I looked at it more closely. The white wire is exposed, and there is not enough extra wire to cut the bad piece and still have it reach where the outlet ought to go, so the wire has to go.

Luckily it seems that the bad spot is on the segment that goes from the first output to the second as shown in this picture:

If that is the case I can fairly easily replace just that piece. Any ideas on how I can tell for sure?

If its only the outside jacket you should be able to fix it by covering the areas with heavy duty electrical tape. The insulation on the individual wires is the primary electrical protection, while the outer layer is more for physical protection from abrasion.

The individual wires can be protected with metal or plastic conduit, metal flexible spiral wrap (BX cable, etc.), or by a heavy duty vinyl/rubber jacket (like your NM cable).

Bruce

What I meant is that the cut goes all the way through the insulation of the white wire in the "bad spot"

It is good to know that the other less bad spot can be covered with electrical tape.

If you've cut the insulation down to the wire (and not just the jacket), then I think you should replace it.

I have verified that all of the rest of the outlets listed as being on that circuit are off when the breaker is on, but the piece labeled "to fuse box" is disconnected. I then connected the "to fuse box" wire to the "to rest of circuit" piece turned the breaker on again, and all of those outlets showed as active. Lastly I checked the piece labeled "bad spot" and the ohmmeter shows continuity.

So I'm convinced that that piece can be snipped off at the ceiling, and be replace with a new piece, and the rest of the circuit will work.

You do know that you must make a junction in a junction box, right? (Cannot tell exactly what you have in mind from your post.)

I do know that, thanks for checking. My plan is the following:

Since I cannot see for certain where the damaged wire goes above the ceiling, I was describing the tests I did to deduce that the wire was as I drew it, in case someone saw a flaw in my reasoning.

You might be able to use the old wire to pull the new wire through the ceiling. Depends on if there are staples and/or any other obstacles.

That's a good idea I'll see if that will work. I suspect that the old wire is stapled though.