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I would suspect a worn stem, or a damaged sleeve.
A piece of nylon hose stretched across the end of a shop vac hose might be used to find that missing screw.
I use a pin vise for the job you are describing. I'm glad you were able to find a way to get the screw started .
You don't need to do any counterboring to replace balance jewels. Just remove the 2 screws that are holding the jewel setting in place, push it...
It appears to me that the balance wheel is not true in the flat, and maybe a coil on the hairspring is beginning to rust.
You might want to purchase a E & J Swigart co. catalog. They can be bought relatively inexpensively and contain a wealth of information.
I would begin by removing the hairspring from the balance and putting it on the cock.
That will quickly tell you if it's centered or not.
I looked, but could not come up with a center wheel for your 1904.
I will look tomorrow, I'm fairly certain that I have a few 1904 parts movements.
I think all his bottles have staffs in them. The low numbers should have hands, so I'm thinking someone has repurposed some empty bottles. It may...
On new stems the length is usually not important, because it will need to be cut to match the old stem.
Yes, I reassemble those movements upside down.
I would remove the jewels, and run them along with the balance in the ultra sonic. A sharpened piece of pegwood can be used to remove any oil or...
Dave's Watch Parts and Tools - Material Assortments
I agree with John Runciman, the watch is out of beat. I'd correct that first, then a good cleaning and see where you're at.
179 is correct, and I'll add that your case would also be correct for a pendant set movement.
You don't squeeze the bow when it's on the pendant, you squeeze it before it's placed on the pendant. The pliers then slightly open the bow enough...
If you would pull the hands and dial, then show us a picture of what she looks like inside we can probably get you started down the right path.
That watch takes a 817 mainspring. Daveswatchparts.com should have those .