164; Mr. Ivoi-y on the Method of the Least Squares. 



These expressions show that e, e', e", &c. are independent of 

 one another. Every one is singly derived from the quantities 

 of its proper experiment, without being influenced by the mag- 

 nitudes of the rest. Let us now suppose that S (a e) has any 

 arbitrary value ; then, according to the foregoing formulae, we 

 shall have, 



x = £ + 



S(ae) 



e = e + a 

 e> = s' + a 



S (a') > 

 S (a e) 



S(a*)' 



S (a e) 



S(ae) 



S (a«) ' 



e" = e" + a". 



&c. 



It is evident, from these expressions that x and all the errors 



e, e 1 , e" &c. depend upon one and the same arbitrary quantity 



S (a e). If we take the value of S {a e) in the formula for e, 



and substitute it in the expressions of e', e" &c. we shall get, 



a i \ 



e =H (e — s) 



e' = e' + — (e — s) 



a v ' 



c" = s'' + — (e - g) 



&C. 



Which proves that all the errors are determined when one only 

 is known. The errors therefore are not independent on one 

 another ; and this is true of every possible system, excepting 

 only the system e, g', e" &c. which is deduced from the rule 

 of the least squares. It is therefore this last system of errors 

 alone that can occur in a set of experiments or observations 

 in which there exists no bias tending regularly one way, and 

 where the error in one case is supposed to have no influence 

 whatever on the error in any other case. 



If we make the coefficients a, a', a", &c. all equal to one 

 another, or rather all equal to unit, we shall have the case of 

 a repetition of the same experiment or observation, viz. 



e = x — m 



e' = x — m! 



e" = x — m" 



&c. 

 where x represents the unknown magnitude to be found, and 

 w, m\ m", &c. the experimental values of it. Here the only 

 system of independent errors is determined by the condition 

 S(e) = 0, or, 



s + 



