528 
J\IE. B. G-OMPEETZ OJf THE SCIEXCE 
Multiply the first and third of the equations together, and square the second, and we 
shall have by reduction, putting 
that is. 
that is. 
consequently 
and therefore 
AMmXXMm+ 2 » ^ m—h.m + 2n—h ^ 
{m + n—h—n) x {rn + n—h + ri) 
m 
+ n—}if 
=Q„ 
m + n—h]^ — w® Pj 
— ^tn ? 
{m+n — hy 
m-\-n—h]^ X 1 — 
h='m-\-n- 
and m, n, >iL,„+ 2 « being taken from the Table for the given or assumed 
values of m and and C and S having been found, we have] the value of h ; and from 
the equation 
. a; — A . X$'o = 
we have, by putting m and m+w for x, the two equations 
e”* X >^2'o X m — A = 
and 
+ w— A= 
and consequently we have 
XM.^+n'X.m — h 
xMm xm + n—li 
and therefore 
+ X( A — m) — X( — XM„) — X(A — m — %), 
because XM^, &c. is negative, and therefore we cannot take its logarithm. Now having 
found A and e, we find from the equation 
. m — A . Xg-o = XM,„, 
which will give 
XXg-o = X( — XM,„) — X(A — m) — rrike ; 
and taking for m and w 20 and 30 respectively, which will give the data for the selec- 
tion (by the vital rule of three) 20, 50, 80, we are to expect, if the theorem is an 
approximation to the law of mortality, that it will very nearly agree with the Tables of 
mortality for every age, from the age of 20 to 80, which, by the example I am about to 
give, will be found to be the case. And to proceed, from having found 
XC='59461, Ag=l-999746, 
we have 
XCg'"=: -589443 ; XCg'"= -581882 ; XCg“=-57431.; 
a''’= 3-88631; a"’=3-81882; Ce'“=3-75249, 
