330 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 



cian. Therefore we give the following deduction, which is based 

 on proposition 



III. 



Any number of equally good direct observations of an unknown 

 magnitude being given, the arithmetical mean of all the observed 

 values determines the most probable value of the unknown magni- 

 tude, so far as it is determinable by these observations, without 

 requiring or universally admitting any other condition. 



Let there be m equally good observations of the unknown 

 magnitude ae, and let them have given for it the values M, M', 

 M", &c. According to the last proposition, if 



M + M' + M" 



p = 



m 



the most probable value of x in every case, so far as it can be 

 concluded from these m observations, will be the magnitude 

 p. Consequently M— ^, M'—p, M." — p must be regarded as 

 errors of observation; or the equation from which the most 

 probable value of x proceeds according to the arithmetical mean, 



is 



M - X + W — X + W - X + M'" -X .. . = 0. (4.) 



If we apply to the same case the general formulae of the cal- 

 culus of probabiUties, we have 



v = M.—x, v =W — X, v" = M" — a? . . . . ; 

 consequently the only equation of condition of the most pro- 

 bable value is 

 <^'(M-a?) + ^' {W-x) + 4)'(M"-^) + cf>'(M"'-«'). .. = 0, 



to which the following form may also be given : 



,-_ ,<t)' (M— a?) ,-,, , ($>' (M'— 07) 



+ (M X) j^„_^ O. 



It follows immediately from this latter form, that the above 

 equation deduced from the arithmetical mean, will universally 

 satisfy this last equation only when 



<^'{ M-x) _ ^' [W - x) _ »' (M"-,r) 

 M-o? ~ W-x W'-x ' ' 



i. e. when — - — is independent of the value of A, or when —r— 



