364 J. F. ENCKE OX THE METHOD OF LEAST SQUARES. 



The most probable value of X— V and its limits are given im- 

 mediately by the most probable value of X and its limits, and 

 conversely, because the two magnitudes X — V and X differ only 

 by a constant; so also the probable eiTors of A <r, A x' , Ax", 

 &c. will be the given magnitudes r, r , r", &c. and the most pro- 

 bable values of Ax, A x' , A x", &c. will be nothing by virtue of 

 the equations x =: a + A x, 8cc. Hence follows, according to 

 (20.), the most probable value of X, 



X - V = 0, 1 



and the probable error of X —V, I 



or the most probable value of X is V, and the probable error of 

 this determination is equal to the above-determined F ; a solu- 

 tion which is rigorously true for linear functions, but only ap- 

 proximately so for higher ones. 



It is a different case, supposing that we have found for one 

 and the same unknown x, by different examinations, the values 

 a, a', a'' . . ., with the probable errors r, r, r" . . ., or the weights 

 •p, p', p" . . ., and that we seek to find from them all the most 

 probable values. The definition of the idea of weight, according 

 to which a, a', a" must be considered as respectively found by 

 the number jo, p', p", of equally good observations, gives here, 

 by virtue of the arithmetical mean, the most probable value of x 

 ap -\- a' p' + a" p" + , &c. 



^ ~ P + p' +p" +, &c. ' 

 with the weight 



p-\-p' +p" +, &c.; 

 or, which is the same thing, the most probable value of 

 a a' a' ^ "^ 



X = 



L L J^ Rr 



with the probable error 



1 



\/(^ + ^, + ^ + .-&c.)_ 



(24.) 



