J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 337 



the constants h are therefore in the inverse proportion oT the 

 equally probable errors of two kinds of observations. This is 

 true for all errors, consequently for the probable errors of 



each kind, as already shown by the equation h = ^-, because 



D is here an absolute number. If, therefore, there is an even 

 chance that an error falls in one kind within one limit, and in the 

 other kind within the other limit, for which generally the pro- 

 bable errors r and / are chosen, we have also the reciprocal 

 proportion of the constants h and K, from the inverse propor- 

 tion of the limits, or from the probable errors r and r . Hence 

 may be derived a preliminary estimation of the proportion of h 

 and h'. If in two measurements of angles there is reason to 

 fear that an error of w" may have been made in one as easily as 

 an error o( mco" in the other, then, if h be taken for the latter, 

 m h must be put for the former. On account of this constant 

 proportion between the increase of exactness and the magnitude 

 of h, Gauss calls this constant the measure of precision. 



The geometric representation of the curve of probability may 

 be also applied to this consideration. Take any unity as the 

 general measure of A, or of the abscissas ; then, by means of the 

 equation 



h -/i'2A2 



cj) A = -7— e , 



the whole curve could at once be drawn if the value of h were 

 known. Consequently, if we only know an ordinate belonging 

 to a determinate A, the whole curve will be fully given. If the 

 ordinate for which A = be chosen, by comparing its value 



. with —. — , we have at once the value of h. If the ordinate 



were chosen, which divides the superficial contents of the curve 

 into two equal parts on either side of zero, we should obtain h 

 from the abscissa belonging to this ordinate, by means of the 



equation /* = — . If we even knew merely the relative propor- 

 tion of two ordinates which correspond to any abscissae, A 

 and A', then as this proportion is as e~ '' "^ :e~^'^', or as 



1 :e~ '' ^'^"'^ , we should be able to determine h from hence. 

 It is most convenient to choose for the one ordinate that 

 which corresponds to the value A = 0. Hence follows a 



