MATHEMATICS AND ACTUARIAL SCIENCE. 283 



deterioration, the other a deterioration, or an increased inability to 

 withstand destruction." The formula given by Gompertz is an 

 exponential one, A'iz. : — , 



L = k (g) 



which gives for the "force of mortality" 



that is, the values of the force of mortality at successive ages form 

 a geometrical series. 



These formula; of Gompertz, by assigning suitable values to the 

 constants, A-, y, y, represent the results of observations for a series 

 of consecutive ages for about thirty years, but require the intro- 

 duction of new sets of constants at certain periods of life to com- 

 plete the table of mortality. The modification of Makeham on 

 these formula; is — ^ —. j. ^ -r / , q'' 



whence 



^t, = A H- B ?' 



A few words from Gompertz's writing show that the hypothesis 

 itself was derived from an analysis of the experience disclosed in 

 various published tables of mortality, so that he would seem to 

 have taken the observed results, and thence, by the differential and 

 integral calculus, derived the philosophical hypothesis already re- 

 ferred to. Makeham' s modification is, we are told, really included 

 in the two suppositions or conclusions of Gompertz — the one being 

 reproduced in the A of the jn^ and the other in the B y^. We are also 

 told that Makeham's modification, M'ith only one change of constants 

 for the childhood ages, will hold good for the whole term of life. 



We are now ready to consider the question of graduation of mor- 

 tality tables — that is, to modify and adjust the actual results, so as 

 to make them reliable for future probable experience. I can only 

 indicate some of the various methods — 



1. Woolhouse's, which seems to be to take the number living at 



any age of the unadjusted data for intervals of five years, 

 commencing at 10, 11, 12, 13, 14, to interpolate by finite 

 differences tor the intermediate ages, thereby producing, 

 so to say, five curves of life. The arithmetical mean of 

 the ordinates of these five curves is at each age then taken 

 as the adjusted number living at that age. 



2. The graphic method, which appears now to be championed 



mostly by Mr. Sprague, and which appears to be to plot 

 the values of the function to be graduated by means of 

 abscissae and ordinates as points in a curve, and then by 

 the eye and hand to draw a curve which shall, to the best 

 of the operator's judgment, make a smooth and even 

 representation of the original facts, or as they would be if 

 due weight were given to all and the effects of further 

 data duly allowed for. 



