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 William Morgan, Esq.F.R.S. Read Dec. 12, 

 1799. [Phil. Trans. 1800, p. 22.] 



Mr. Morgan having already communicated to the Society the so- 

 lutions of seventeen different problems in the doctrine of contingent 

 reversions depending upon three lives, has been induced, from a wish 

 to complete the subject, to investigate in the present paper seven 

 more problems in which the same number of lives are concerned in 

 the survivorship. These, he tells us, include, as far as he can perceive, 

 all the remaining cases involving those complicated contingencies. 



In examining the investigation of these problems, it appears that 

 the determination of the reversion in some of them depends in each 

 year on the happening of twelve or thirteen different events. These 

 numerous contingencies being all expressed by separate fractions 

 (each of which is resolved into two or more different series) renders 

 the operations exceedingly intricate and laborious. From an appre- 

 hension, it seems, of becoming tedious and diffusive in his demonstra- 

 tions, the author has in general contented himself with merely giving 

 the fractions denoting the contingencies on which the reversion de- 

 pends, without specifying in words at length the nature of those con- 

 tingencies. He has, however, in these as in all the other problems 

 he has investigated, given different demonstrations, both by solving 

 each independent of any other problem, and by deriving the solution 

 from those of two or more problems, which had been already investi- 

 gated; so that from the exact agreement in the results, proofs are de- 

 duced of the perfect accuracy of the demonstration, not only of the 

 problem investigated, but also of those which are applied to the solu- 

 tion. 



In all these problems, a contingency is involved, which having never 

 been accurately determined, had hitherto rendered even an approxi- 

 mation to the solution of them impossible. This contingency is that 

 of one life's failing after another in a given time. This appears to 

 have been ascertained with sufficient accuracy to enable the author 

 to surmount a difficulty in the solution of these problems, which he 

 owns he had once considered as insuperable. 



Having thus accomplished the investigation of every case in which 

 he conceives it possible that the contingency may be varied between 

 these lives, he conceives that he has now exhausted the subject ; and 

 concludes his paper with observing, that those cases in which four 

 lives are involved in the survivorship are not only too numerous and 

 complicated to admit of solution, but that they occur so seldom in 

 practice as to render the labour of such solution (if it were practicable) 

 both useless and unnecessary. 



