ON ERRORS OF OBSERVATION. 151 



serrations ; in the second, an adaptation of the instru- 

 ment which reproduces an equal likelihood of positive 

 and negative error, by making the fixed error itself as 

 often positive as negative. What we have called a 

 fixed error is in fact a part of the phenomenon, styled 

 an error because it is not a part of the result we wish 

 to observe. The errors which a simple application 

 of our theory removes are those of which no account 

 whatever can be given, and of which nothing can be 

 previously known. Such is not the case when positive 

 error is more likely than negative, or vice versa: for 

 this very circumstance is itself a phenomenon, which 

 must arise from some unvarying cause. 



Having stated that it is indifferent, in a mathematical 

 point of view, whether the law of the facility of error 

 above explained be true or not, because any law what- 

 soever which falls within the widest permission of 

 common sense leads to the same results as above ex- 

 plained, when the number of observations is consider- 

 able I will now point out, in some simple cases, 

 how different laws of error are to be reduced to the 

 preceding. 



1. Let all errors, positive and negative, between 4- E 

 and E be equally probable, and all others impossible. 

 Treat large numbers of such observations as in pages 

 187. and 144.j on the supposition that the weight is 3 

 divided by twice the square of E. 



2. Let the probability of error decrease uniformly 

 as the magnitude increases, the greatest possible errors 

 being -f- E and E ; which implies that the chance 

 of an error lying between go and -\- x is the product 

 of x and the remainder of 2 E divided by the square of 

 E. For instance, if E be 10, and this law of facility 

 prevail, the chance of an error lying between 2 and 

 -j- 2, is the product of 2 and 18 divided by 10 times 

 10, or -ff-Q. In this case a large number of observ- 

 ations must be treated as if the weight of each observ- 

 ation were 3 divided by the square of E. 



3. If the weight of the observations be considered 



