545 
expressive of the law of human mortality, &c. 
I observe that I have not given any table of the logarithm 
of the accommodated ratios for an unlimited term, except 
that calculated with 5 per cent, as a radix ; but by the assist- 
ance of a table of life annuities, for single life at different 
rates per cent., this will enable us, independent of certain 
exceptions, to derive the quantity for the same rates per cent, 
for any radix at the per cent, contained in the second table ; 
thus to find R Carlisle mortality, radix 8 per cent. I look to 
the Carlisle table of single lives at 8 per cent., and I find the 
value of the annuity on the life of 50 = 8.987, I search the 
age to which this will correspond at 5 per cent, and I find 
sufficiently nearly 59,82 for the age corresponding, to which 
from my table (with the radix at 5 per cent.) for the log. 
of ratios I find T.97536 ; to this I add log. of ; that is, 
,01223, and we get 1.98759^ the same as given on the other 
side. This method is accurately consistent with the defi- 
nition of accommodated ratios for unlimited periods ; and if 
this description of accommodated ratios at a certain rate per 
cent, be given for one table, for which at the same rate per 
cent, we have the value of single lives, we may find the 
same description of accommodated ratios for any other table 
of mortality for which, at the same rate per cent, we have a 
table of the value of single lives : thus, suppose the logarithm 
of this description of accommodated ratios be given for the 
Carlisle table at five per cent., and the same be required for 
1,05 
-1 
60 
the Northampton for the age 60, at the same rate ; 
Northampton = 8,392, this being sought in the Carlisle 
