Page:BatemanElectrodynamical.djvu/24

 Next, suppose that the formula of transformation of a quadratic differential form is known, e.g.,

then taking two independent differentials dx, dy, &delta;x, &delta;y, and writing $$dx+\lambda\delta x,\ dy+\lambda\delta y,\dots$$ in place of dx, dy, we get the formula of transformation of the bilinear form

We may multiply this equation by itself, multiplying both sets of differentials according to Grassmann's rule. The resulting equation is

This gives

or

Multiplying (6) by (1) according to Grassmann's rule, we obtain

$$\begin{array}{l} \left[(Eb-Fa)\delta x-(Ga-Fb)\delta y\right]dx\ dy\\ \qquad=\left[(E'b'-F'a')\delta x'-(G'a'-F'b')\delta y'\right]dx'\ dy'.\end{array}$$

This gives the formula of transformation of a linear form

which may be called the reciprocal of the first.

Multiplying this equation by

respectively, we get