Smoluchowski attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are equal. If the probability of m gains and n − m losses follows a binomial distribution, P m , n = ( n m ) 2 − − n , with equal a priori probabilities of 1/2, the mean total gain is E [ 2 m − − n ] = ∑ ∑ m = n 2 n ( 2 m − − n ) P m , n = n n ! 2 n [ ( n 2 ) ! ] 2 .