Subject arms that are anywhere from a little woggly to downright falling
off can be tighted.  You need two allen wrenches, one about 6mm and one
about 3mm.  Buy or borrow a metric set.  Pull the cap off the "axle"
of the chair arm, and if it is dark brown send it to me.  If there is
no socket-headed bolt visible, feel around under the vinyl till you find
it.  If it is visible, undo it with the larger allen wrench and remove
the arm.  Be careful not to lose the bolt or washers under the vinyl.
The axle is held to the seat back with two little socket headed bolts.
Fanatics would loctite them; the rest of us just tighten them down.
Replace the arm on the axle, and put the retaining bolt back in, again
being careful not to drop washers or bolts inside the vinyl.  Replace
the cosmetic cap unless it is one of my three missing brown ones.
B.
Wm Kennedy
kennedy@admin.njit.edu