30 Mr. T. R. Edmonds on Vital Force according 



In applying my formula to the determination of the probability 

 of living one year at the precise ages 20, 30, .... 80, and 90 

 years, Dr. Farr substitutes the ambiguous quantity m for the a 

 of my formula, at the same time* describing m as equal to the 

 annual rates of mortality at the "precise ages" 20, 30, . . . 80^ 

 and 90 years. According to this definition, the m of Dr. Farr's 

 formula is identical with the a of my formula as described by me 

 in 1832. But this definition is immediately followed by the 

 inconsistent and contradictory statement that m at the age 20 

 years is represented by the mortality "ruling" from the age 19^ 

 to 20J years. This statement is elsewhere confirmed by Dr. 

 Farr, and made to extend to quinquennial and to decennial inter- 

 vals of age. The erroneous principle adopted in the construc- 

 tion of the English Life Table is, that the annual rate of mortality 

 at the middle point of any interval of age is identical with the 

 annual ratio of the dying to the living during that interval, 

 whether such interval is one year, five years, or ten years. This 

 erroneous principle may otherwise be described as resting on the 

 erroneous and gratuitous assumption, that the area of the curve 

 of surviving population, or \~Pdt, between limits t and £ + 1 (in 

 age) is always represented by the ordinate (multiplied by unity) 

 corresponding to the abscissa t-\-\, whatever be the unit of age, 

 whether one year, five years, or ten years. 



In its second and principal signification, as adopted by Dr. 

 Farr, the apparently simple quantity m, which has been substi- 

 tuted for the constant quantity a of my formula, is in reality a 

 variable quantity of great complexity, and more unknown than 

 the quantity P, which is to be expressed in terms of m. For m, 

 which represents the "mean mortality" at any or the (^4-l)th 

 interval of age, is of the form following : 



jPA 



The numerator of the above fraction, expressed in terms of the 

 variable t and constants, is 



loto-^-lO^'-^ 

 whilst the denominator for integration between limits t aud^-f- 1 is 



J 



io^ ( " 'dt. 



If the above fraction could be expressed in finite terms, there 

 would be no ground for supposing that the value of m for the 

 first interval of age, if multiplied by p*, would represent the 

 value of m for the (£+l)th interval of age. No more is there 

 any ground for supposing that the differential of log e P is equal 



* See Introduction to 'English Life Table/ pp. xxiii & xxiv. 



