Adjustment of Observations. 193 



method of L.S., according to which 6=^—. In this case 



° z<xy 



actual practice would always follow Z.S. and not L.S. 



And I do not think the reason is merely simplicity. This 



example calls attention to a fundamental ambiguity in L.S. 



If the relation between x and y were written ax=y, then L.S. 



indicates that a = ^ — . But then, in general, it \\ould not 



he found that a = 1/6; in our example the specific volume 

 would not be the reciprocal of the density. On the other 

 hand, according to Z.S. that relation always would be 

 fulfilled. It is probably the realization of this ambiguity 

 which prevents anyone using L.S. in dealing with such 

 •observations. 



But the ambiguity is not confined to this simple example. 

 If, in place of 



x ■= by + cz + . .. +m, (8) 



we write 



ax = y + c f zA- . . . 4- m', .... (9) 



we shall not find in general that (a, c , . . . m) = (1, c, . . . ni)/b, 

 if we calculate the constants by L.S. The reason is that the 

 method of L.S. assumes that only one of x,y,z,... is affected 

 by any error at all, and that this one is x, of which the constant 

 coefficient is 1. We get different results by using (8) and (9) 

 because in one case we are attributing the residuals to errors 

 in x, in the other case to errors of y. It is generally re- 

 cognized that L.S. is only applicable strictly when one 

 variable alone is liable to error — though it is often applied 

 •when that condition is certainly not fulfilled. But it appears 

 not to be recognized generally that, even when it is known 

 -that only one variable is liable to error, it is very seldom 

 known which of the variables is this one. For instance, 

 I am observing the position of a pointer at various instants 

 of time. Are my errors due to observing the wrong position 

 at the right time or observing the right position at the wrong 

 time ? Indeed is there any sense in asserting one of these 

 statements rather than the other ? But according as I adopt 

 one statement or the other I shall obtain different values for 

 the constants of the equation relating position and time, if 

 I calculate according to the method of L.S. 



This appears to me the most fundamental objection to L.S. 

 as a means of obtaining unique values in the solution of 

 problems of the third kind. It is important to realize that 

 it does not arise in dealing with problems of the second kind. 

 To ascertain how the ambiguity is introduced, we must 



Phil. Mag. S. 6. Vol. 39. No. 230. Feb. 1920. 



