Final Solution for me
After reading all the advice on this site, I decided to let the pros decide what was causing the drive line hum.
First, the technicians drove the car and heard the noise. It was a hum, close to the driveline area at around 40 MPH. They took it into the shop and put the car on a hoist and ran the engine and listened with a stethoscope. Using this method was not conclusive. Next the placed two sensors, one on the transmission and one on the differential and drove the car down the street. The transmission sounded normal but the differential sounded a little odd. They finally had to remove the differential cover and look at the gears. Apparently there was a lot of wear and tear on the gears, enough to cause the hum. They showed me exactly what was wrong by shining small penlights with a blueish type light into the gear box.
The techncians installed a used differential ($250) plus labor that solved the problem.
Thanks, to everyone who offered advise. I am glad that I took the car in and they were able to accurately diagnose the problem. They also found my front brakes needed repacement, which I just went ahead and had done.