```Hi Steve

Thanks for the derivation in your email of 8/28.  I'll save it as
a reference to help understand Lorentz dot-product manipulations
if I ever run across them again.  Actually, the description of the
space-time diagram in your 8/26 email was more effective in con-
veying the relation to my intuition.  Having gone through that
thought process, I can now rewrite the derivation more succinctly:

The distance, x, of the sail after accelerating at a level, a, for
a proper time, t', is
x = [cosh(a * t') - 1]/a   .

The apparent (Earth) time, t, at which the sail reaches x is
t = [sinh(a * t')]/a   .

The time, delta-t, that it takes light from Earth to reach the
sail at x is
delta-t = t - te
= x/c
or           = x          for c = 1 lt-yr/yr,

where te is the Earth time at which the light was emitted.

Solving for te gives
te = t - x
= [1 + sinh(a * t') - cosh(a * t')]/a   .

Replacing sinh and cosh with their exponential forms gives
te = [1 - exp(-a * t')]/a   .

And solving for t' gives
t' = -[ln(1 - a * te)]/a   .                          Q.E.D.

I'll be looking forward to the way you'll treat the "dreaded
derivation" for the SD Group.  I'd feel honored if you used any
part of the above approach.  You started a very interesting
learning process with your email of 8/20 to the Group.

"If mankind is ever to achieve the ultimate goal of traveling to
the stars, we will need an engineering understanding of relativ-
ity" (RGF, IDA Paper P-2361, Oct 1990).

Regards, Rex

```