ON ERRORS OP OBSERVATION. 153 



much more erroneous than the observations, or much 

 less so ; both of which possible cases are considered, in 

 all their extent, in the investigations which give the 

 following results. In all of them, except the first, the 

 mean risks are supposed to be small. 



1. To find the mean risk of the sum or difference 

 of any number of quantities determined by observation, 

 add together the squares of all their mean risks, 

 and extract the square root of the result. Thus, if 

 the mean risks of two quantities be 3 and 4, that 

 of their sum or difference is the square ro s ot of 16 + 4, 

 that is, 5. If the mean risks be all equal, the rule 

 may be simplified into that of multiplying the mean 

 risk of one by the square root of the number of 

 quantities. Thus the mean risk of the sum of 100 

 observed quantities of equal risk is 10 times that of 

 one of them. 



EXAMPLE. Given the mean risks of A, B, and C, 

 namely, 1, 2, and 3, required that of 10 A + 9 B - 4 C. 

 Here every error which can happen in A is made 

 tenfold in 10 A, and the mean risk oflOAislOxl 

 or 10. Similarly the mean risks of 9 B and 4 C, are 

 9x2 and 4 X 3 or 18 and 12. The squares of 10, 

 18, and 12, added together, give 568, the square root 

 of which is 23*8, the mean risk required. 



It may seem strange at first sight that, cceteris paribus, 

 the mean risk of a sum and difference should be the 

 same. But a little consideration will show that, posi- 

 tive and negative errors being equally likely, the errors 

 of a difference may be as large as those of a sum : 

 and that no combination of errors can affect a sum, 

 without an equal probability of another equally pro- 

 bable combination affecting the difference in the same 

 way. 



2. To find the mean risk of the product of any 

 number of quantities A, B, C, &c. Take the fraction 

 which each mean risk is of its quantity : add the 

 squares of these fractions^ and multiply the square 

 root of the result by the product itself. This rule is 



