Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/441

348.] conductors $$a,\,b\, \mbox{and}\, c.$$ Let the currents along $$OA,\, OB\, \mbox{and}\, OC$$ be $$\xi,\, \eta,\, \zeta$$ and the resistances $$ \alpha,\, \beta\, \mbox{and}\, \gamma.$$ Let an electromotive force $$E$$ act along $$BC.$$ Required the current $$\xi$$ along $$OA.$$

Let the potentials at the points $$A,\, B,\, C\, \mbox{and}\,O$$ be denoted by the symbols $$A,\, B,\, C\, \mbox{and}\, 0.$$ The equations of conduction are with the equations of continuity

By considering the system as made up of three circuits $$OBC,\, OCA\, \mbox{and}\, OAB$$ in which the currents are $$x,\, y,\, z$$ respectively, and applying Kirchhoff's rule to each cycle, we eliminate the values of the potentials $$O,\, A,\, B,\, C,$$ and the currents $$\xi,\, \eta,\, \zeta,$$ and obtain the following equations for $$x,\, y\, \mbox{and}\, z,$$ Hence, if we put

348.] The value of $$D$$ may be expressed in the symmetrical form, $$D=\alpha bc+bc(\beta+\gamma) +ca(\gamma+\alpha)+ab(\alpha+\beta)+(a+b+c)(\beta\gamma+\gamma\alpha+\alpha\beta)$$ or, since we suppose the battery in the conductor $$a$$ and the galvanometer in $$\alpha$$, we may put $$B$$ the battery resistance for $$a$$ and $$G$$ the galvanometer resistance for $$\alpha$$. We then find

If the electromotive force $$E$$ were made to act along $$OA,$$ the resistance of $$OA$$ being still $$\alpha$$, and if the galvanometer were placed