In this paper we derive a simpler form of Particle Swarm Optimiser (PSO) which still retains the key properties of the original model. We do so by progressively altering the original model via mathematical transformations which have clear and understandable probabilistic effects. Then we study this new PSO mathematically and compare it to the original PSO. In particular we analyse the stability and the sampling distribution for the new PSO during stagnation. Finally, we test it on 14 standard benchmark problems and with two different communication topologies, with very encouraging results.