Page:The principle of relativity (1920).djvu/104

 §11.

In the statement of the fundamental equations, our leading idea had been that they should retain a covariance of form, when subjected to a group of Lorentz-transformations. Now we have to deal with ponderomotive reactions and energy in the electro-magnetic field. Here from the very first there can be no doubt that the settlement of this question is in some way connected with the simplest forms which can be given to the fundamental equations, satisfying the conditions of covariance. In order to arrive at such forms, I shall first of all put the fundamental equations in a typical form which brings out clearly their covariance in case of a Lorentz-transformation. Here I am using a method of calculation, which enables us to deal in a simple manner with the space-time vectors of the 1st, and 2nd kind, and of which the rules, as far as required are given below.

A system of magnitudes a_{h k} formed into the matrix


 * a_{1 1}a_{1 q} |
 * a_{p 1}a_{p q} |
 * a_{p 1}a_{p q} |
 * a_{p 1}a_{p q} |
 * a_{p 1}a_{p q} |

arranged in p horizontal rows, and q vertical columns is called a p × q series-matrix, and will be denoted by the letter A.

If all the quantities a_{h k} are multiplied by C, the resulting matrix will be denoted by CA.

If the roles of the horizontal rows and vertical columns be intercharged, we obtain a q × p series matrix, which