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



two determinations by the proportions of their respective values 

 of h and r, it is more convenient to bring in the new idea of 

 weight. By the weight of a given value we understand the num- 

 ber of equally good observations of a determinate kind (of which 

 the exactness is to be viewed as the unit of exactness), which are 

 required to furnish, by their arithmetical mean, a determination 

 of equal exactness to that of the given value. According to 

 this, in the present case, if the weight of the single observation 

 be regarded as unity, the weight of x= m-, — if /t be the mea- 

 sure of the exactness of the single observations, the measure 

 of the exactness oi x = h \/ m, — and if the probable error 

 of an observation be designated by r, the probable error of x 



= £- z= — L — = . The weights of two determinations 



H h ^ m V »i ^ 



are to each other in the direct proportion of the squares of their 

 respective measures of exactness, and in the inverse proportion 

 of the squares of the probable errors. 



If Ave substitute in the equations of condition the most 

 probable values of x, then the diiferences, between the result 

 calculated with this value, and the actual observation, are to 

 be regarded as the errors of observation which appi'oximate 

 most nearly to the truth ; therefore, so long as we have no 

 means of determining the value of x more nearly, the errors 

 thus obtained are to be regarded as the true ones. The sum 

 of their squares must, according to the whole deduction hi- 

 therto, be equal to the minimum just determined, or it must 



be = [n^^ — ^^^ . In order to obtain, generally, a more con- 

 *- -* m 



venient expression for this sum, we introduce a new idea, 

 that of the mean error. By mean error we understand the 

 magnitude which is obtained, if the sum of the squares of 

 the true errors of observation be divided by the number of 

 observations, and the square root of the quotient be extracted. 

 Consequently, in the present case, the mean error being desig- 

 nated bv Eg, 



V(i=#), 



m ' 



inasmuch as -we are at present obliged to regard the errors re- 

 sulting from the most probable hypothesis as the true ones. We 



