Gentlemen,—This Society was privileged during a recent Session to listen to an eloquent and highly original address by Professor Chrystal:— ‘On some fundamental Principles in the Theory of Probability.’ The value of such addresses, dealing, as Professor Chrystal's did, with questions on which wide differences of opinion exist, is not to be measured simply by the amount of information they contain ; they serve a much more useful purpose, by leading those who hear them, or afterwards read them, to think out for themselves the various questions discussed. Speaking for myself, I have to thank the Professor, not only for a pleasant evening spent in listening to him, but for causing me to consider carefully the fundamental principles of the theory with which, as Actuaries, we are all supposed to be so much concerned. I have found the subject a most interesting one, and I propose to lay before you this evening, some of the reflections that have occurred to me, and the conclusions at which I have arrived, after carefully studying what has been written on the subject by a number of authors.