J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 351 



m J A" 4/ (A) dA = m k^''> 



A = — CO 



will express generally the sum of the ni\\ powers of all the 

 errors which, according to the law of probabilities, should occur 

 in m observations. The magnitude A'("), in which the index n 

 depends on the power of A, or the integral taken between 

 the widest limits, cannot be merely an absolute number, but 

 will contain one or more constants, having respect to the 

 class of observations. If, therefore, we knew truly the form of 

 ■^ {A), but were still uncertain of the exact value of the con- 

 stants contained in it, then any number of w observations, when 

 the pure errors of observation are found thereby, would lead ua 

 to the knowledge of the constants. For, let the errors a, /3, y, S, 

 be given immediately up to the number m, then the most probable 

 value of k^"'^ will be found by 



^('0 _ «"+/3" + y" + S"... ^ £^ . 

 m in 



Any other hypothesis as to the value of k^"^ would not suppose 

 the errors distributed according to the lawrj/ ( A) ; consequently, 

 it would assume an error in one or several values of «", /3", y"> 

 &c. The value of k^'"\ which, in its conditions, involves no eri-or, 

 must be the most probable, according to these m observa- 

 tions. 



This form gives also at the same time the limits of certainty 

 of the determination thus obtained oik'^"\ With k^'"^ the principle 

 of the arithmetical mean holds rigorously good, by which, for 

 each m, we find, from the results given by the observations 

 singly, the most probable value of one and the same unknown 

 magnitude. The magnitudes a", /3", y", come consequently into 

 the series of direct observations of the magnitude A:^"), and the 

 differences /t^") — a", k'^"'^ — /S", k^"^ —y^"\ are to be viewed as 

 the errors of one such single determination. The above-deter- 

 mined form <p {A) holds good for them, apart from the original 

 form vf/ ( A) in every case. Hence the mean deviation of such 

 a single determination 



= / / (^^") - «")^ + jk^"^ - ^y -H (k^"^ - y")^ + • • ' • Y 

 instead of which, by substitution of 



