282 PROCEEDINGS OF SECTION A. 



two. One is that of equal decrements of life in equal times, known 

 as DeMoivres' hypothesis, and the formulae and tables giving the 

 values of annuities and assurances on one or more lives on this 

 h}^othesis have been obtained and calculated. From a mathe- 

 matical point of view these formulse are simple, and the 

 calculations of the annuities and assurances are for several lives 

 comparatively easy. It is therefore to be regretted that the 

 hypothesis is very far removed from the truth, so that these formulaB 

 and tables are practically useless ; as;, if you take any of the tables 

 • — as Northampton, Carlisle, or indeed any table of actual results 

 of duration of life — you will see that the decrements of life are 

 anything but equal. 



These decrements of life are now known as d^, i.e., the number 

 who die between age x and x -\- \, and just as these decrements are 

 the first differences of the column 4, the number alive at age x, so 

 they in turn may be differenced so that DeMoivres' hypothesis is 

 that the second differences vanish and the first are constant. 

 Although this is not true in any ti'ue table, yet if the second and 

 subsequent dift'erences, whether positive or negative if only small ,^ 

 be taken out. some of the advantages of the simplicity of the 

 DeMoivres' formulae may be retained. I can here speak of my 

 own knowledge, having differenced out the H^j table to the fifth 

 difference, and the Carlisle to the fourth, one starting with 100,000 

 lives, and the other with 6,460 at age 10, and have used these 

 differences to construct the C, D, M, N, R, and S columns for 

 each table. The late Peter Gray gives a mode of constructing the 

 D column given C, and while not knowing his mode 1 had extended 

 it to the construction of the C column which involves d^, or the first 

 difference from a column we may call C'^, including the smaller 

 numbers of the second dift'erence, and these again from, say, 

 C"^, involving the third difference, and so on. This I regard as a 

 good mode of constructing the C and D columns to a large number 

 of places of decimals, or even to the usual six or seven figures, 

 more especially as the M, N, R, and S columns are easily obtained 

 at the same time. 



It may be remembered that these C, D, M, N, R, and S columns, 

 are the commutation columns, and that a^ A^,, the annuity and 

 assurance on life, aged x, are given by the formulae — 



N, , M, 



fl., = d; ^' = d: 



while the R and S columns are needed for the values of increasing 

 annuities and assurances. 



The only other attempt, and it is a brilliant one, to obtain the 

 law of mortality to which I shall refer is that of Benjamin 

 Gompertz. His principles seem to be that " death may be the 

 consequence of two generally co-existing causes, the one chance 

 without, i.e., independently of any previous disposition to death or 



