expressed by a New Formula. 3 



trial interval of age, in the period of life between the ages of 15 

 and 80 years. These three quantities are (1) the mean ratio of 

 the dying to the living throughout the interval, (2) the differ- 

 ential coefficient of the hyperbolic logarithm of the living at the 

 middle point of the interval, and (3) the finite difference between 

 the hyperbolic logarithms of the numbers living at the ages 

 /and (* + l). That is, 



AP 



p-^=« m =Alog c P f , 



t + i 



if ot t+i be used for differential coefficient — — J **' • 



The first and second of the above quantities have already been 

 shown to be equal to one another very nearly. The near coin- 

 cidence in value of the second and third quantities will be obvious 

 on consideration that, if the rate of decrement throughout the 

 given unit interval of age varies continuously and equably from 

 *t at the beginning to u t +i at the end of the interval, the total 

 effect produced in the diminution of the number living (P*) will 

 be very nearly the same as the total effect in the same interval 

 which would be produced by the rate of decrement ot t+ $ at the 

 middle of the interval, assumed to be constant for the whole in- 

 terval. On reference to Table V. hereunto annexed, it will be 

 seen, for example, how slightly the three quantities above men- 

 tioned differ from perfect equality when the unit interval of age 

 is five years. The remarkable property now mentioned, being 

 possessed in common by all good Tables of mortality, is of great 

 practical importance. For observations, hitherto supposed to 

 give ratios of dying to living only, may henceforth be used as 

 giving directly the finite differences of the logarithms of the 

 living (A log e P,), which finite differences are the essential parts 

 of the Tables of mortality sought to be constructed. In columns 

 5 and 6 of another Table (VII.) hereunto annexed, the reader may 

 see how closely the values of A log P in the English Life Table 

 No 2 for males approach the approximate values obtained as 

 above stated directly from observation. 



When for a particular population the rates of decrement at 

 every age are known, a Table of mortality may be constructed 

 therefrom which will correctly represent the number living or 

 surviving at the end of any entire number of years from birth, 

 out of a given number born alive. Similarly, when the Table of 

 mortality is given, and the number of survivors at every year of 

 age is known, there may be deduced from such Table the rate of 

 decrement for every age. 



Observations for the purpose of determining the laws of mor- 

 tality according to age of particular populations are made in one 



B2 



