Adjustment of Observations. 219 



probable value. It is therefore (as also on practical grounds) 

 absurd to speak of the possibility of establishing it directly 

 by experiment. 



The second type of problem is the familiar one of direct 

 conditioned observations. Dr. Campbell makes the indefinite 

 statement * that in one type of this problem the method of 

 least squares is not applicable : the statement is supported 

 by no reasoning, and his objections are not clear ; the point 

 will not be discussed here, but it will appear from the 

 discussion of the general case below that the allegation is 

 incorrect. 



It is, however, to the problem of what he calls the third 

 type that he pays more particular attention. As enunciated f, 

 this is the perfectly general case of indirect observations ; 

 as such, it includes not only the case of the arithmetic mean, 

 but also those of both direct and indirect conditioned obser- 

 vations : in fact, the first and second types are merely 

 particular cases of the third ; the solution given J, however, 

 and stated to be that of the method of least squares, is 

 entirely incorrect. It may be worth while to consider this 

 general problem in some detail, and to indicate the correct 

 method of solution, more especially since the treatment 

 in the text-books is less general, and does not appear to 

 cover all cases §. 



It is assumed that measurements have been made on 

 quantities whose true values are # l3 y u z x . . . #2, y> 2 , z 2 . . < 

 # 3j 7/3, z z . . . a?N, y&i zn ■ • • , and that these quantities are 

 known to be related by the equations 



/1O1, V\, Zi . . . a,b, c ...)== 0, 1 

 f-2 U'2, */2> * 8 . - . a, 6, c . . .) = 0, 



fxU's>//s, ~x • • • a, #, c ■ • •) — 0, j 



• (i) 



the forms of the £ unctions f i9 f 2 ... /i being known, but 

 not the values of the constants a, b, c . . . . It is required 

 to determine the most probable values of a, h, e . . . , the 

 number of which we shall suppose to be q(q<~N). If 

 the number of the observed quantities in each of the 

 equations (1) be n, there are in all evidently N« observed 



* Loc. cit. pp. 1«2 and 191. 



t Loc. cit. p. 179. 



% Loo. cit. p. 180. 



§ See, for example, the two specific problems solved below. 



Q2 



