Page:The principle of relativity (1920).djvu/259

 to see that lor S will be a four-vector. Let us find the component of this four-vector in any direction s. Let S denote the three-space which passes through the point Q (x_{1}, x_{2}, x_{3}, x_{4}) and is perpendicular to s, ΔS a very small part of it in the region of Q, dσ is an element of its two-dimensional surface. Let the perpendicular to this surface lying in the space be denoted by n, and let f_{s n} denote the component of f in the plane of (sn) which is evidently conjugate to the plane dσ. Then the s-component of the vector divergence of f because the operator lor multiplies f vectorially)  = Div f_{s} = Lim ([integral] f_{s n}dσ)/ΔS.     Δs = 0  Where the integration in dσ is to be extended over the whole surface.  If now s is selected as the x-direction, Δs is then a three-dimensional parallelopiped with the sides dy, dz, dl, then we have  Div f_{x} = 1/(dy dz dl) { dz. dl. [part]f_{x y}/[part]y dy + dl dy· [part]f_{x z}/[part]z dz

+ dy dz [part]f_{x l}/[part]l dl } = [part]f_{x y}/[part]y + [part]f_{x z}/[part]z + [part]f_{x l}/[part]l,

and generally

Div f_{j} = [part]f_{j x}/[part]x + [part]f_{j y}/[part]y + [part]f_{j z}/[part]z + [part]f_{j l}/[part]l (where f_{j, j} = 0).

Hence the four-components of the four-vector lor S or Div. f is a four-vector with the components given on page 42.

According to the formulae of space geometry, D_{x} denotes a parallelopiped laid in the (y-z-l) space, formed out of the vectors (P_{y} P_{z} P_{l}), (U_{y}^* U_{z}^* U_{l}^*) (V_{y}^* V_{z}^* V_{l}^*).