719 



time. Now if y represent the living at a definite age, and r the rate 

 at which the mortality increases at that point of age, then mr z will 

 be the rate of mortality after the lapse of z units of time. The 

 decrement of y in an infinitely short time will be dy=ymr x dz. 

 This was pointed out by Mr. Gompertz, and Mr. Edmonds subse- 

 quently extended the theory. This expression can be integrated, 



W> 

 and the final equation of the corrected integral is y=10 ; 



where \ is put for the common logarithm, and k for its modulus. 



Either of the hypotheses gives a close approximation to the exact 

 result, within short intervals of time ; and the results by the two 

 hypotheses agree at the principal ages after 20, when they can be 

 fairly tested. Thus, if the rate of mortality in any year of age (x to 

 x-}- 1) is m, then 



It is here assumed that m is known ; and putting p x for J~ , 



we have p x l x = l x +\ ; and have thus the means of passing from the 

 numbers living at the age x to the numbers living at the age x + 1 . 

 But upon the other hypothesis, y being =1, then 



Wm 



*,= yi =10^ 



Upon the two hypotheses, 



Xp ao =T'9966528 by the one, and T'9966527 by the other; 



^40= ^'9956263 by the one hypothesis, and T'9956264 by the other. 



At 80 there is some divergence. I have adopted the latter hypo- 

 thesis generally ; but the other hypothesis is at some of the earlier 

 ages preferred. I have only adopted these hypotheses within the 

 safe limits of a single year in determining eleven values of \p x , which 

 I have afterwards interpolated by the method of finite differences ; 

 thus assuming that the third difference was constant. This gives, I 

 conceive, as near an approximation as we can obtain in the present 

 state of the observations. 



\p x is the first difference of the series \4; and consequently it 

 can be constructed by four orders of differences, on the assumption 

 that no error of consequence is caused by assuming that within given 

 limits the fourth difference is constant. 



