391 



The other part of the Paper of 1825 is devoted to a mode with 

 accompanying tables, to enable a calculator with any given table of 

 mortality to calculate the value of an annuity on any number of 

 joint lives to within any degree of accuracy at pleasure ; provided 

 the periods, if long, are divided into parts. 



The present supplement, written so many years after the original 

 papers were communicated to the Society, appeared to the author to 

 be called for in order that he might further illustrate the subject, and 

 add his later speculations and improvements. The supplement the 

 author commenced writing about two years ago from collections of 

 his manuscripts, but was prevented by continued ill-health from pro- 

 ceeding with it ; but having received an invitation last year from the 

 International Congress to assist in their meritorious labours, and being 

 unable from indisposition to attend the Meeting, as he stated to the 

 Congress, he offered to send them some hints respecting his recent 

 labours, the results of which he intended to present to the Royal 

 Society, if his health should permit him to finish the papers. The 

 hints which he gave, which were honoured by a place in the Reports 

 of the Congress, were, he believes, deemed interesting, but he thinks 

 they were sufficiently separated from the strictly mathematical part 

 of the subject, and also from the most important portions of the 

 result, to allow him to consider the present paper a new work, or suf- 

 ficiently new, to be thought worthy of presentation to the Royal 

 Society, and especially because since the notice above alluded to was 

 furnished, the author has been able to introduce important improve- 

 ments. The original formulae give one uniform law of mortality from 

 birth to the utmost limit of the table ; but in the case of the table 

 presented to the Congress, and of the present further improved for- 

 mulae, if a comparison be made, for instance, with the Carlisle table 

 from birth to the age of 100, and even from birth to one month, to 

 two months, to three months, to six months, and to one year, where 

 the tables appear to be so irregular, and formed on no law, the result 

 appears to show that a law really exists and is available. 



The author shows the vast use and applicability to solutions of all 

 intricate inquiries of life contingencies of his present deductions. He 

 gives a theory, which he believes to be quite new, which he calls 

 special, single, and specially influenced contingencies ; a subject 

 which he states he had not lost sight of when officially engaged in 



