Mystery circuit: latching relay control with transitor / capacitor - why doesn't it work?
The circuit, as drawn, does not work.

Furthermore, even if it did work, it is doing so by horribly abusing the transistors base-emitter junction diode, which can potentially damage the transistor. 



It's worth noting that it is possible to make the circuit work. As far as I can tell, the person drawing the diagram simply forgot a diode:



The pull-up resistor also needs to be somewhat smaller, in order to turn Q1 on hard enough that the generated current waveform is appropriately symmetric. It's 10K in the above image, but 1K works much better.