480 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1788. 



Ages. 



61., 



62., 



63. 



64., 



65.. 



66., 



67.. 7 



68.. 8 



69.. 9 



71.. 1 



72.. 2 



73.. 3 



74.. 4 



75.. 3 



76". . 6 



77.. 7 



Probabilities. 



34.94 

 24.13 

 19.52 

 16.19 

 14.28 

 12.37 

 10.92 

 9-82 

 9.12 



30.49 

 I9.+5 

 14.87 

 11.59 

 9.80 

 7.98 

 6.60 



65.06 



75.87 

 80.48 

 82.81 

 85.72 

 87.63 

 89.08 

 90.18 

 90.88 



69.51 

 80.55 

 85.13 

 88.41 

 90.20 

 92.02 

 93.40 



Ages. 



70. . 10 

 71.. 11 

 72.. 12 

 73.. 13 

 74.. 14 

 75.. 15 

 76. . 16 

 77.. 17 

 78.. 18 



Sixty Years Difference. 

 Probabilities. Ages. Probabilities. 



8.73 

 8.4<) 

 8.25 

 8.05 

 7.88 

 7-75 

 7.6i^ 

 7.59 

 7.43 



91.27 

 91.54 

 91.75 

 91.95 

 92.12 

 92.25 

 92.32 

 92.41 

 92.57 



79.. 19 

 80. . 20 

 81. .21 

 82.. 22 

 83. . 23 

 84. . 24 

 85.. 25 

 86. . 26 

 87.. 27 



7.19 

 6.90 

 6.55 

 6.16 

 5.81 

 5.56 

 5.32 

 5.11 

 4.90 



92.81 

 93.10 

 93.45 

 03.84 

 94.19 

 94.44 

 94.68 



94.89 

 95.10 



Seventy Years Diff'erence. 



3.68 

 3.70 

 3.73 

 3.75 

 3.67 

 3.45 

 3.09 



96.32 

 96.30 

 96.27 

 96.23 

 96.33 

 96.55 

 96.91 



88. .28 

 89. . 29 

 90. . 30 



91..31 

 92. . 32 

 93. . 33 

 94-. . 34 

 95. . 35 

 96. . 36 



92. . 22 

 93. . 23 

 94. . 24 

 95.,. 25 

 96. . 26 



Probabilities. 



4.73 

 4.48 

 4.13 

 3.63 

 3.10 

 2.48 

 1.94 

 1.40 

 0.95 



95.26 

 95.52 

 95.87 

 96.37 

 96.90 

 97.52 

 98.06 

 98.60 

 9905 



2.63 



2 



1.64 



1.18 







1097 



80 99 



97.37 

 .90 



98.36 

 98.82 

 20 



98.75 

 99.03 

 99.30 

 99.51 



Ages. 



91.. 

 92., 

 93. 



Ninety Years Diff'erence. 

 Probabilities. Ages. Probabilities. 



18.93181.07 

 9.1390.82 

 5.5394.46 



94.. 

 95.. 

 96.. 6 



3.12 

 2.12 

 1.15 



96.88 

 97.88 

 98.58 



Prob. 1. Supposing the ages of a and b to be given ; to determine, from 

 any table of observations, the present value of the sum s payable on the con- 

 tingency of one life's surviving the other. 



Solution. — Let r denote l/. increased by its interest for a year, and let all the 

 other symbols be the same as in the preceding problem. Let the life of b also 

 be supposed to be the older of the 2 lives ; and then it will follow, by reasoning 

 as in the solution of that problem, that the present value of s to be received on 

 the death of a, should that happen in the life time of b, will be expressed by the 



,b + c , , c-]r d „ . e + e ,„ . c -\- f of/ o \ mi . . i 



series s X \-^:n cc -J- — 1„2« + ^rm, « + T7zi:!i ^ &c.) Ihis series may be re- 



labr ' ^.ahr"^ 



solved into the 2 following 



da'" 

 abr^ 

 ca—ca 



abr 



,ba' 1^ ca" 

 ^abr abr'^ 



+ 



+S^'=-) 



;-x 



' 2abr^ 

 , ca' . da! 



+ ^, +{":. &c.) + 



into 



-Xf-- 



^br 



d_ 

 br'' 



"t" U-Jl 



^nbr ' abr'^ ' abr^ ' abr'^ 



The first of these 2 series may be again resolved 



da — da' — da" ^^ e ea — ea' — ea" 



abr"^ br^ abr^ 



&C.) - 



