THE 



LONDON AND EDINBURGH 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[THIRD SERIES.] 



NOVEMBER 1839. 



L. On the Rule for fading the Value of an Aiiniiity on three 

 Lives. By Augustus De Morgan, Professor of Mathe- 

 matics in University College.^ 



npHE rule which was given by Thomas Simpson for finding 

 -*- the vahie of an annuity on three lives is as follows: Let 

 A, B, and C be the three lives and let K be a single life, the 

 annuity on which is equal to an annuity on the joint lives of 

 B and C; then the value of an annuity on the joint lives of 

 A and K is the required annuity on the joint lives of A, B, 

 and C. The life A should be the youngest of the three. 



No reason has ever been given for this rule, simple as it is ; 

 and the only considerations of which I know, partaking of 

 the nature of demonstration, and connected with it, are those 

 by which Mr. Milne has shown that A should be the youngest 

 of the three, and also that, instead of interpolating the age of 

 K to a fraction of a year, the complete year of age next above 

 the real age of K should be chosen. 



If the rule be examined by the hypothesis of equal decre- 

 ments, and if 1 —a, 1 —2a, &c. be the chances of A's living 

 one, two, &c. years, and 1 — /3, &c. 1 — y, &c. the same for 

 B and C, the error of the rule, or at least its order, will be 

 represented by the fraction 



— ^, (r = interest of £l for one year) 



whenever a, |3, y are so small that « j3 y is small compared 

 with ;•*. As far as my investigations go, I doubt if, on the 

 hypothesis of equal decrements, the rule would be a good ap- 



* Communicated by the Author. 

 Phil. Mag. S. 3. Vol. 15. No. 97. Nov. 1839. Z 



