362 J. F. ENCKE ON THE METHOD OF LEAST SaUARES. 



the members which contam x. Now, for temporary abbrevia- 

 tion, let 



jp' X + j9 a — p' d _ 



X : ; — Xc\^ 



then 



j9 +y 



X — a — a' = Xq ; 



/K-=;^v/C^;') 





hV {p+p') 



+ 00 

 »-' 00 



e-'''(p+p)''o^dx^ 



the value of the factor which contains the integral will, accord- 

 ing to (5), = 1 ; consequently the probability of X 



_ h /( PP' \ -h^ ^^x-a-a'T- 



~~ -v/ttV \p+p') -^ P+P' 



is a maximum, if 



X = c + c', 



and the weight of this determination will be given immediately 

 by the form 



p^ PP' . 

 P +/' 

 consequently the probable error 



a/F V PP V \p p } 



= \/ {r^ + r'-^). 

 The simple proposition thus found is this : If a and a!, the most 

 probable values of x and a?', independently of each other, toge- 

 ther with the probable errors r and r', are given, then the most 

 probable value of X = a? + a/ 

 =■ a + d, 

 and the probable error of this value 



= V {r^ + r'% 

 Combining this with the preceding proposition, we obtain 

 consequently for any linear function 



X = ux + ^x' + 'yx".... 

 the most probable value 



= xa +^a' + <y a" . . . . 

 with the probable error 



> . . (20.) 



