the closed surface $S$ consist of three parts $S_{1}$, $S_{0}$ and $S_{2}$. Let $S_{1}$ be a surface of any form bounded by a closed line $L_{1}$. Let $S_{0}$ be formed by drawing lines from every point of $L_{1}$ always coinciding with the direction of $R$. If $l,m,n$ are the direction cosines of the normal at any point of the surface $S_{0}$, we have

$R\cos \epsilon =Xl+Ym+Zn=0.$

(12)

Hence this part of the surface contributes nothing towards the value of the surface-integral.

Let $S_{2}$ be another surface of any form bounded by the closed curve $L_{2}$ in which it meets the surface $S_{0}$. Let $Q_{1}Q_{0},Q_{2}$ be the surface-integrals of the surfaces $S_{1},S_{0},S_{2}$, and let $Q$ be the surface-integral of the closed surface $S$. Then

$Q=Q_{1}+Q_{0}+Q_{2}=0$;

(13)