is a homogeneous linear function of . Under the special assumptions specified at the beginning of this § these are every where, the same functions. Let us thus consider a definite component of (15) e.g. that which corresponds to the direction of the coordinate . We can represent it by an expression of the form
where are constants. It will therefore be sufficient to prove that the four integrals
In order to calculate we consider an infinitely small prism, the edges of which have the direction . This prism cuts from the boundary surface two elements and . Proceeding along a generating line in the direction of the positive we shall enter the extension bounded by through one of these elements and leave it through the other. Now the vectors perpendicular to , which occur in (15) and which we shall denote by and for the two elements, have the same value. If, therefore, and are the absolute values of the projections of and on a line in the direction , we have according to (14)
Let first the four directions of coordinates be perpendicular to one another. Then the components of the vector obtained by projecting on the above mentioned line are and similarly those of the projection of . But as, proceeding in the direction of we enter through one element and leave it through the other, while and are both directed outward, and , must have opposite signs. So we have
- From § 10 it follows that if the length of a vector that is represented by a line (§ 17) coincides with a radius-vector of the conjugate indicatrix, it is always represented by an imaginary number. We may however obtain a vector which in natural units is represented by a real number e.g. by 1 (§ 13) if we multiply the vector by an imaginary factor, which means that its components and also those of a vector product in which it occurs are multiplied by that factor.