Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/356

324 The volume of this layer is

where $$dl$$ is the length of wire in this layer.

This gives us $$dl$$ in terms of $$dx$$. Substituting this in equations (3) and (4), we find where $$dG$$ and $$dR$$ represent the portions of the values of $$G$$ and of $$R$$ due to this layer of the coil.

Now if $$E$$ be the given electromotive force, where $$r$$ is the resistance of the external part of the circuit, independent of the galvanometer, and the force at the centre is

We have therefore to make $$\frac{G}{R + r}$$ a maximum, by properly adjusting the section of the wire in each layer. This also necessarily involves a variation of $$Y$$ because $$Y$$ depends on $$y$$.

Let $$G_0$$ and $$R_0$$ be the values of $$G$$ and of $$R + r$$ when the given layer is excluded from the calculation. We have then and to make this a maximum by the variation of the value of $$y$$ for he given layer we must have

Since $$dx$$ is very small and ultimately vanishes, $$\frac{G_0}{R_0}$$ will be sensibly, and ultimately exactly, the same whichever layer is excluded, and we may therefore regard it as constant. We have therefore, by (10) and (11),

If the method of covering the wire and of winding it is such that the proportion between the space occupied by the metal of