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Photos always help in dating an old Smith & Wesson, but here are a few guidelines that will help you date your revolver.
The front sight changed from a half moon shape to a ramp in 1952.
The upper sideplate screw was discontinued in 1955.
The model designation was stamped on the frame inside the yoke in 1957.
The screw in front of the trigger guard was eliminated in 1962.
From what I can tell from the SCS&W, the C prefix began in 1948 with C1 and ran to 1967, ending with C999999. If you own a four screw gun (no upper sideplate screw, but having a screw in front of the trigger guard) we know by your screw count that your gun was produced between 1955 and 1962. If it has MOD 10 stamped inside the yoke, we can narrow that down to 1957 to 1962. Mr.Borland is correct on the serial number time frame, so I have to assume you have a 1954-1956 gun, probably a 1955-1956 by the screw count, that it lacks the MOD 10 stamp in the yoke, and somebody just told you it was a "Model 10." That being said, only
Roy Jinks, the S&W historian can tell you for certain when your revolver left the factory. It is entirely possible that you could have an earlier serialed frame that left the factory at a later date, receiving the MOD 10 stamp. It has been known to happen.
If you do not have MOD 10 stamped inside the yoke, you technically own a Model of 1905 4th Change, also known as a Smith & Wesson Military and Police revolver, not a Model 10.
The walnut magna grips with the diamond surrounding the screw were changed to diamondless magnas in 1968. Thus, we can say with certainty the correct grip for your revolver would be diamond magnas. Diamond magnas came in flat bottom styles and later in a rounded bottom "plainsclothes" style. The lower Model 10-5 in the photo below is a C serialed 1964 gun, (C7946XX) with correct "plain clothes" diamond magnas. These are the correct grips you seek, except yours were likely flat bottom grips. FWIW, each of these Model 10s in the photo wear aftermarket period grip adapters, the black item behind the triggerguard. Diamond magna grips for K frame revolvers occasionally show up on ebay. They usually sell for $25-$35, depending on condition. Just search for "K frame grips."
Here is a set.
Here is another set.
Here is another.
Here is one more set. Voila!
Here is another with a 338XXX serial number. If you purchase a set off ebay, make certain you get the screw. The original ones for your revolver would have had the serial number stamped inside the right grip.