MAGNETIC SURVEY OF GREAT BRITAIN. 9& 



The object proposed in the preceding method has been at- 

 tained by Major Sabine by a different process, which will be 

 applied by him in the sequel. It is therefore unnecessary to 

 make any application of that here laid down. 



In combining the equations of condition by the method of 

 least squares, it is manifest that we cannot, in general, allow 

 equal weight to all. The result obtained at one station may be 

 derived from a single observation only ; while, at another, it 

 may be the mean of several observations, made at different 

 times, and with different instruments. In a former discussion 

 of the observations in Ireland, weights were assigned to the 

 results at each station, but on arbitrary and uncertain princi- 

 ples. I now proceed to remedy this defect ; and I do so the 

 more willingly, both on account of the great importance of 

 this branch of the theory of probabilities in Physical science, 

 and because the results to be referred to are connected wath 

 researches not as well known as they deserve. 



Let x^, a-g, x.^, &c., x^, be n values of the quantity x, ob- 

 tained by separate and independent observations ; and let a 

 denote their arithmetical mean, so that 



a— -(^1 + or. + 0^3 + &c. -f- x^ ; 



then the probable error of this mean, i. e. the limit on either 

 side of which there are equal chances of the actual error lying, 

 is given by the formula 



n{n-l) ' ^^^ 



in which S (j: — «)^ denotes the sum of the squares of the dif- 

 ferences of the several partial results and the mean, or the va- 

 lue of 



{x, - af + {x^ - af + &c. + (07,^ - af ; 



and in which, also, p is the number which satisfies the equa- 

 tion 





'' e ''dt^i ^- 







Numerically, p = 0"4769 ; and substituting in (10) 



E« = -iS^lifl^' (11) 



n [n — 1) 



The pi'obable error of a single result, as deduced from com- 

 parison with the rest, is in like manner given by the formula 



