expressive of the law of human mortality^ ^c. 525 
interpolations. Thus because ^ (L.o) - (L.) = ,02561, the 
first term of the differences, and \ 
the fifth term of the differences : take the common ratio 
= il 7 and m = , 0256; x(w)= 2,40824. These will give 
X (/>) = ,126 ; p = 1,3365 ; a = io,r=io,X(q)— ,0126, 
^ (0 = ^ te) negative ; XXg = 2,40824. 
— X(, 3365)— ,126 = 2,75526; and = X = 
3,88631, and accordingly, to interpolate the Carlisle table of 
mortality for the ages between lo and 6o, we have for any 
age x, 
N=^ ,88631 — number whose logarithm is (2,88i26+,oi26 jc). 
Here we have formed a theorem for a larger portion of 
time than we had previously done. If by the second method 
the theorem should be required from the data of a larger 
portion of life, we must take r accordingly larger ; thus if a 
be taken lo, r = 12, then the interpolation would be formed 
from an extent of life from 10 to 58 years ; and referring to 
Mr. Milne's tables, our second method would give X N= 
3,89063 — the number whose logarithm is (2,784336 
120948 x); this differs a little from the other, which 
ought to be expected. 
If the portion between 60 and 100 years of Mr. Milne's 
C arlisle table be required to be interpolated by our second 
method, we shall find p = 1,86466 ; X (;?z)=: 1,30812 ; m == 
,20329, &c. and we shall have X =3,79657 — the num- 
ber whose logarithm is (3,74767 -|- ,02706 x). 
This last theorem will give the numbers corresponding to 
the living at 60, 80, and 100, the same as in the table; but 
for the ages 70 and 90, they will differ by about one year : 
