PHILOSOPHICAL TRANSACTIONS. 



^77 



VOL. LXXVIIlJ 



would be necessary for computing one of the expectations of 2 joint lives. To 

 exemplify this, I shall just set down a few operations for determining the proba- 

 bility of survivorship, according to the Northampton Table of Observations, be- 

 between 2 persons whose common difference of age is 10 years. 



Age ! Age 



of B.of A. 



96 



9^ 

 94 



93 

 92 

 91 

 90 



86 



85 

 84 

 83 

 82 

 81 

 80 



Probability of b's surviving a. 



1 X 145 

 1 • 



X (-^— X 34) = 1173 



2 

 4 + 1 

 4 X 180 '" "^ 2 



9 + 4 



L_x i 



9 X 234 ^ ^ 



X 41 + 17) = .1606.. 

 2 X 48+ 11 9.5) = .2049 



WTW9 X ( -i^ X ^^ + ^3^-^) = -^^2^ 

 <ir7-3:^x(^-x^7 + in9) = .2720 



3TT4-56 X (-f- X 60 + 2259) = .2897 

 46 x-l69X('-^X 63 + 3999) = .3022 



Probability of a's survi- 

 ving B. 



I — .1173 = .8887 

 1 — .1606 = .8394 

 1 — .2049 = .7951 

 1 — .2420 = .7580 

 1 — .2720 = .7280 



1 — .2897 = .7103 



1 — .3022 = .6978 



It may easily be seen, from these specimens, in what manner the probabilities 

 of survivorship between 2 younger lives are deduced from the probabilities be- 

 tween 2 older lives, provided their common difference of age be the same; for 

 the numbers 17 . . 119-5 ... 431.5, &c. in the 2d, 3d, 4th, &c. series, are the 

 sums of the series next preceding. Thus 17 is = 34 X -3- ... 1 19-5 is = 41 X -f- 

 + 17 .... 431.5 is = 48 X V + 1 19-5, &c. It may be necessary to observe 

 further, that if the ages of the 2 persons be equal, the probability of survivor- 

 ship between them being likewise equal, is expressed by the fraction -i-; and that 

 this affords an instance of the accuracy of the foregoing investigation; for the 

 series expressing the probability in this case is the same with this fraction, the 

 chance of survivorship Decoming then (since a =z b\ a =■ b — c\ a" •=. c — d^ 

 &c.; and f^ + c^ X a = T^ + cj X T^ - cj = i^ - c\ &c.) = ^^^ + 

 cc — dd 



2bb 



&C. = 



Mr. Simpson, in his Treatise on Annuities and Reversions, (Lemma 2, p. 100,) 

 has given a curve whose area determines the probability of survivorship between 

 2 persons according to any table of observations. If one of the lives be not 

 very young, so that the equidistant ordinates may not be too few, this area is 

 sufficiently correct. But if the elder of the 2 lives is under 20 years of age, it 

 becomes necessary to assume so many equidistant ordinates to render the solu- 

 tion accurate, when the decrements of life are unequal, that the operation is 

 rendered much too laborious for use; nor do I know that it can be necessary to 



