DE, FAEE ON THE CONSTEHCTION OE LIFE-TABLES. 
845 
In the equation -^=v, where s indicates space, t time, v velocity, the units of measure 
must be fixed before numbers can be inserted in the general expression ; and then v will 
express, in the measure that has been applied to space, the number of such units of 
space described in one unit of time. Here v is a ratio ; it is the rate at which the body 
d 
moves : and in the same manner m, in the equation is the rate of dying, that is, 
as I shall express it, the mortality ; or it is the ratio of the dying to the living in a given 
unit of time, the time during which the deaths occur being of precisely the same dura- 
tion as the time during which the living are under observation, 
I (li\ing duiing 1 year) : d (dying durmg a year) : : 1 (year of life) : m. 
If for I the number 100,000 is substituted, it is assumed that immediately a death 
occurs another life is substituted; and as the time is a year, then 760 will represent the 
value of d at the age 20, according to the preceding Table ; -00760. If the time, 
instead of one year, be the thousandth part of one year, then m= -0000076; and if the 
time be infinitely short, m will be infinitely small : w is a ratio ; the quantity of life 
existing during the time is represented by 1, and the quantity of life destroyed by a 
fraction, m. Whether the life inheres in the first organic molecule after conception, in 
the infant, or in the man, the vital action has a certain force of continuance, which is 
constantly varying ; and the amount of this force that is extinguished at a given instant 
of time will be represented by the force of mortality, namely, by m at that instant. 
Then let the age x—z-\-a, where a represents the number of years up to the age at 
which a given rate (r) of increase oim begins; then z=x—a. And the mortality at 
any instant of age, in an instant of time at the end of z years or parts of years, will be 
mr*. Now let y represent the li-ving at that precise age ; then the decrement of y in an 
infinitely short time will be — dy=ymr^dz\ the dy being negative as it is taken in a 
direction opposite to that in which the ordinate y of the curve is assumed to be drawn. 
Transferring to the other side of the equation, this becomes ———mr^dz-, and inte- 
grating both sides, we have ig^y being put for the hyperbolic logarithm of y, and \fi 
for the difference between the constants of the two integrals) — 
and 
(3.) 
( 2 .) 
( 1 -) 
When z is made zero, let y—\’, then \y will also disappear, and X,c=—. 
Upon 
substituting this value of in equation (2.), it becomes 
( 4 .) 
5 T 
MDCCCLIi. 
