340 A Letter to William Morgan, Esq. F.R.S., 



86 only. The mean of the numbers here computed is 137: 

 and nearly in the proportion of 86 or 87 to 137 does the ex- 

 pectation of life, as exhibited by this table, exceed the estimate 

 of the Northampton table: a result not materially differing 

 from the proportion of 2 to 3, which you assign. 



2. But a more remarkable peculiarity of the decrements, or 

 rather the decremental quotients, derived from your table, is 

 the regularity which is observable in their progress after the 

 period of middle life; each of the numbers, which express 

 them, being precisely or very nearly the half of the preceding 



number. Thus, disregarding fractions, we have — m 67 for 



69, ^- = 34, 5^- = 17, and —■ = 8 for 85, which is equiva- 

 lent to 7 at 87^. 



3. We may therefore continue this series with perfect con- 

 fidence, until the whole number of lives is exhausted, taking 

 the annual decrement at 95, J, at 105,J, and at 115,1, which 

 may be supposed to be a sufficient age for the termination of 

 our computations. 



4. The decremental quotient in your table, , is very 



nearly — ; y being — — — , x the age, and z the number living : 



for this expression gives us, from 25 to 85, (512, 256,) 128, 

 64, 82, 16, 8 : and if we wish to modify the formula, we may 



make it more generally £. — = «y, and y = b — c x; so that 



the computation might be adapted to the earlier ages, if we 

 had sufficient documents for the purpose. 



5. We might at once form a table of mortality from the 

 quotients thus computed, proceeding downwards from a single 

 life at the age of 1 15 : but it will be much more convenient, and 

 perhaps equally accurate, to employ the method of fluxions. 



6. Since ~ = a y A x, A x being = 1, y = b — ex, and 

 Ay =—cAx, we have -r = a y . :-—•; whence, substitu- 

 ting, as usual, -r- for —— , we have the equation -— = a y . —?-■> 

 of which the fluent is hlz = a y . — -j — j- /; which becomes, for 



chla •* ' ' 



the values a = \ and c = T \,,/- -^7, or hi* =/- 14 ' 4 ^ 695; 



which, when y is 1 1*5, becomes^/— '0049926; and if we suppose 

 the number born to be 100,000, 1 1-5129254 =/— -0049926, 



and 



