CONNECTED WITH HUMAN" MOETALITY. 
517 
j+2n 
m+2n, we obtain 
.q”", XL„+„=X(Z4'^5''2’"^”’ 
whence we get 
l)=?^L«+«— 1)=XL^+2„— XL„+„; 
and therefore, by division, 
xL„ 
-aL„ 
and 
and we have 
X(XL^+,j XLni+2ra) XLm+n) 
X^= ; 
. ^Dm+7t~ 
^5'— 1) ’ 
and Xy in the application I am about to make will turn out to be a negative quantity, 
or in other words, g ■will be a positive fraction less than unity ; and consequently to find 
g, as a negative number has no logarithm in the positive scale, if we are to proceed by 
logarithms, we must take the logarithm of the positive quantity —Xy, and say 
X(--Xy)=X(XL,„-XL,„+„)~X2™-X(f-l); 
and then g being found, we find d by the equation 
Xd='kL^—'kg.g”'. 
This is the way I have generally proceeded since I discovered the above approximative 
formula of mortality ; but if we only consult this formula in order to find Xd, which it 
contains, which is the only part of it necessary to find C, € of the formula 
XL^—C^—X~^{e^.Xgo-x—h), 
we need not take the trouble to find g ov g of the preceding or old formula, and from 
the equations above put in the form 
Xg.g^=XL^-Xd, Xg.g"^^-=KL^^^-Xd, Xg.g”^^^’^=xL^^,,-xd, 
multiplyiag the first and last together, we have 
(Xy)' 2 “-“=(XL,_X^) X (xL.+.-X<?) ; 
and squaring the middle, we have also 
(Xy)Y-^-=(XL^^„-.XcZ)^; 
these put equal to each other, give 
(Xli^— X(^) X (ALot+2»i — — xdy , 
X^(XL,„ + XL^+2n) = (^hi 
m+B )®— 2XL„,+„X<?; 
and therefore 
and consequently 
\d=- 
2ALm+re+ ^Lct+2b 
Art. 6. The information which we have that the formula 
— X '(X( — \g^-\-x\g^. 
