The content of this article is indicated by what could be its full title: "An Explanation of the dependence of the rate of the cationic polymerizations of alkenes and of the DP of their products, on the reaction variables, especially the size of the anionic moiety of the initiator." We continue here the discussion started in 1965 and show mathematically how the theory of dieidic polymerizations by unpaired and paired cations can explain why some of these polymerizations become faster with falling temperature, why the Arrhenius plot of the DP of the polymers obtained from most such systems shows a discontinuity or kink, and also how the temperature of minimum rate, T-M, and that at which the kink occurs, T-K, depend on the reaction variables, namely the concentrations of monomer, m, and of initiator, c, and the a, D, and T (interionic distance in the ion-pair, dielectric constant of the reaction mixture and temperature). Our treatment explains why the most effective way of achieving the economically desirable aim, to make the longest polymers at the highest possible temperatures, is by maximizing the product a.D, so as to increase the T-K, preferably by the use of polar solvents and initiators with large anions. The choice of such combinations by several investigators, but for other, vaguer, reasons, is given here a theoretical basis. Our argument is illustrated by Literature examples and is presented in the form of a new diagram (the Plesch-Austin plot) which shows the T-K as a function of a.D for several systems. (c) 2008 Wiley Periodicals, Inc.