W. Palin Elderton 2(>1 



illustrate this proposition wo niay consider Gompertz' " Law of Mortality," whicli 



assiiiiu's that tlu> loivo oC uu>vU\\\ty {- , j k, where ^^ is Ü\v niiinber of jiersons 



living at age .r in a statiniiai-v p.'pulalioii) is of fho form Ac" and from the point 

 of viow taken above \ve can describc the " law " as assuniing that if the exposed 

 to lisk be represoiitcd by the normal curve of error then the deaths will bo 

 represented by a curve of the «amo tyiie haviiig the same Standard deviation. For 

 the force of mortality is eqiial to the ordinale of the deaths dividcd by the ordinale 

 of the exposed ; this gives us ^e-^^" -'"'+'■''- '"'^ 1-"'' = Bc''. 



If possible, I think it woidd be a good prinoiple in graduation to work on the 

 exposed to risk and deaths (or weeks' sickness), for these functions are actual 

 frequencies, but it is often difficuit to do this in practice. In the present case, 

 for instanee, the exposed to risk for ages 18—22 are 3114; 4029; 5850; 23485; 

 33534 and this siidden chaiige in the size of tlie numbers is awkward to deal 

 ■with; nor is this the ouly difficulty, for though the exposed curve rises to a 

 maximum at age 28 and then steadily falls, the sickness curve has two distinct 

 niaxima at ages 38 and 67. 



A first examination of Sutton's table was made by assuming the normal curve 

 of error as a hypothetical exposed to risk. The rea.son for choosing a fi equency 

 curve was that I have found it possible to represent (approximately) the exposed 

 and deaths statistics in actual mortality investigations by such curve.s. The values 



for the integral (-7= I e~''' dt) wcre found by taking equal intervais of -0(5; this 



used the whole table given in Galloway's Pruhahüity. It was then necessary to 

 decide on the origin for this curve. I wished to obtain, if possible, a frequency 

 curve for the number of weeks' sickness derived from the hypothetical expo.sed and 

 to avoid the two maxima of the sickness distribution referred to above. It was 

 desirable, therefore, to have the mode of the exposed curve noar the mininium 

 of the sickness curve and this was obviously convenient for it was near the 

 middle of the ränge. Age 52 was chosen as the origin. The amount of sickness 

 was obtained by multiplying the areas of the normal curve y = 07703 g-^' ^f"™"" by 

 the rates of sickness and the resultiiig figures were grouped about every fifth 

 age to render the calculation of the moments less laborious*. The grouped 

 figures ran smoothly aud were fitted closely by the Type I. frequency curve 



y = 248-34 



1 + 



88Ü7125 



1 



37-6809 



with an origin at 64-2G940G. The graduated rates were unsatisfoctory above 

 age 85, as they decreased very rapidly owing to the sickness curve ending at 

 101-9.50. 



The rates of sickness given in Fig. 1 were obtained by dividing the areas of the 

 above-mentioned Type I. curve corresponding to each age by the areas of the 



* An attempt to split up the distribution iuto two normal curves failed to give satisfactory results. 



