J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 32'J 



of equal absolute aniouatj are to be viewed as equal errors. If 

 a fundamental principle is necessarily required, this supposi- 

 tion appears to be the simplest of all. It rests on the conscious- 

 ness of having exercised the greatest possible care, so that no 

 reason exists for assuming that an error has been made, either 

 in the positive or in the negative sense. But let it even be 

 granted that an en-or tends to occur more frequently in one 

 sense than in the other, still, so long as we do not know in 

 which sense it occurs, the value ^ {a + b) is the only one which 

 in this uncertainty will make the error of the result the smallest ; 

 or, at least, which will most securely avoid the danger of in- 

 creasing the error. 



Now let three observations have been made. On account of 

 the fully equal worth of the observations, the values found, a, b, c, 

 must be so combined that no one shall influence the result more 

 than another, independently of their numerical values. Or it 

 must be assumed that 



X = symmetric function {a, b, c). 



But we may consider the subject in another point of view. If 

 two of the observations alone be taken, we should have, ac- 

 cording to which two were selected for that purpose, one of 

 the three following results, which in each case would be the 

 only result that could be chosen : — 



i{a + b), i (a + c), 1 (6 + c). 



To this the third observation adds c, b, a. It is true, that 

 we can no longer combine the two values in each arrange- 

 ment symmetrically, because one rests on two observations, the 

 other on one. But whatever may be the form for the combina- 

 tion of both, it must unquestionably be that, which would pro- 

 duce the result to which the preference is due as derived from 

 the three observations ; and this form, w^hich may be arbitrarily 

 designated by -v/r, must be the same for all three. Hence we 

 have for x the three values 



«■ = ^ (i (« + b)> c), 

 = ^lr (^ (a + c), b), 

 = ^ (1 (6 -F c), a). 



If we introduce here the sum of a, b, c, or if we say 

 a + b + c = s, 



