Again Robert, you have a notational problem. You've been trying
to learn from idiots who want to look clever, but they are still idiots.
The correct FORM of the cuckoo transform is

	tau = t  * sqrt(1-vē/cē)
	xi  = x' /  sqrt(1-vē/cē)
	eta = y
	zeta= z

There is no t'.

Think of it this way.
A welding rod or a train moves along a track, so the tip or engine is always
increasing x.
The length of the rod (don't strike an arc) or the train doesn't change.
Now Einstein calls the length of the rod (or train) x', he says x' = x-vt.
 That's actually true, we can create a table of values and prove it.
I'll use a spreadsheet.

         x   t   v = x/t  x'=x-vt
        10   0      2        10
        12   1      2        10
        14   2      2        10
        16   3      2        10
        18   4      2        10
        20   5      2        10
        22   6      2        10
        24   7      2        10
        26   8      2        10
        28   9      2        10
        30  10      2        10

         x   t   v = x/t  x'=x-vt
        10   0     200       10
       210   1     200       10
       410   2     200       10
       610   3     200       10
       810   4     200       10
      1010   5     200       10
      1210   6     200       10
      1410   7     200       10
      1610   8     200       10
      1810   9     200       10
      2010  10     200       10

Now that is the first operation, and we've used up x' and v.
The SECOND operation is gamma to change the length of the
rod or the train.
	xi = x' * gamma
	tau = t / gamma
to make the ant crawling along the rod have the same speed
on the rod as it does on the plate being welded, which is silly.

> Well, that is enough for today.

Ok. Ask again if you have any trouble.

Androcles.