ON ERRORS OP OBSERVATION. 14?7 



as negative. The average of a large number of observ- 

 ations will in such a case be extremely near the truth, 

 and provided this condition can be fulfilled, the absolute 

 amount of the tendency to error is comparatively un- 

 important. Let two observers, A and B, have instru- 

 ments the average error of the first of which is double 

 of that of the second. A given number of observations 

 made by A is not as likely to be within a given amount 

 of the truth in the average as the same number made 

 by B. But the former may more than make the 

 difference, by -taking a larger number of observations. 

 The rule is, that the square roots of the numbers of 

 observations must be in proportion to the average errors 

 of the instruments. That is, if A's instrument have an 

 average error double of that of B, he must make four 

 times B's number of observations before he can place 

 the same reliance upon his own observations which he 

 ought to do upon those of B. And the same is true 

 if for average error we read mean risk, or probable 

 error. But if the weights of the observations be known, 

 the numbers of observations (and not their square roots), 

 must be inversely as the weights. 



When, however, there is a fixed error in the instru- 

 ment, independently of casual errors, or of such as are 

 as likely to be positive as negative, there are two 

 modes of proceeding. The average of the observations 

 will now be too great or too small, according as the fixed 

 error is positive or negative. 



1. If the truth can be found by any other means, 

 in any one instance, a large number of observations, 

 such as would be made if the truth in that one in- 

 stance were the object of inquiry, will serve to detect 

 the fixed error, with a high degree of probability that 

 the result shall be correct. If a result should be 29 

 and the average of 100 observations give 28, then it 

 must be presumed that instead of errors -j- x and x 

 being equally likely \-\-oc and - 1 an are equally 

 likely, or there is a fixed error of 1. If A be the 

 true result, and if P be the fixed error, then A -f- P is 

 L 2 



