Page:BatemanElectrodynamical.djvu/11

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1909.]
283
The transformations of the electrodynamical equations.

have and eighteen relations of the types

There are clearly nine relations of the first type and nine relations of the second type. We shall now show that when these relations imply that there is a relation of the form

for this purpose we shall require the following lemma.

Lemma. — Let the sixteen quantities

be connected by the eighteen relations of type

which imply that conjugate minors of the determinant are equal. The identity

then gives

or

Introducing the notation

we may obtain in the above way the equations

Hence, either

or all the quantities of type are zero. In the same way we can prove that either