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The math involved is pretty straightforward, but I wonder if anybody can clarify his reasoning for his equation 3 (below)

He has two frames, with motion along x and x’ coordinates. The path of light shined along the positive direction of these axis is described by:

[tex]

\begin{align*}

x -ct = 0 & \ \ \ (1)

\end{align*}

[/tex]

[tex]

\begin{align*}

x' -ct' = 0 & \ \ \ (2)

\end{align*}

[/tex]

He states:

“Those space-time points (events) which satisfy (1) must also satisfy (2). Obviously this will be the case when the relation “

[tex]

(x' -ct') = \lambda(x -ct)

[/tex]

is fulfilled in general, where [itex]\lambda[/itex] indicates a constant; for, according to (3), the disappearance of (x – ct) involves the disappearance of (x' – ct')”

His “Obviously” isn’t so obvious to me. Given what he described I don’t see how the concurrent disappearance implies that these are linearly related by a constant. This step isn't terribly suprising given that the whole point of the appendix is to find the linear transformation between these (ie: the Lorentz tx.).

If I pretend that I didn't know that such a linear relationship was being looked for, I don't follow is argument of why to expect these should be linearly related. Is this obvious to anybody else?

--

ps. For reference I found an online version of this appendix here:

http://www.bartleby.com/173/a1.html

(I didn’t look to see if the whole book is there … I’ve got a copy from the public library).