LAPLACE. 601 



continually increases with h. Let x^ —y\ then we have to shew 



that 



(/-1)(/+1) 



2/"+^ - 1 



continually increases as y increases from unity : and this is what 

 we have already shewn. 



1034. Laplace's eighth Chapter is entitled Des durees moyennes 

 de la vie, des mariages et des associations quelcoiiques: it occupies 

 pages 408 — 418. 



Suppose we have found from the tables of mortality the 

 mean duration of the life of n infants, where n is a very large 

 number. Laplace proposes to investigate the probability that the 

 deviation of this result from what may be considered to be the 

 true result will lie within assigned limits : by the true result is 

 meant the result which would be obtained if n were infinite. 

 Laplace's analysis is of the same kind as that in his fourth Chapter. 



1035. Laplace then examines the effect which would be 

 produced on the laws of mortality if a particular disease were ex- 

 tinguished, as for example the small-pox. Laplace's investigation 

 resembles that of Daniel Bernoulli, as modified by D'Alembert : 

 see Arts. 402, 405, 483. 



"We will give Laplace's result. In Art. 402, we have arrived 

 at the equation 



dx n mn ' 



fc . 1 1 



where o' = - . Put i for - , and r for — : and let i and r not be 

 ^ s n m 



assumed constant. Thus we have 



da 



-y- =iq — ir. 



ax ^ 



Let V denote e"/^^; thus 



-gv = -{rv; 



therefore qv— constant — Itrv dx. 



