Considering four distinct events in a spacetime region in which values of Synge's world function $\sigma$ are defined (up to a common non-zero factor) for each pair of events, and specificly considering
- three events $\mathcal A$, $\mathcal P$ and $\mathcal Q$ which are straight wrt. each other, i.e. in terms of the values of Synge's world function
$$ 0 = \begin{vmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & \sigma[ \, \, \mathcal A, \, \mathcal P \, ] & \sigma[ \, \, \mathcal A, \, \mathcal Q \, ] \\ 1 & \sigma[ \, \, \mathcal A, \, \mathcal P \, ] & 0 & \sigma[ \, \, \mathcal P, \, \mathcal Q \, ] \\ 1 & \sigma[ \, \, \mathcal A, \, \mathcal Q \, ] & \sigma[ \, \, \mathcal P, \, \mathcal Q \, ] & 0 \end{vmatrix}, $$
and a fourth event $\mathcal B$ such that
events $\mathcal B$ and $\mathcal P$ are lightlike related to each other, i.e. $\sigma[ \, \, \mathcal B, \, \mathcal P \, ] = 0$,
events $\mathcal B$ and $\mathcal Q$ are lightlike related to each other, i.e. $\sigma[ \, \, \mathcal B, \, \mathcal Q \, ] = 0$, and
all four events are flat (or more correctly, of course: plane) wrt. each other, i.e.
$$ 0 = \begin{vmatrix} 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & \sigma[ \, \, \mathcal A, \, \mathcal B \, ] & \sigma[ \, \, \mathcal A, \, \mathcal P \, ] & \sigma[ \, \, \mathcal A, \, \mathcal Q \, ] \\ 1 & \sigma[ \, \, \mathcal A, \, \mathcal B \, ] & 0 & \sigma[ \, \, \mathcal B, \, \mathcal P \, ] & \sigma[ \, \, \mathcal B, \, \mathcal Q \, ] \\ 1 & \sigma[ \, \, \mathcal A, \, \mathcal P \, ] & \sigma[ \, \, \mathcal B, \, \mathcal P \, ] & 0 & \sigma[ \, \, \mathcal P, \, \mathcal Q \, ] \\ 1 & \sigma[ \, \, \mathcal A, \, \mathcal Q \, ] & \sigma[ \, \, \mathcal B, \, \mathcal Q \, ] & \sigma[ \, \, \mathcal P, \, \mathcal Q \, ] & 0 \end{vmatrix},$$
then (as follows by explicit calculation):
either event $\mathcal A$ is between events $\mathcal P$ and $\mathcal Q$, i.e. (consistent with the straightness-relation above): $\sigma[ \, \, \mathcal P, \, \mathcal Q \, ] = \sigma[ \, \, \mathcal A, \, \mathcal P \, ] + \sigma[ \, \, \mathcal A, \, \mathcal Q \, ] + 2 \, \sqrt{ \sigma[ \, \, \mathcal A, \, \mathcal P \, ] \, \sigma[ \, \, \mathcal A, \, \mathcal Q \, ] } $, and $$ \sigma[ \, \, \mathcal A, \, \mathcal B \, ] = -\sqrt{ \sigma[ \, \, \mathcal A, \, \mathcal P \, ] \, \sigma[ \, \, \mathcal A, \, \mathcal Q \, ] } \, \left( \text{Sgn}[ \, \sigma[ \, \, \mathcal P, \, \mathcal Q \, ] \, ] \right), \tag{*} $$
or otherwise: $$ \sigma[ \, \, \mathcal A, \, \mathcal B \, ] = \sqrt{ \sigma[ \, \, \mathcal A, \, \mathcal P \, ] \, \sigma[ \, \, \mathcal A, \, \mathcal Q \, ] } \, \left( \text{Sgn}[ \, \sigma[ \, \, \mathcal P, \, \mathcal Q \, ] \, ] \right). \tag{**} $$
Equations $(*)$ or $(**)$, in various choices of "language" and notation, are "here and there" attributed to A. A. Robb, A Theory of Time and Space (1936).
My question:
Exactly which (outright) theorem, or which (supplementary) formula, in Robb's book corresponds to either equation $(*)$, or $(**)$, or to both ?
