The Principle of Relativity · Albert Einstein
[M. N. S.]
Where the integration in _d_σ is to be extended over the whole surface. If now _s_ is selected as the _x_-direction, Δ_s_ is then a three-dimensional parallelopiped with the sides _dy_, _dz_, _dl_, then we have $$ Div f_{x} = \frac{1}{dy dz dl} {dz. dl. \frac{\partial f_{xy}}{\partial y} dy + dl dy \frac{\partial f_{xy}}{\partial z} dz + dy dz \frac{\partial f_{xy}}{\partial l} dl} = \frac{\partial f_{xy}}{\partial y} + \frac{\partial f_{xy}}{\partial z} + \frac{\partial f_{xy}}{\partial l} $$ and generally Div _f__{_j_} = ∂_f__{_j_ _x_}/∂_x_ + ∂_f__{_j_ _y_}/∂_y_ + ∂_f__{_j_ _z_}/∂_z_ + ∂_f__{_j_ _l_}/∂_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_}^*). D_{_x_} 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 (D_{_x_}, D_{_y_}, D_{_z_}, D_{_l_}) is perpendicular to the parallelopiped formed by (P U^* V^*). Generally we have (P_f_) = PD + P^*D^*. ∴ The vector of the third type represented by (P_f_) is given by the geometrical sum of the two four-vectors of the first type PD and P^*D^*. ● Transcriber’s Notes: ○ The book's idiosyncratic spelling, emphasis, punctuation, and symbology especially in mathematical formulas, have been retained. ○ Text that was in italics is enclosed by underscores (_italics_). Text that was in bold face is enclosed by equals signs (=bold=). ○ Footnotes have been moved to follow the chapters in which they are referenced.
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