Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/271

 Graphic Representation of a Function which is the Sum of Two Given Functions.

Let a function ($$\alpha $$) of $$x$$ and $$y$$ be graphically represented by a series of curves in the plane of $$xy,$$ each of these curves corresponding to a value of a which belongs to a series of such values increasing by a common difference, $$\delta $$.

Let any other function, $$\beta $$, of $$x$$ and $$y$$ be represented in the same way by a series of curves corresponding to a series of values of $$\beta$$ having the same common difference as those of $$\alpha $$.

Then to represent the function $$\alpha+\beta $$ in the same way, we must draw a series of curves through the intersections of the two former series from the intersection of the curves ($$\alpha $$) and ($$\beta $$) to that of the curves $$(\alpha+\delta) $$ and $$(\beta-\delta) $$, then through the intersection of $$(\alpha+2\delta) $$ and $$(\beta-2\delta) $$, and so on. At each of these points the function will have the same value, namely $$\alpha+\beta $$. The next curve must be drawn through the points of intersection of $$\alpha$$ and $$\alpha+\delta $$, of $$\alpha+\delta $$ and $$\beta$$, of $$\alpha+2\delta $$ and $$\beta-\delta $$, and so on. The function belonging to this curve will be $$\alpha+\beta+\delta $$.

In this way, when the series of curves ($$\alpha $$) and the series ($$\beta $$) are drawn, the series $$(\alpha+\beta) $$ may be constructed. These three series of curves may be drawn on separate pieces of transparent paper, and when the first and second have been properly superposed, the third may be drawn.

The combination of conjugate functions by addition in this way enables us to draw figures of many interesting cases with very little trouble when we know how to draw the simpler cases of which they are compounded. We have, however, a far more powerful method of transformation of solutions, depending on the following theorem.

185.] THEOREM II. If $$x$$ and $$y$$ are conjugate functions with respect to the variables $$x'$$ and $$y'$$, and if $$x'$$ and $$y'$$ are conjugate functions with respect to $$x$$ and $$y$$, then $$x$$ and $$y$$ will be con jugate functions with respect to $$x$$ and $$y$$.

For

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