Does this linear current source driver deliver constant current?
Q1 sucks enough of the base current (coming through R1) from Q2 to keep the emitter at around 0.6V higher than ground, because when the transistor is in the active region, that's Vbe. More voltage means the transistor diverts more current from the base of Q2, less voltage and it diverts less. 

There are limits- if the value of R1 is chosen too high, then Q2 won't have enough base current to drive the LED fully on. If the value of R2 is chosen too low, then Q1 won't be able to divert the base current as well, and it won't regulate very well. 

If you assume Vbe of Q1 is 0.6V-0.7V (not a great approximation) and ignore the base current of Q2 in relation to the LED current (a good approximation), the LED current will be a constant current of ~0.65V/R2

It's easy to simulate this with the simulator here: 







simulate this circuit &ndash; Schematic created using CircuitLab



As you can see, once the supply voltage exceeds 2.5 or 3V, the current is pretty constant. Similar plots could be made for LED voltage drop and for temperature and you'd see that the current is "fairly" constant, not by instrumentation standards, but visually (LED brightness) it's pretty decent, especially for such a simple circuit, and the current is close to the predicted 0.65V/50R = 13mA.