Mr. Ivory on the Method of the Least Squares. 163 



the same elementary quantities sought, the principle of the in- 

 dependence of the experiments will still hold good ; we must 

 still suppose that the error committed in one case has no in- 

 fluence whatever on the error committed in any other case ; 

 and I shall prove that this principle, combined with the equa- 

 tions of condition, leads necessarily to the method of the least 

 squares, of which the rule for the arithmetical mean is only a 

 particular case. 



Multiply all the terms of every equation of condition by the 

 coefficient of x ; then, having added all the results, we shall get 



S [ae) = x x S (a?) — S (a m), 

 the symbols used for the sake of abridging being thus ex- 

 plained, viz. S {a e) = a e + a'e' + a"e" + &c. 



S (a 9 ) = a 2 + a' 2 + a" 2 + &c. 



S(am) = am + a'?n' + a"?n"+ &c. 

 Find x from the equation just obtained, then 



S(ftm) , S(ae) 

 X ~~ S (a*) ~*~ S(a2). » 



and by substituting this value in the expressions of the errors, 

 we further obtain, 



f$ (am) S(ae) 



e = — m + a. v + a. \ ' . , 



, i.i S(om) f S(ae) 



e = — ?n' + a'. -57-^- + «' • tk{' 



S (a") S (a 2 ) 



&C. 



All the errors as well as x now depend upon one and the same 

 quantity, namely, S(ae). If we make this arbitrary quantity 

 go through every gradation of magnitude, we shall obtain 

 all the possible systems of the errors and every possible value 

 of x. Let a, e', s" &c. denote the particular set of errors found 

 by the condition, S(«e) = 0; 



which equation may be otherwise written thus, 



dt . . di' .. d s" Q 



s — -t- s' — + s" — + &c. = 0, 

 dx - d x dx 



and it determines the minimum of the expression s 2 + g' 2 4- 

 e" 2 + &c. : it also fulfils the equilibrium of the weights a, a', a!' 

 &c. as noticed in the beginning of this paper. Now put £ for 

 the particular value of x when the errors are s, s', s" &c. ; 

 then, by making S (a e) = in the foregoing formulae, we get 



£ S (a m) 



% ~ S (o 2 ) » 



S (a m) 



s =- m + a "sk*7> 

 £ > ^-^r + a'.i^, 



£ " - - m 'i 4. a" S(am) 



&c. These 



