Page:The principle of relativity (1920).djvu/254

 frequent occurrence in this paper, it will be better to form an idea of their geometrical meaning. The following is taken from the above mentioned paper of Sommerfeld.

"We can also form a vectorial combination of a four-vector and a six-vector, giving us a vector of the third type. If the six-vector be of a special type, i.e., a piece of plane, then this vector of the third type denotes the parallelopiped formed of this four-vector and the complement of this piece of plane. In the general case, the product will be the geometric sum of two parallelopipeds, but it can always be represented by a four-vector of the 1st type. For two pieces of 3-space volumes can always be added together by the vectorial addition of their components. So by the addition of two 3-space volumes, we do not obtain a vector of a more general type, but one which can always be represented by a four-vector (loc, cit. p. 759). The state of affairs here is the same as in the ordinary vector calculus, where by the vector-multiplication of a vector of the first, and a vector of the second type (i.e., a polar vector), we obtain a vector of the first type (axial vector). The formal scheme of this multiplication is taken from the three-dimensional case.

Let A = (A_x, A_y, A_z) denote a vector of the first type, B = (B_{y z}, B_{z x}, B_{x y}) denote a vector of the second type. From this last, let us form three special vectors of the first kind, namely—

B_x = (B_{x x}, B_{x y}, B_{x z}) } B_y = (B_{y x}, B_{y y}, B_{y z}) } (B_{i k} = - B_{k i}, B_{i i} = 0). B_z = (B_{z x}, B_{z y}, B_{z z}) }

Since B_{j j} is zero, B_j is perpendicular to the j-axis. The j-component of the vector-product of A and B is equivalent to the scalar product of A and B_j, i.e.,

(A B_j ,) = A_x B_{j x} + A_y B_{j y} + A_z B_{j z}.