different Tables ofMortalify. 349 



It appears, from this comparison, that we approach very near to 

 the expectation of life at Carlisle, by taking the mean of e and e ', 

 or by making o = 88. 5 : and from 10 to 70 the formula appears to 

 represent the mortality more correctly than the tables, which are 

 extremely irregular in their differences^ probably on account of the 

 very small population on which the observations were made. 



16. The only remaining determination to be considered, that is 

 exempt from the effect of interest, is that of the probability of sur- 

 vivorship between two live j ; a probability which is made up of the 

 sum of the probabilities of survivorship for every year, or every 

 portion of a year, throughout the full range of the life of the eldest; 

 that is, the probability that the one will die within the element of time 

 considered, while the other survives : so that the fluxion of the pro- 



s ds' 



babihty is : k being the number surviving at the age of the 



k k' 



eldest, qy and k' at that of the youngest, while s and / represent the 



variable number of survivors. 



17. In the arithmetical hypothesis we have constantly ds= — _f , 



c 



and the fluxion of this probability is — . — .; which is equal to the 



k ck 



fluxion of the expectation e, divided by ck\ and the fluent being 



taken between the same limits x = q and j; = c in both cases, it 



follows that—- is, in this hypothesis, the value of the probability 

 ckr 



that the younger life will fail first; and, since e'=: jL, (14) and 



A:' = 1 — _2_ , we have — ::^ — : a. very simple consequence of 

 c ck' 2e 



this hypothesis, which appears hitherto to have escaped observation. 



The PROBABILITY, therefore, that the younger of two lives 



WILL FAIL before THE ELDER, IS EXPRESSED BY THE EXPECTATION 

 OF THE ELDER DIVIDED BY TWICE THAT OF THE YOUNGER. And 



it is obvious that by taking the several expectations as directly 

 computed from the tables, this determination may be extended, as 

 a good approximation, to the utmost limits of the observations. 

 OCT.— DEC. 1828. 2 B 



