XXIV APPENDIX THE FOURTH. 



correction will have the positive sign when & c bears a 

 greater proportion to a b than does q r to p q: 

 that is, when the mortality in A's part of the table is in- 

 creasing faster than in B's. Now, ceteris paribus, the 

 larger the comparative mortality of the year succeeding 

 a given year, the more likely are the deaths of the latter 

 part of that given year to predominate over those of the 

 former ; consequently, the more likely is the death of 

 A, if it happen in that year, to be towards the end of it. 

 But any thing which shows that the death of A is more 

 likely than before to take place later in its year, increases 

 the probability that a survivorship commencing in that 

 year shall be in favour of A, and not of B. 



APPENDIX THE FOURTH. 



ON THE AVERAGE RESULT OF A NUMBER OF OB- 

 SERVATIONS. 



THAT I might not further embarrass the most ab- 

 struse chapter of this work, by the introduction of an 

 isolated point of difficulty, I have chosen here to men- 

 tion some considerations connected with the value of 

 the average of observations. There is a remarkable 

 difference of principle between two problems which at 

 first sight appear identical; namely, where it is re- 

 quired to invent a method of treating observations be- 

 fore they are made, and after they are made. Positive 

 and negative errors being equally likely, and no observ- 

 ations having been made, it is easily proved that there 

 is a high probability in favour of a large number of 

 observations giving exactly or nearly the same total 

 amount of one as of the other. The case is analogous 

 to that of an urn filled with black and white balls in 



