ON ERRORS OF OBSERVATION. 139 



square of the number of observations divided by twice 

 the sum of the squares of the reputed errors (the 

 average being reputed correct) ought to stand in the 

 place of the weight in the preceding rule, whenever 

 different averages are to be combined together to form 

 one general average. If, for instance, one average of 

 100 observations gave 10 and another of 50 gave 11, 

 and if the squares of 100 and 50 respectively divided 

 by twice the sums of the squares of the errors gave 1*5 

 and 1*1, the most probable average of these averages 

 would not be 10^, but the product of 10 and 1*5 

 increased by that of 11 and 1*1, and divided by the 

 sum of 1-5 and 1*1, which gives 10-4. Hence the 

 term weight is now applied to the quotient above de- 

 scribed. 



When the law of error is of the kind figured in 

 p. 17., the mean risk of either sort of error, the probable 

 error, and the weight of a single observation, are con- 

 nected by very simple relations, as follows: 



1. The mean risk is 200 divided by 709 times the 

 square root of the weight ; more nearly, "2820953 

 divided by the square root of the weight. 



2. The probable error is 62 divided by 130 times 

 the square root of the weight ; more nearly, -476936 

 divided by the square root of the weight. 



3. The weight is 113 divided by 1420 times the 

 square of the mean risk. 



4. The probable error is 1 T 7 , T of the mean risk; 

 more nearly 1 -690694 of the mean risk. 



5. The weight is 5 divided by 22 times the square 

 of the probable error ; more nearly, '227468 divided 

 by the square of the probable error. 



6. The mean risk is -^4f of the probable error ; more 

 nearly, '591473 multiplied by the probable error. 



We can thus find the remaining two, when either 

 of the three is given. Of the three, I apprehend that 

 the probable error refers to the most instructive notion ; 

 but the mean risk and the specific weight enter more 

 usefully into formulae .of .calculation. The average 



