VOL. LXXXI.j PHILOSOPHICAL TRANSACTIONS. 73 



which it is added, I have now given general rules for determining the values of 

 reversions depending on 3 lives in every case which, as far as I can discover, will 

 admit of an exact solution. The remaining cases, which are nearly equal m 

 number to those I have investigated, involve a contingency for which it appears 

 very difficult to find such a general expression as shall not render the rules much 

 too complicated and laborious. The contingency to which I refer is that of one 

 life's failing after another in any given time. The fractions expressing this pro- 

 bability are every year increasing, so that the value of the reversion must be repre- 

 sented by as many series at least as are equal to the difference between the age ot 

 one of the lives, and that of the oldest life in the table of observations. I have 

 indeed so far succeeded in the method of approximation as that the reversion 

 may be generally ascertained within about -^V part of its exact value; but I shall 

 not trouble the r. s. at present with these investigations. 



The 34th, 35th, and Sdth problems in Mr. Simpson's Select Exercises involve 

 this contingency, and, by the assistance of M. de Moivre's hypothesis, admit of 

 an easy solution. But such is the fallacy of this hypothesis, that it renders Mr, 

 Simpson's conclusions obviously wrong, though his reasoning is perfectly correct; 

 for it cannot surely be an equal chance in all cases that one life shall die after an- 

 other. In the short term of a single year the chances are indeed so nearly equal, 

 that it would be wrong to perplex the solution by attempting greater accuracy. 

 But when the number of years, and the difference between the ages of the 2 

 lives are considerable, those chances must vary in proportion; and therefore, un- 

 less the contingency is blended with another which shall very much diminish the 

 probability of the event, the solution, by thus indiscriminately supposing the 

 chances to be equal, must be rendered extremely inaccurate. In Mr. Simpson's 

 36th problem the solution by this means appears to be absurd: for, in the parti- 

 cular case in which c is the oldest of the 3 lives, the value of the reversionary 

 annuity becomes = ^c — ^ac; that is, the value of an annuity in this case 

 during the life of c after b and A, provided a dies first, is the same whatever be 

 the age of b; for no mention is made of his life in the foregoing expression. It 

 should be observed however, that the rule itself is strictly true, and that the error 

 arises from Mr. Simpson's having been misled by the hypothesis in determining 

 the probability of b's dying after a in his investigation of the 34th problem, which 

 is applied to the solution of this problem.* 



I have declined giving specimens of the different values of the reversions as 

 deduced from the foregoing rules and those which have been hitherto published, 

 not only from an apprehension of becoming tedious, but also from the conviction 

 that at present they are unnecessary; those which I have formerly given being, 

 I think, sufficient to prove the inaccuracy of M. de Moivre's hypothesis. In 



* It is proper to observe, that I have followed Mr. Simpson's method of determining this contin- 

 gency in the 23d, 27th, 28th, and 29th problems in my Treatise on Annuities. — Orig. 

 VOL. XVII. L 



