exhibited in Tables of Statistics. 367 



The numbers for the figures above 242 may easily be similarly 

 worked out, and will present an appearance somewhat symme- 

 trical with the above. It will be thus seen that there are about 

 a dozen numbers, 236 to 248 say, all nearly equally likely to 

 occur with the average one ; about thirty which are at least half 

 as likely to occur as that one. And the above Table shows with 

 what enormous rapidity the numbers increase when we get far 

 away from the limit. 



The above figures are of course obtained by the use of loga- 

 rithms ; it is therefore easy from the logarithms to obtain their 

 reciprocals, which are inserted above in the small figures oppo- 

 site the respective numbers. 



These reciprocals will give a key in any particular case to find 

 the absolute probability of a given number occun'ing ; for it is 

 obvious that the probability of 242 occurring 



1 



1 -I- sum of their reciprocals' 

 Now the sum of the reciprocals may be found approximately 

 without very great labour ; for it will make little difierence in the 

 result if we neglect all that come after such a number as that 

 opposite 202. 



The probability of 242 occurring having been found, we have 

 the key to any problem such as " to find the probability of a num- 

 ber between 222 and 232 occurring ; " " to find the probability 

 of a number less than 222 occurring," &c. : e. g., to find the pro- 

 bability of a number between 212 and 220 inclusive occurring 

 in a given year, "add the reciprocals opposite the numbers, 

 divide by 9, and multiply the result by the probability of 242 

 occurring," &c. 



I should suggest as likely to be a very useful application of 

 the above, the solution of problems relating to the amount of 

 capital requisite to ensure the stability of an insurance office 

 with regard to fortuitous fluctuations, problems with regard to 

 laying aside of bonus additions to policies, &c., especially where 

 the statistics already obtained with regard to the particular 

 class of risks are scanty. 



The fundamental expressions receive another simplification if a 

 is large, that is, supposing us to be supplied with tables extending 

 through a great number of years. The expressions then become — 

 _n—b+\ b 



_[n-b + \){n-b + %)f J)\^ 

 "^- ' lj[b-\)~ \n-b)' 



_[n-b + \){n-b + 2){n^b + Z)( b V 



