J, F. ENCKE ON THE METHOD OF LEAST SQUARES. 319 



kind of observation) be designated generally by ^ A, then 

 among m observed errors there will be, according to proba- 

 bility, m (j> A errors of the value A ; and this determination 

 will be so much nearer the truth as m is greater ; so that if 

 m be indefinitely increased, there can be no assignable dif- 

 ference between the value of m </> A, and the true number of the 

 errors A. 



Even with this indeterminate designation some of the pro- 

 perties of the function (}> A can be shown. We know that in 

 each kind of observation the errors can in no case go beyond a 

 certain, though not precisely definable limit ; consequently, if a 

 denote the value of this limit, for A > a (abstracting signs) ^ A 

 becomes impossible, or = 0. In like manner, on the supposi- 

 tion of the greatest possible care in the obsei-vations, and \^dth 

 the assumption, which is the only warrant of certainty in ex- 

 perimental science, that a greater number of observations gives 

 hope of a more exact result, — it is implied that ^ A is a maxi- 

 mum for A = 0, and is equal for equal positive and negative va- 

 lues. If indeed this were not the case in a continued repetition 

 of the observations, the erroneous values of the quantity to be 

 determined would prevail so much on either the positive or the 

 negative side, that we should find ourselves in the impossibility 

 of attaining the truth, and should be in danger, even with an in- 

 finite number of observations, of taking an erroneous value for 

 the most probable one. We have then as the most probable 

 value resulting from our observations, that for w-hich ^ A is a 

 maximum with A = 0, and which is besides a direct function of 

 A ; and as we have no other means than the obsen^ations of 

 determining the true value, this value must be to us the true 

 one. 



In these assumptions, however, the distinction between con- 

 stant and irregular errors requires consideration. By constant 

 errors, are generally understood those of which the sources are 

 not general, but belong to the particular observations, some- 

 times to a particular instrument, or to the individuality of the 

 observer. Irregular errors, on the other hand, are those which 

 occur under all circumstances, and which are therefore properly 

 subject to the calculus of probabiUties. The causes of the 

 smaller constant errors are in themselves analogous to those 

 which produce the irregular errors, and the total avoidance of 

 them may even be regarded as impossible. Our aim should be 



