1. The subject of analysis of surplus has been given much study by actuaries both for the intellectual satisfaction it affords and for the practical value of the results of such inquiry. One of the earliest papers in our literature is that of G. H. Ryan in 1892 (J.I.A. 30), the first half of which was entitled ‘On a Method for determining the Gain or Loss from Mortality in an Annuity Company’. Since that date, however, attention has been devoted almost entirely to the analysis of surplus of a life fund, in view of the problems involved in its distribution, and Ryan's restricted treatment remains the only reference for students with regard to the analysis of surplus of an annuity fund. The purpose of the following notes, therefore, is to illustrate symbolically a full analysis of such a fund comprising immediate, deferred and contingent annuities, obtaining a complete reconciliation, and at the same time to indicate briefly the practical difficulties and the consequent steps in the analysis at which curtailment would be necessary in practice. Discussion of the relative importance of sources of surplus is outside the scope of these notes.