1357 
[Re . Nix 
is a homogeneous linear function of X,,...X,. 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 
vq. We can represent it by an expression of the form 
fee X,+...+ a, X) do, 
where «,,.... @, are constants. It will therefore be sufficient to 
prove that the four integrals 
[Xie Xoo CREE ERE ds Wad €00) 
In order to calculate fx do we consider an infinitely small 
vanish. 
prism, the edges of which have the direction v,. This prism -cuts 
from the boundary surface o two elements do and do. Proceeding 
along a generating line in, the direction of the positive w, we shall 
enter the extension 2 bounded by o through one of these elements 
and leave it through the other. Now the vectors perpendicular to 
6, which oceur in (15) and which we shall denote by N and N for 
the two elements, have the same value. *) If, therefore, Sand S are 
the absolute values of the projections of N and N on a line in the 
direction z,, we have according to (14) 
Gis SO ee ee ye wk Be ED 
Let first the four directions of coordinates be perpendicular to one 
another. Then the components of the vector obtained by projecting 
N on the above mentioned line are Xj, 0, 0,0 and similarly 
those of the projection of N: X,,0,0,0. But as, „proceeding in the 
direction of z,, we enter 2 through one element and leave it through 
the other, while N and N are both directed outward, X, and ix 
must have opposite signs. So we have 
S18 =X, 3 Sa 
and because of (17) we may now conclude that the elements X,d6 
1) From § 10 it follows that if the length of a vector A 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 ($ 15) if we multiply 
the vector A 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. 
