Black Label, here is a link which will makes available many, many manuals, including one for the model 39: http://stevespages.com/page7b.htm
This link will download a PDF file for the model 39 manual, which includes a parts list in the event you cannot identify certain parts based upon verbal nomenclature.
In the slide, mounted under the rear sight, are two parts: 6013 Ejector-depressor Plunger and 6014 Ejector-depressor Plunger Spring. I believe that the Ejector Depressor Plunger presses down on the ejector when the slide is in battery. A magazine in place in the weapon pushes up on the ejector. When the ejector is up, the trigger when pulled will drop the hammer. When the ejector is down, the trigger will not drop the hammer. So, if there is no magazine seated in the pistol, the hammer will not drop. As Felix FR said, check that the Ejector-depressor Plunger and Spring are not gunked up, and also the 6015 Ejector and 6017 Ejector Spring. Again, as Felix FR noted, you must have a magazine fully seated in this pistol for it to fire.
By carefully twisting the rear sight around to (I believe, as it has been a while) the right, you can expose the Ejector-depressor Plunger and Spring. It then is easily removed, thereby disabling the magazine disconnect feature, and your gun will fire without a magazine in place. I dislike magazine disconnects and always disable them on my semi-auto pistols which have them. Others like them.