Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/272

 and

$\begin{array}{ll} \frac{dx}{dy} & =\frac{dx}{dx'}\frac{dx'}{dy}+\frac{dx''}{dy'}\frac{dy'}{dy}\\ \\ & =-\frac{dy}{dy'}\frac{dy'}{dx}-\frac{dy}{dx'}\frac{dx'}{dx}\\ \\ & =-\frac{dy''}{dx}; \end{array} $

and these are the conditions that $$x$$ and $$y$$ should be conjugate functions of $$x$$ and $$y$$.

This may also be shewn from the original definition of conjugate functions. For $$x+\sqrt{-1}y $$ is a function of $$x'+\sqrt{-1}y' $$, and $$x'+\sqrt{-1}y' $$ is a function of $$x+\sqrt{-1}y $$. Hence, $$x+\sqrt{-1}y $$ is a function of $$x+\sqrt{-1}y $$.

In the same way we may shew that if $$x'$$ and $$y'$$ are conjugate functions of $$x$$ and $$y$$, then $$x$$ and $$y$$ are conjugate functions of $$x'$$ and $$y'$$.

This theorem may be interpreted graphically as follows:–

Let $$x', y'$$ be taken as rectangular coordinates, and let the curves corresponding to values of $$x$$ and of $$y$$ taken in regular arithmetical series be drawn on paper. A double system of curves will thus be drawn cutting the paper into little squares. Let the paper be also ruled with horizontal and vertical lines at equal intervals, and let these lines be marked with the corresponding values of $$x'$$ and $$y'$$.

Next, let another piece of paper be taken in which $$x$$ and $$y$$ are made rectangular coordinates and a double system of curves $$x', y'$$ is drawn, each curve being marked with the corresponding value of $$x'$$ or $$y'$$. This system of curvilinear coordinates will correspond, point for point, to the rectilinear system of coordinates $$x', y'$$ on the first piece of paper.

Hence, if we take any number of points on the curve $$x$$ on the first paper, and note the values of $$x', y'$$ at these points, and mark the corresponding points on the second paper, we shall find a number of points on the transformed curve $$x$$. If we do the same for all the curves $$x, y$$ on the first paper, we shall obtain on the second paper a double series of curves $$x, y$$ of a different form, but having the same property of cutting the paper into little squares.