An algebraical Expression of the Values of Lives. 5 
The advantage of this method of calculation is strongly ex- 
emplified in the determination of the value of the joint continu- 
ance of two or more lives, which may be obtained by means of 
‘the quadrature of a curve having its ordinates in the joint ratio 
of the survivors at the time expressed by the absciss ; the cor- 
rected area being divided by the product of the numbers of sur- 
Vivors at its commencement, which obviously expresses the num- 
ber of all the possible combinations of their lives, as the area 
does the aggregate duration of those combinations. 
For this purpose it will be convenient to represent the num- 
ber of deaths by the formula -38 + -0022*—-00000137.23, which 
will be sufficiently accurate for any age exceeding 10 years; and 
if we make a= 62, } =-0002, and c=-00000137, and call the 
two ages x and «+7, the product will be d + ex + fx*+ gui + 
hat + ix + kx®, where d=a?— abn? + acn}, e= 3acn?—2albn, 
S = b’n* —2ab + 3acn —ben3, g = 2b’n—Abcen* + 2de + c*n3, 
h= l?—Sdben + 3c*n*, i= 3c*n—2bce, and k= c*; and the area 
of the curve will be dz + = en" + = fai + 2 gxt+ 3 has +- 
as ‘ ; 
= ae + = kx, which must be found for the given age, and for 
x + n= 98 or 100, and the difference, divided by the product, 
will show the value of the joint lives. 
But in pursuing the calculation for a greater number of lives, 
it would be necessary to assume a still simpler expression for 
sittey. 1 
the number of deaths, such as ma, m being <—, or from — to 
1 : 4 see 
4p according to circumstances, retaining the more accurate ex- 
pression for the elder lives ; and taking for the age of the younger 
x—p, and for the number of the survivors |-—mx + mp, which 
may be called g—mzx: and the former product might be mul- 
tiplied by this for the case of three lives, and the area found as 
before. Indeed this expression may be employed for the younger 
- of two lives without material indecuracy, the product becoming 
ag —amx — lgx* + (cq + bm) «} — cmx*, and the area aqgx— 
1 i 1 1 
= amx* — = hq#? “3 eg =F bm) «+— = emu’; whence, for 
example, when the ages are 10 and 20, supposing m = ‘012, 
we obtain the mean joint value 22°9: nor would the result in 
this case be materially different if we employed the same simple 
estimate of the mortality with respect to both lives, though it 
would vary more at other ages. We may however safely make 
the value of m ='012 between the ages of 20 and 60 in Lon- 
don, even for the case of three joint lives, the number of sur- 
vivors being called 1—mv, 1 + mp—mx, or g—mwx, and 
A3 1 + 
