Why there is transistor on LM317 ADJ pin?
If that diagram matches real circuit it's just a huge mistake. 

Q1 base is just grounded via lower part of R2.
Even taking into account R2 wiper resistance (a few ten ohms at worst) current from R1 (approx 5mA) cannot get base voltage above a few ten millivolts, so no way to turn Q1 substantially on.

So what remains is just Q1 Icbo, again several orders of magnitude from other currents involved at room temperatures. 

What about using Q1 as an remote(*) over temperature sensor to drop voltage? To have any appreciable effect Q1 Icbo should be in the milliamperes range: again really far from what any silicon BJT can reach. And even if could, it would be so affected from Q1 parameters dispersion to do it in a practically impredictable way.

(1) remote since LM317 has its own built in overtemperature protection which does work. 

I'd just add that circuit will be working anyway, it's just Q1 to be totally meaningless.

I'd also deprecate PTC fuse R4 being wired on the output, there's no need since LM317 has it's own current limiter, but what's worst fuse spoils voltage regulation adding its own resistance series to the output and, finally it would not protect foregoing mains transformer from LM317 or rectifiers failures.

It could be profitably moved on the input side where it wouldn't spoil voltage regulation and effectively protect foregoing source.