Method of computing the Values of Life Contingencies, 269 



ledge," where, for the first time, we see tables for the problems 

 which involve two lives. 



An improvement in the method of calculating tables for 

 two lives has suggested itself to ine, which I have no doubt 

 will be adopted as soon as proposed, and will be carried into 

 effect in future tables. 



If a be the number living at the age m, and v the value 



of 1/. due in a year, the construction of Barrett's tables (in 

 the improved form suggested by Mr. Griffith Davies) is as 

 follows : — 



D = fl v\ N = D , 1 + D„, , o + ■ 



m m ' »i m-j- 1 ' m-\-2 ' 



with other results, not here necessary to be specified. When 

 we come to construct tables for two lives it is necessary that 

 the column answering to D^, say D^^^ ^^, should be of the form 



a ct v\ where k is a function of m and n, which increases 



by a unit when ?« and 7i are both increased by a unit. In 

 Mr. Jones's tables, k is the greater of the two, m and n ; I 

 propose that it should be the half sum of the two, thus : 



In problems in which the order of survivorship does not 

 enter, I do not see any particular advantage in this change; 

 but in all such problems there is a decided and obvious gain. 

 At present two formula must be used, accordingly as the life 

 which is to survive the other is elder or younger of the two ; 

 while with tables calculated in the manner proposed by me, 

 one only would be necessary. And even supposing that no 

 new tables were calculated, it would still be better to teach the 

 method in the manner I propose, while two very simple 

 transformations would reduce any result to those which the 

 present tables require. 



For example, the present value of 1/. payable on the de- 

 cease of a life aged y, on condition that a life aged x is then 

 subsisting, according to the method I propose, is in all cases 



X— \,y — \ x,y ' ^ x,y — I x — l,t/' 



D 



To convert this into the common method (if it can yet be 

 called common, as far as two lives are concerned) the rule is, 



divide every term of the form N or D by v-^^~'^' or 



J x,yx,r/J 



v' '•'~^\ according as x or y is the greater. If then x — 1 be 



