W. Palin Elderton 2(;:3 



noniuil ciirvf wliich liad beeii ;ilre;idy u.swl for finding the actual amouiit of 

 «ickncss to wliicli tlie Type I. curve was Htted. Stiictly speakiiig, I t.liink it woidd 

 havc bcen morc corrtct to read <>tV (irdiiiatcs in tlic dciioininators, but the aualysis 

 of the poiiit call haidly be dealt wilh in a t'ew lines, and would bc of a tcclinical 

 character. The iise of arcas makes it impossiblc to writc down an algebiaic 

 expression for the rates of sickiiess ; the quotient of the ordinates of the two 

 curves would not give these rates, but a function that might be called the " foice 

 of sickness," for it is analogous to the force of mortality raentioned above. 



This le.sult was so good that I decided to make some further trial.s in the hope 

 of ohtaining other interesting representations. Symmetrical limited-range curves 

 were tried with the same origin but though nearly all the Type I. curves I 

 obtained represented the amount of sickness closely they failed at the ends of the 

 table to give a satisfactory grailuatiou of the sickness rates. A fcw trials were 

 also niade with different origins. 



The next attempts were made from the opposite point of view, that is, a 

 hypothetical curve was assumed for the sickness. The .same normal curve was 

 tried and the exposed to risk were then represented by a Type IV. curve 



y = 273-3649 [1 +a,'Y(66-9.ö76)=]--''""»"e-'-'™'"'"'"'w*'"="'», 



with an oiigin at .52-5043939. This curve is very nearly .synimotiical but on 

 tryiug a normal curve it was fouiid to fail above age Ö6 (i.e. for 34 ages. Compare 

 Makeham's " Law of Mortality " which is unsuitable for the first 2.5 ages). In 

 Order to see if the quotient of two normal curves would graduate the rates I 

 tried to fit the parabola y = a -\- hx + ex- * to their logarithms but the result 

 was unsatisfactory. Trials were then made with limited-range curves and the 

 corresponding exposed to risk represented by Type I. curves. The results were 

 unsatisfactory as regards the rates for at least 15 consecutive ages, in the case of 

 the Type IV. and normal curve, for iustauce, the rates were considerably too low 

 from age 67 to age 82. A trial with the Type I. curve to the rates of sickness 

 also gave a poor result. 



Viewed as processes of graduation the trials mentioned above are, as a rule, 

 satisfactoiy for the main part of the sickness table aud the practical conclusion to 

 be drawn is that a rapid graduation can be made with reasonablc results as far as 

 age SO or 85. The assuinption of a hypothetical curve for the e.xposed is prefer- 

 able to the opposite course, for, if the latter be adopted, there is some doubt as to 

 the correct values to be given to the exposed to risk at ages whcre no persons 

 have been sick. In the present case this only occurred at age 4, but in small 

 experiences it would be a matter of greater importance. Another reasou for the 

 preference is that the old age end of the experience is relatively smaller wheii the 

 hypothetical curve is used for the sickness than when it is used for the exposed ; 

 that is to say, the former method gives less weight to the old .-ige values than the 



* Rate of sickness = Ae"'-^"'''°''''' -r «"^^""^'-'S therefore log rate =n.r= + 6x + c (see Pearson, " On the 

 Systematic Fitting of Curves," Biometrika, Vol. ii. Pt. i.). 



