A year ago I had the honour of submitting to the Institute a paper on “A Short Method of Constructing Select Mortality Tables”, which has been published in the Journal, vol. lii, pp. 286–341. For convenience it may be referred to as “the former paper.” A definition of the short method was given as follows:

Many investigations have been made in the past to ascertain the surplus, or the bonuses, which would arise with a given valuation basis, from different assumptions as to the actual experience and premiums of an office ; but there is little on record regarding similar efforts which may have been made to see how the surplus year after year would be varied by the introduction of different valuation bases, the other two factors remaining constant.

The remainder form of Newton's formula of interpolation with divided differences has been known for at least 37 years; yet actuaries have taken hardly any notice of the complete formula which ought to be one of their principal tools. It will, therefore, not be superfluous to begin by reproducing the very elementary proof from first principles.