XX11 APPENDIX THE THIRD. 



where A + AA is the value of a life one year younger 

 than the office age of the party at entry. 



When an annuity is granted upon condition that the 

 executors of the party are to receive such a proportion 

 of payment for the year in which the annuitant dies, as 

 corresponds to the portion of the year during which he 

 is alive, the addition to A the value of the annuity is 

 the present value of \ + ^ r of a year's purchase, pay- 

 able at the end of the year of death. The formula 

 which then very nearly represents the result of the pre- 

 ceding correction is 



A (1-5) +4-5 



or to the tabular value of the annuity add the excess of 

 half a year's purchase over half a year's interest of the 

 tabular value, together with one- third of the interest of If. 



APPENDIX THE THIRD. 



ON THE BULB FOB DETERMINING THE PROBABILITIES 

 OF SURVIVORSHIP. 



IT does not seem to have been noticed, that this rule 

 is considerably more correct than its framers could have 

 anticipated, supposing them to have contemplated no 

 higher degree of exactness than their demonstration en- 

 titled them to assert. Let< and -fyt represent the pro- 

 babilities that A and B, now alive, shall be alive at the 

 end of t years, t being whole or fractional. Then 

 \l/'t.dt is the probability that B shall die between 

 t and t + dt and ftyt. tyt dt from t=n to t=n+l 

 is the chance of a survivorship of A beginning to take 

 place somewhere in the (n-fl) th year after the present 

 time. Let a and a + Act and /3 and /3 + A/3 be the 

 chances that A and B shall be alive at the end of n and 

 years from this time : then, in the demonstration 



