I made a mistake (it should be x-vt in the equation for t'), the full
transformation between ether frame (unprimed [x, y, t]) and moving frame
(primed [x', y', t']) is:

       x' = (x - vt) / sqrt(1 - v^2/c^2)
       y' = y
       t' = t - ((v * (x - vt)) / (c^2 - v^2))
          = (t - vx/c^2) / (1 - v^2/c^2)

Let me show how this is consistent with what Poincaré says on pages 565-566
of the article http://www.soso.ch/wissen/hist/SRT/P-1908.pdf. He considers
two events: A (emission of a light wave) and B (detection of the light
on an arbitrary point of the wave front at some later time). Let the
space-time coordinates of A and B in the ether frame be:

       x_A = 0
       y_A = 0
       t_A = 0
and
       x_B = ct cos(a)
       y_B = ct sin(a)
       t_B = t

(the angle a is arbitrary and serves as parameter to denote any point on
the wave front.)

With the help of the above formulas, I find that the space-time coordinates
in the moving frame are:

       x_A' = 0
       y_A' = 0
       t_A' = 0
and
       x_B' = (ct cos(a) - vt) / sqrt(1 - v^2/c^2)
       y_B' = ct sin(a)
       t_B' = (t - (vct cos(a) /c^2)) / (1 - v^2/c^2)

Then I can check that the circular (in the ether frame) wave front goes into
an ellipse

       (x_B' * sqrt(1 - v^2/c^2) + vt)^2 + y_B'^2 = (ct)^2

with ellipcity e = v/c and center [-vt * sqrt(1 - v^2/c^2), 0], corresponding
to the picture on page 566.

Next I compute Poincaré's AB and AB':

        AB  = sqrt(x_B'^2 + y_B'^2) = t * (c - v * cos(a)) / sqrt(1 - v^2/c^2)
        AB' = x_B' = t * (c cos(a) - v)/ sqrt(1 - v^2/c^2)

and check that indeed:

        AB + v/c * AB' = ct * sqrt(1 - v^2/c^2)

which is the second equation on page 566. Then I can find the next-to-last
equation for the local time by rewriting:

         tau = t_B'
             = (t - (vct cos(a)/c^2)) / (1 - v^2/c^2)
             = (t - t* v^2/c^2 + t * v^2/c^2 - (vct cos(a)/c^2))/(1 - v^2/c^2)
             = t - ((ct cos(a) - vt)*v/c^2)/(1 - v^2/c^2)
             = t - ((AB' v/c) / c sqrt(1 - v^2/c^2))

Finally, it's easy to check that:

        AB = c tau sqrt(1 - v^2/c^2)

which is the last equation (with t corrected to tau -- that this is
a misprint can be seen by comparing the next-to-last and the second
equation).

As AB is the distance travelled by the ligth wave and tau the time of
travel, *as measured by an observer in the moving frame*, I deduce that
the speed of light for that observer is:

        c' = AB / tau = c sqrt(1 - v^2/c^2)

While it is true that "you don't find such a nonsense" (as Homo Lykos
says) in the article, it's just a division away :-/

Now I'd like to follow-up to some remarks by Harry and Homo Lykos.

In article <4332845...@epflnews.epfl.ch>,
       Harry <harald.vanlin...@epfl.ch> writes:

] > you seem to be saying that after discovering Special Relativity in 1905,
] > he changed his mind... It is easy to see that what is missing in his 1908
] > article is time dilatation:
]
] I am flabbergasted - "time dilatation" is represented by the tau on page
] 566.

there is "time delay" but no time dilatation in the equation linking t and
tau: for an observer at rest in the moving frame (hence: constant AB')
a phenomenon won't start at the same time as for an observer in the ether
frame (because of the second term -- that's time delay), but its *duration*
will be the same (t and tau come with the same unit factor in the equation.)
Hence, in Poincaré's theory time durations have an absolute meaning.
Not so in Einstein's theory.

In article <4335d73e$...@news.bluewin.ch>,
              Homo Lykos <ly...@lykos.ch> writes:

] Especially strange for me:
]
] " Supposons que la différence entre le temps vrai et le temp local en un
] point quelconque soit égale à l'abscisse de ce point multipliée par la
] constante: e/(V sqrt(1-e^2)) "
]
] Poincaré gives no explanation for this (and speaks only of supposons(=we
] suppose)) and I - at least in this moment - can't understand it.

if you compute in the ether frame the correction needed to synchronize
moving clocks, you get a term vL/(c^2 - v^2) where L is the distance
between the clocks projected on the direction of the motion measured
in the ether frame -- you get Poincaré's expression if you use the distance
as measured in the moving frame but corrected for the length contraction.

Note that Einstein uses the same synchronisation procedure as Poincaré
(who must indeed be credited for this nice idea) but comes to different
conclusions because he postulates that the speed of light is the same in all
inertial frames.