VOL. LXXVIII.] 



PHILOSOPHICAL TRANSACTIONS. 



481 



S X 



c 

 2br 



X ( h — ^ ~ — — ^%--^ &c.) Let B denote the value of 



^cT acr ' cr* acr* ' 



an annuity on the life of b, c the value of an annuity on a life 1 year older than 

 B, AB and AC the values of annuities on the joint lives of a and b and of a and c, 



and these series will be = - ^ T — —"VaT"""" • -^S^^"* ^^^ ^^ series 



2br 



above mentioned, or ^ X (-^j^ -\ — 7- H — 7-7 &c.), by pursuing the same steps, may 



^abr 



/3 X 3 



be found = —xj— X (k — ak) — 



s (b — ab) 



where |3 denotes the number of 



2b '^ ^'~ ' 2r 



persons living at the age of a person 1 year younger than b, k the value of an 

 annuity on that life, and ak the value of an annuity on the joint lives of a 



and K. The whole value of the survivorship is therefore = s X 



/r — 1) . (b — ab) , /3.(k — ak) c.(a — ac) 



Having now the value of the sum s payable on the contingency of b's surviving 

 A, the value of the same sum, payable on the contingency of a's surviving b, is 

 easily obtained by the well known method of subtracting the value found above 

 from the whole value of the reversion after the extinction of the joint lives of a 

 and b. It is evident that the exactness of the above rule must depend on the ac- 

 curacy with which the values of the single and joint lives are computed. Being 

 possessed of such tables for all ages, even with respect to the joint lives, I have 

 computed the following values, that it may be seen how far Mr. Simpson's ap- 

 proximation*, the only rule now in use, may be depended on. 



< 



10 

 10 

 20 

 20 

 20 

 30 

 30 

 30 

 40 



+ Value of 100/. 

 payable on the 

 death of a if b 

 survives him. 



True 

 value. 



32.67 

 24.74 



29-99 

 22.11 

 27.95 

 28.79 

 19.84 

 30.22 

 26.65 



Simp- 

 son's 

 value. 



2605 

 24.75 

 24.73 

 23.50 

 27.96 

 22.60 

 21.47 

 30.21 

 20.07 



bfl 



40 

 40 

 50 

 50 

 50 

 50 

 60 

 60 

 60 



Value of 100/. 

 payable on the 

 death of a if b 

 survives him. 



True 

 value 



10 17 



40 32 



2 23 



10 14 



20 18, 



50, 35 



2 21, 



10' 10, 



30, 17. 



,10 

 .86 

 36 

 10 

 65 

 SO 

 52 

 65 

 51 



Simp- 

 son's 

 value. 



19.07 

 32.87 

 17.06 

 16.21 

 19-29 

 35.85 

 13.61 

 12.93 

 18.19 



• It must be remembered, that the correction explained by Dr. Price, in vol. i. p. 39, &c. of his 

 Treatise on Reversionary Payments, must be applied to Mr. Simpson's rule ; that i,s, when the re- 

 version is a sum and not an estate, the value found by the rule must be divided by 1/. increased by its 

 interest for a year. 



+ These values have been computed at 3 per cent, and from the Northampton Table of Ob- 

 servations. • / 

 VOL. XVI, 3 Q 



