The nice thing about forums like this is that sometimes merely asking the question to someone else stimulates the mind and the solution comes to you. After I asked the question last night I started thinking about the way I asked it. That led me to better understand how to solve the problem, although the solution still evades me.
Let two light pulses, separated by a distance d, be moving parallel to the x-axis. At t = 0 let the leading pulse be at x = 0 and the trailing pulse be at x = -d. After a time delta t = T the trailing light pulse reaches S’s origin clock, which has moved a distance vT in the mean time.
Define the distance between the pulses as measured in S’ to be d’ = cT’ where T’ is the time it takes for the second pulse to reach the clock from its initial position. Determine T’ by the Lorentz transformation. Define the following two events
Event 1: First light pulse arrives at clock. Coordinates of event (x,t) = (0,0)
Event 2: Second light pulse arrives at clock. Coordinates of event (x,t) = (vT, T)
The temporal displacement between these two events is delta t = T. The spatial displacement between these two events is delta x = vT. We wish to find d’ = cT’. The Lorentz transformation for time is
t’ = gamma(t – vx/c^2)
The temporal difference between the two events is then
delta t’ = gamma(delta t – v delta x/c^2)
Substitute delta t’ = T’, delta t = T and delta x = vT
T’ = gamma(T – v(vT)/c^2) = gamma(T – v^2T/c^2) = gamma*T*(1-v^2/c^2)
d’ = cT’ = gamma*cT*(1-v^2/c^2)
cT = vT + d -----> T = d/(c – v)
d’ = cT’ = gamma*c[d/(c – v)](1 – v^2/c^2) = d*gamma*(1 – v^2/c^2)
d’ = d*sqrt[(1-v/c)/(1+v/c)]
which is the wrong answer. [:-'(]