Page:BatemanElectrodynamical.djvu/29

 (4) by (1) three times and rejecting the factor $$dx\ dy\ dz\ dt\ \sqrt{\Delta}$$. The relations expressing the quantities $$u_x$$ in terms of the quantities $$u_x$$ are exactly the same as those obtained by solving equations (5).

Next, suppose we are given an integral form of the second order

which is an invariant. Then multiplying it twice by (1), and rejecting the factor $$\sqrt{\Delta}dx\ dy\ dz\ dt$$, we obtain a reciprocal integral invariant

where

{{MathForm2|(9)|$$\left.\begin{array}{rc} \sqrt{\Delta}D_{x} & =\left(BC+F^{2}\right)E_{x}+(FG-CH)E_{y}+(HF-BG)E_{z}\\ \\ & +(HW-VG)B_{x}+(BW-VF)B_{y}+(FW-CV)B_{z}\\ & \dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\\ \\-\sqrt{\Delta}H_{x} & =\left(HW-VG\right)E_{x}+(GU-AW)E_{y}+(AV-HU)E_{z}\\ \\ & +(AD-U^{2})B_{x}+(HD-UV)B_{y}+(GD-UW)B_{z}\\ & \dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\end{array}\right\} $$}}

The relation between these two invariants is evidently a mutual one.

Multiplying them together, we obtain the integral invariant

and the absolute invariant

The theory of reciprocal invariants can evidently be extended to the case in which there are n variables, and a bilinear integral form is known to be invariant.

If we suppose that a biquadratic integral form

is invariant for a transformation, we may multiply it by itself and obtain the invariant

$\left(\kappa_{11}\kappa_{44}+\dots+2\kappa_{23}\kappa_{14}+\dots\right)dx\ dy\ dz\ dt\ \delta x\ \delta y\ \delta z\ \delta t.$