the decrement of human life. 289 
The terms of this formula have some remarkable relations 
to the different periods of life. Halley’s first approximation 
was y = 1000, throughout life. Demoivre’s arithmetical hy- 
pothesis wasy = = 1163 : but of the present formula 
the principal foundation, as extending to the whole of life, 
is, y = 368 -j- 10 x. In infancy the term containing the re- 
ciprocal of the powers of x has a preponderating value : in 
3 _ 
youth, the term — (i56-|-2o.r — xx ) 2 , which diminishes 
the mortality, ends somewhat abruptly at 25, and would be 
incapable of being employed with safety in algebraical cal- 
culations, from its having a negative as well as a positive 
value. Old age is expressed almost exclusively by the high 
powers at the end of the formula, which terminate the series 
with great and increasing rapidity. It is obvious that for 
many purposes of calculation, the terms belonging to youth 
and to old age might be neglected without inconvenience, 
and that, for the middle portion of life, the terms 368 -f- 10 x 
alone, with some little modification, might be employed as 
sufficiently correct ; or certainly as much nearer to the truth 
than either the arithmetical or geometrical hypothesis of 
Demoivre. The relations of the different parts of the for- 
mula will be best appreciated from their developement in the 
following tables. 
