Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/380

 CHAPTER VII.

CONDUCTION IN THREE DIMENSIONS.

Notation of Electric Currents.

285.] any point let an element of area $$dS$$ be taken normal to the axis of $$x$$, and let $$Q$$ units of electricity pass across this area from the negative to the positive side in unit of time, then, if $$\frac$$ becomes ultimately equal to $$u$$ when $$dS$$ is indefinitely diminished, $$ u$$ is said to be the Component of the electric current in the direction of $$x$$ at the given point.

In the same way we may determine $$v$$ and $$w$$, the components of the current in the directions of $$y$$ and $$z$$ respectively.

286.] To determine the component of the current in any other direction $$OR$$ through the given point $$O$$.

Let $$l, m, n$$ be the direction-cosines of $$OR$$, then cutting off from the axes of $$x, y, z$$ portions equal to

$ \frac, \frac,\;\;\mbox{ and }\;\; \frac $

respectively at $$A, B, \mathrm{ and }C, $$ the triangle $$ ABC $$ will be normal to $$ OR.$$

The area of this triangle $$ ABC $$ will be

$ ds=\frac \frac$

and by diminishing $$r$$ this area may be diminished without limit.

The quantity of electricity which leaves the tetrahedron $$ ABCO $$ by the triangle $$ABC$$ must be equal to that which enters it through the three triangles $$ OBC, OCA,$$ and $$ OAB.$$

The area of the triangle $$ OBC $$ is $$ \frac \frac,$$ and the component of