^{page 218 note *} As we are now about to make a number of new columns calculated on new functional values and with new combinations, it is necessary t o have new symbols to distinguish, them. To avoid the use of new letters, where the columns are of a similar character to those in use, it is proposed to use the old symbols, D, N, C, M, R, with an index showing th e table on which the calculations are made, and a prefix, on the top left-hand corner, to show the function on which the columns are based. Although ^dC_x above, is the same as the ordinary C_x, ^dM_x, and ^dR_x adopted to show tha t they are not the ordinary columnar values, but the results of summations from age 64 only.

^{page 220 note *} If there were any necessity to distinguish the limiting age, as may possibly happen sometimes, such age might be put as a prefix at the bottom of the symbol, thus, , but, as there is only one limiting age throughout the whole of this paper and the formulas are general, I see no advantage in complicating the notation.

^{page 224 note *} I shall not repeat that the annual premium is due at the beginning but paid at the end of the year, or that the benefits are payable at the end of the year of exit. That is to be understood throughout the paper.

^{page 230 note *} I have purposely omitted the v in this symbol for the same reason as before, namely, to avoid confusion, as its appearance in a formula might be taken to indicate an operation to be performed, while the use of a single symbol with the index s, shows that it is a value from a specially constructed column.