6 An algebraical Expression of the Values of Lives. 
1+ mr—mx, or s—mx, respectively; p and r being the dif- 
ferences between the eldest and the two younger lives: the pro- 
duct of these will be gs—m(s + 4 + gs)x +m?(1+9+5s)a*— 
mx3, and the area gsv— a m(s + q+ qs)x? + = m* (1 + 
1 A 
gq +5s)x3— —mix+, which must be found for mz = 1, and for 
the given age, and the difference divided by the product. 
When two lives are equal, the mean value of their joint con- 
tinuance, thus approximated, becomes exactly two-thirds of 
that of a single life; of three, two-fourths or a half; of 4, two- 
fifths, of 5 two-sixths, and so forth : whence also we obtain, for 
: 4 ; ’ 
the value of the longest of two lives, > of that of a single life, 
6 i 3‘ 
and for the longest of three aa and we may continue the series 
at pleasure by adding at each step 2 to the numerator and | to 
the denominator. 
According to the usnal method of estimating the value of three 
joint lives from that of two lives, one of which is of equal value 
with two of the three taken together, the result, in the case of 
equal lives, is about = of the value of a single life, instead of 
half; that is, almost 4 percent. too much, an error by no means 
to be neglected in practice. It will be easy to obtain a mere 
correct approximation, from the principles here explained, em- 
ploying any tables of the value of lives that may be preferred. 
Let m be found for the eldest life, by making ~ equal to twice 
its value, increased by the age: and let ¢ and z be found in the 
same manner for the other two lives, so that the numbers of 
survivors may be denoted by 1—ma, a—tr, and s—ux, q being 
t+ én, and s1 + ur: the area will then be (qs — 5 (qsm + 
qu + st)v + 5 (fu + qmu + smt)a*— ——mlux) x, which must 
be taken for the given age of the eldest life and for mv=1, and 
the difference divided by the product of the survivors will give 
the value of the three joint lives, with much greater accuracy 
than it can be determined in the manner directed in the Legacy 
Duty Act. 
This remark is, however, only strictly correct, with regard to 
the precise amount of the error in question, when the age is so 
great, that the different effects of the operation of interest on 
the relative pecuniary values of the lives may be disregarded. 
It is obvious that the preceding calculations are wholly indepen- 
dent of this consideration, giving us only a theoretical mean 
value, 
