Just picked up a S&W M&P (pre model 10 I believe) and struggling to date it. S/N 370xxx
I just wanted to add to Jim's note -- his suggestion that it is from the early 1920's is very much supported by the one that I own that is factory lettered.370XXX would date it around early 1920's.
It did become the model 10 when S&W began model numbers around 1957.
It actually should have a 4th side plate screw hidden behind the grip.
Mine bears SN 357xxx and S&W historian Mr. Roy Jinks tells me that S&W shipped it on May 30, 1921. Yours with a 370xxx "should" be late 1921 or early 1922. Keeping in mind that S&W didn't always do everything in a predictable fashion.