326 J. F. E.VCKE ON THE METHOD OF LEAST SQUARES. 



and designate the logarithmic differential by 0' A, so that 

 clcbA ... 



jid-^ = ^^' 



the equations of condition of the maximum become the following: 



dx^ d x^ dx ^ dx ^ 



^f „ + ^f»' + 4^V'v" + '^' *'»"'•. .. = 0, 

 dy^ dy^ dy ^ dy ^ 



dv ., dv\, , dv" ,, ,, d v'" ,, ,., 



whence the values of x, y, z, which satisfy them, and which con- 

 sequently are the most probable values, must be determined. 



These general propositions can, however, only be applied 

 when the function <^ is known in each separate case. Instead 

 of making different hj^jotheses as to its most appropriate form, 

 and then tndng which of these corresponds best with experi- 

 ence, we shall attain our object more directly, by considering in 

 a converse manner the simplest case, — examining for it what 

 values experience (apart from the general formulae of the cal- 

 culus of probabihties) teaches us to prefer, — and then trying 

 to determine from thence the form of by means of the general 

 formulae. 



Let us suppose any arbitrary number of observations, all 

 made under equal circumstances, so that beforehand no pre- 

 ference can be given to any one above the rest. Let us say 

 that these observations are to be applied to the detei-mination 

 of the value of an unknown quantitj^, of which the true value 

 would be given directly by each single observation, if there 

 were no errors of observation. An examination of the differ- 

 ence between two right lines may serve as an example. 



First, if one observation has been made, giving the value a, 

 there is no choice but to put 



X ■=■ a. 



If two observations have given the values a and b, and if 

 neither of these is to be preferred to the other, then from these 

 observations alone the value of x must be determined in such 

 manner that the differences x — a and x — b may come out 

 equal. This gives 



X = \ {a + b), 



under the supposition that a positive and a negative deviation, 



