Page:The principle of relativity (1920).djvu/107

 Then (36) f* f = f_{34} f_{22} + f_{42} f_{32} + f_{32} f_{24}

ie. We shall have a 4 × 4 series matrix in which all the elements except those on the diagonal from left up to right down are zero, and the elements in this diagonal agree with each other, and are each equal to the above mentioned combination in (36).

The determinant of f is therefore the square of the combination, by Det^{1/2}f we shall denote the expression

Det^{1/2}f = f_{32} f_{14} f_{13} f_{24} + f_{21} f_{34}·

4^o. A linear transformation

x_{h} = α_{h1} x_{1}´ + α_{h2} x´ + α_{h3} x_{3}´ + α_{h4} x_{4}´ (h = 1,2,3,

which is accomplished by the matrix

A = | α_{11} · α_{12}, α_{13}, α_{14} | |  |    |   |    |   |
 * α_{21}, α_{ 2}, α_{23}, α_{24} |
 * α_{31}, α_{32}, α_{33}, α_{34} |
 * α_{41}, α_{42}, α_{43}, α_{44} |

will be denoted as the transformation A

By the transformation A, the expression

x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} is changed into the quadratic for m [<=>]?] α_{hk} x_{h}´ x_{k}´,

where α_{hk} = α_{1k} α_{1k} + α_{2h} α_{2k} + α_{3h} α_{3k} + α_{4h} α_{4k}. are the members of a 4 × 4 series matrix which is the product of Ā A, the transposed matrix of A into A. If by the transformation, the expression is changed to

x´_{1}^2 + x_{2}´^2 + x_{3}´^2 + x´_{4}^2,

we must have Ā A = 1.