516 
ME. B. GOMPEETZ ON THE SCIENCE 
functions of x to be discovered to meet the cases of comparison of the formula with the 
Tables. Here I retain the d and q with the prefix \ for the sake of convenient reference 
to the old formula; and it appears that these functions and are subject to such 
slow variation by the variation of x, that in forty years in the above examples kd^ only 
varies from 3*88631 to 3*79657, and kq^ from *0126 to about its double; the prefix k 
signifying common logarithm of, and k~^ the anti-common logarithm of. Now supposing 
t and V to be very small quantities, and that kd^ is a function of 1+ w, and kq^ a function 
of l-f^^, then, provided v and t are sufficiently small, whatever these functions may be, 
we consider that if in consequence of the smallness of t and v we may in the develop- 
ment of the functions according to the powers of x be satisfied with the first power, we 
may assume these functions of any form we please to suit our convenience. I will then 
for that end suppose kd^=CC, kq^—kq^.e’’, C standing for kd^^^ and S and e will be 
quantities differing very little from unity, the one being =l-\-v, where, as above shown, 
V must be a very small negative fraction, and the other where ^ is a very small 
affirmative fraction, and then the approximate law of mortality will be 
XL^=C.Q''—k~\e‘.kqo.(x—h)), 
where C, €, e, q^ are all four constant quantities from birth to extreme old age, quan- 
tities to be discovered from the Table of Statements of the living at different ages ; 
C and € first to be determined by two convenient statements, and e, q and h by three 
statements, by the method I call the vital rule of three. First, for finding € and C 
take two values of x, one x=m, the other saying C. S'” and there- 
lore by division £ —\T-> 
kkd 'TifiWdyi, 
n—m 
n—m 
d^ and d^ being found from the Table of data by means of the previously stated original 
formula, 
-X *(X^ — kq^-\-kq, x'^,^ 
which gave when x was 20, kd^^ that is X(?2o=3*88631, and when ir=60, X6?6o=3*79657 ; 
.*. CS'“=3*88631; 0?®"= 3*79657. 
Consequently 
xS"' = *57939-*58954; .*. xS=- ’-^^=-*00025375=1*99974625 
and 
XC=*589540-20xS=*589540+*005075=*594615. 
Art. 5. But instead of proceeding at present with this formula, I will refer to my original 
— ^ 2 ^ • 
formula, L^=^.^j , to show its value, and how q may be found by the vital rule of three 
by a method different from that in my paper of 1825, a mode I then pm*sued before I 
discovered by a general investigation that the above equation with g, q constant quan- 
tities did very approximatively express for a very long period the law of mortality. 
The equation gives XL^=X(?-|-X^.g*, and therefore by taking x successively =w, m-f-w, 
