Div fj = ∂fj x/∂x + ∂fj y/∂y + ∂fj z/∂z + ∂fj l/∂l (where fj, 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, Dx denotes a parallelopiped laid in the (y-z-l) space, formed out of the vectors (Py Pz Pl), (Uy* Uz* Ul*) (Vy* Vz* Vl*).
Dx is therefore the projection on the y-z-l space of the parallelopiped formed out of these three four-vectors (P, U*, V*), and could as well be denoted by Dyzl. We see directly that the four-vector of the kind represented by (Dx, Dy, Dz, Dl) is perpendicular to the parallelopiped formed by (P U* V*).
Generally we have
(Pf) = PD + P*D*.
∴ The vector of the third type represented by (Pf) is given by the geometrical sum of the two four-vectors of the first type PD and P*D*.
[M. N. S.]
Footnotes