Page:BatemanElectrodynamical.djvu/24

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

$E\ dx^{2}+2F\ dx\ dy+G\ dy^{2}=E'dx'^{2}+2F'dx'dy'+G'dy'^{2};$

(5)

then taking two independent differentials dx, dy, δx, δy, 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

${\begin{array}{l}E\ dx\ \delta x+F(dx\ \delta y+dy\ \delta x)+G\ dy\ \delta y\\\qquad =E'dx'\delta x'+F'(dx'\delta y'+dy'\delta x')+G'dy'\delta y'.\end{array}}$

(6)

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

$\left(EG-F^{2}\right)dx\ dy\ \delta x\ \delta y=\left(E'G'-F'^{2}\right)dx'dy'\delta x'\delta y'.$

This gives

${\sqrt {EG-F^{2}}}dx\ dy={\sqrt {E'G'-F'^{2}}}dx'dy',$

(7)

or

$\Delta dx\ dy=\Delta 'dx'dy'$

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

${\frac {Eb-Fa}{\Delta }}\delta x-{\frac {Ga-Fb}{\Delta }}\delta y={\frac {E'b'-F'a'}{\Delta }}\delta x'-{\frac {G'a'-F'b'}{\Delta '}}\delta y'.$

(8)

which may be called the reciprocal of the first.

Multiplying this equation by

$a\ dx+b\ dy=a'dx'+b'dy',$ $dx{\frac {\partial }{\partial x}}+dy{\frac {\partial }{\partial y}}=dx'{\frac {\partial }{\partial x'}}+dy'{\frac {\partial }{\partial y'}},$

respectively, we get

${\frac {Eb^{2}-2Fab+Ga^{2}}{\Delta }}dx\ dy={\frac {E'b'^{2}-2F'a'b'+G'a'^{2}}{\Delta }}dx'\ dy',$ ${\begin{array}{l}\left[{\frac {\partial }{\partial x}}\left({\frac {Ga-Fb}{\Delta }}\right)+{\frac {\partial }{\partial y}}\left({\frac {Eb-Fa}{\Delta }}\right)\right]dx\ dy\\\\\qquad =\left[{\frac {\partial }{\partial x'}}\left({\frac {G'a'-F'b'}{\Delta '}}\right)+{\frac {\partial }{\partial y'}}\left({\frac {E'b'-F'a'}{\Delta '}}\right)\right]dx'\ dy'.\end{array}}$