DE. FAEE ON THE CONSTEUCTION OF LIFE-TABLES. 
867 
NOTE ON THE TWO HYPOTHESES. 
Let 5 be the decrement of the ordinate in a 
unit of time, then the decrement Ay of the ordi- 
nate in the time x, represented by the abscissa, 
will be Ay=—bx, on Demoivee’s hypothesis; 
and as it is always proportional to the time, it 
will be in an infinitely short time dy— — bdx. 
Passing to the integral — hx. And if y—a 
at the origin when 0, c=«, y~a—bx. And 
if 5 = 1 , then y = a — x. This evidently represents 
very closely short portions of the Life-Table 
cnrv^e ; and the smaller x is taken, the nearer is the 
approximation to the corresponding value of y. 
Again, let Ay be the decrement of the ordinate 
y in the indefinite time Aa‘ represented by the 
abscissa; and let the mortality (in) represented 
by the ratio of the area abfg to the area dfg be 
^=mo. Let also nio increase at the rate r in a 
unit of time, so that j^=mi=mor, and 
generally ufithin given limits then 
Ay=— y/n^Aa’ nearly, Aa’ being any small por- 
tion of time. 
The eiTor increases as the time Aa’ is extended, 
from the circumstance that on the one hand 
varies by hypothesis momentarily, and that y, 
from which the varying proportional part is taken, 
constantly grows shorter. But by passing to the 
limit and making the time dx infinitely short, 
and y diu’ing that infinitely short time may 
be considered constant, and dy= —yinjdx will be 
the true decrement. Substituting for 
the equation becomes (Zy= —ym^dx, from which 
the value of y can be derived, as before shown. 
For y = — m^r^dx^ and integrating both sides 
Tfl 
'K^y—\c — Here Xj stands for the logarithm 
having s for its base. 
Ages. 
Numbers k - 
living. 30 31 32 40 41 42 
At the origin of the curve, when a-=0, let y — \, and then Now substituting 
