VOL. LXXXIV.^ PHILOSOPHICAL TRANSACTIONS. 417 



^"^ + '^^ &c 



25.10 ^ 33.10' 



2.9.10 ^ 37.10 ' 



27.10 ^ 35.10 ' 

 3c 4d 5 



7.80 15.10 ' 23.10 31.10 ' 3.9.10 



By which series the arithmetical computation will be much more easy than by 

 the original series, 



XVII. On the Method of Determining, from the Real Probabilities of Life, the 

 Values of Contingent Reversions, in which Three Lives are involved in the 

 Survivorship. By Wm. Morgan, Esq., F. R. S. p. 223. 



In the last paper, says Mr. M., which I communicated to the r. s. on the 

 doctrine of survivorships, I concluded that, as far as my own judgment could 

 discover, I had then given rules for determining the values of reversions depend- 

 ing on 3 lives in every case which admitted of an exact solution, and that the re- 

 maining cases, which were nearly equal in number to those I had already investi-. 

 gated, involved a contingency for which it appeared very difficult to find such a 

 general expression as should not render the rules too complicated and laborious. 

 Since that period I have bestowed much time and attention on this subject, and 

 have at length so far succeeded as to give reason now to hope that it is capable of 

 being entirely exhausted. It is not my present design to enter into the investi- 

 gation of all the problems which still remain to be solved. I shall here confine 

 myself to a few of the most important, reserving the conclusion of the subject 

 for some future opportunity. 



The contingency to which I have alluded, as opposing the great difficulty in 

 those problems which I have not yet solved, is that of one life's failing after an- 

 other in a given time. It becomes necessary therefore, previous to any other 

 investigation, to deduce a general method of ascertaining such an event, and for 

 this purpose I shall subjoin the following lemma : viz. To determine, from any 

 table of observations, the probability that b the elder dies after a the younger of 

 2 lives, either in any given number of years, or during the whole continuance of 

 the life of b. From the analytical solution of this problem, Mr, M, deduces 

 the following table. 



VOL. XVII. 



3H 



