Comparison of different Tables of Mortality. 343 



pearing to be represented by a geometrical progression of divisors ; 

 so that we may suppose the divisor to be doubled once in every 

 ten years that the age falls short of 115 ; while, in the Northampton 

 table, which approaches very near to the law of the arithmetical 

 hypothesis, the divisor requires to be doubled more nearly once in 

 22 years. 



4. The exponential hypothesis affords us, as I have shown by an 

 example, a ready mode of computing the number of survivors at a 

 given age, as required by the supposed law of the divisors ; but if 

 we proceed to compute by it the expectation of life, or the value of 

 an annuity, it leads, in the simplest cases, to a transcendental 

 quantity, which has long served for the amusement or for the 

 torment of the most refined mathematicians, under the name of a 

 logologarithmic integral, without having been rendered the more 

 manageable by all their elaborate investigations. 



5. Still less would it be practicable to make any use of an addi- 

 tional exponential term, which might be made to express with great 

 accuracy the decreasing mortality of early infancy and childhood. A 

 difficulty nearly similar occurs also in computing from an expression 

 which I had deduced from the equable variation of the value of an 

 annuity under certain circumstances; a property which I have lately 

 employed, as you will recollect, for facilitating the valuation of out- 

 standing policies for insurance. This formula for the decrement was 



, which leads to the same hyperlogarithmic series as the ex- 



ponential hypothesis. 



6. We may form a correct conception of the character of the 

 exponential hypothesis by laying down, in the diagram of my paper 

 in the Transactions, the numbers of the table that I have published 

 in my letter to Mr. Morgan, taking f of the quinquennial differences 

 for the comparative annual mortality : and it will be found that the 

 curve thus obtained, approaches, in its general appearance, sur- 

 prisingly near to that of the Carlisle table, and considerably re 

 sembles the curves of Deparcieux and of Finlaison, especially 

 between the ages of 40 and 80. 



