272 ESSAY ON PROBABILITIES. 



equality of the remainders is increased ; it being obvious 

 that any disproportion which exists between two numbers 

 is made larger by taking away the same from both. 



The way to correct the inequality, without altering 

 the actual receipts of the office, is as follows. The pro- 

 portions in which the different ages exist in the office at 

 any one time can be pretty nearly found. Let the office 

 table of premiums be taken, and from it let an average 

 premium be formed, by taking into account as well the 

 several premiums, as the numbers who pay them, Suppose, 

 for instance, that A persons pay the premium a, B pay 

 b, &c. &c. ; then the average premium is found by 

 dividing the sum of the products of A and a, B and b, 

 &c., by the sum of A, B, &c. Let the actual average 

 premium be called P ; and let the average premium, 

 formed in the same manner from a true table of mor- 

 tality (in which a, b, c. are different, but A, B, &c. 

 the same as before), be Q. Let P exceed Q by k per 

 cent, of Q ; then the premiums given by the true table, 

 increased by k per cent, are those which should be sub- 

 stituted for the existing premiums, in order that all 

 inequalities may be corrected, without diminishing the 

 receipts of the office. It matters nothing, in the pre- 

 ceding rule, whether the premiums of what has been 

 called the true table are correct or not, so long as their 

 proportions are correct ; and one office might, by this 

 rule, adopt the proportions of another, without altering 

 its own receipts. 



If such a process as the preceding were performed, 

 deducting from the receipts required by the office the 

 whole expense of management, and afterwards adding 

 the last-mentioned item in equal shares to all the poli- 

 cies, the distribution of the premiums would be theo- 

 retically perfect. It remains to consider the more difficult 

 part of the question, the method of dividing the profits. 



Hitherto, I have had no occasion to speak of a most 

 important difference of system which distinguishes one 

 office from another ; the distinction of mutual and pro- 

 prietary. The former have no capital, except what arises 



