Have been away, hence the late response.
Your suggestion that it may be the connecting cable that was at fault, was helpful.
The connecting eSata cable was plugged into an Addonics Port Multiplier. I tested the connection a couple of times with three different eSata cables, plugged directly into my Mac. Each time, the MyBook icon popped up on my monitor.
So it would appear that the Port Multipier is malfunctioning.
