There is 3 main reasons the lamp will not stay put.
1 The bulb is to heavy, eg. energy saving or large LED.
2 The lock nut on the arm is missing or loose.
3 The springs have stretched and may need replacing.
The first two are easy to fix tighten nut and change bulb, the replacement springs are a little harder to do mostly because they are not available from Anglepoise. There is two places that re-manufacture them (but seem to sell out fast) and is a simple job to clip the new springs on.
Just checked today and they are in stock link form site below. (they sold so well the manufacture made a new batch.. joy)
So just though I should let you know as they sell really fast. I’ll be doing a review on them soon so you can see what your getting as soon as my sets arrive.