Adjustment of Observations, 181 



The procedure is so obvious that it would be the first to 

 occur to anyone to whom the problem was presented. It has 

 doubtless not been adopted mainly because the alternative 

 method of Least Squares was held to be the only one that is 

 justifiable by the theory of errors. But this procedure can 

 also be based on theory. When we select a group of the 

 equations and add them to obtain a normal equation, we are 

 assuming that this equation is absoiutel}' correct, and that 

 the sums of the measured magnitudes are the same as the 

 sums of the true magnitudes to which those measured 

 magnitudes approach. In other words we assume that, if 

 we take the sum of a group of measured magnitudes, the 

 •errors of measurement cancel out; we assume that the sum 

 of the errors in any group is zero. Now if the group is 

 sufficiently large, this assumption will be true, even if we 

 believe in the Graussian law of errors, but it will also be 

 true if we adopt any other reasonable law of errors ; for 

 it is an assumption more fundamental than those on which 

 the Graussian law is based that positive and negative errors 

 are " equally probable." Accordingly, if the groups into 

 which the equations are divided are sufficiently large, the 

 assumption that their sum will be free from error is based 

 on theory much more firmly than the assumption of the 

 Gaussian law : for the first assumption is part of the second, 

 and the part which is least dubitable. The only question 

 which can arise is whether the assumption, and the pro- 

 cedure founded on it, is justifiable when the groups which 

 are added are not very large. In a later part of this paper 

 I shall argue that, even in this case, the procedure, though 

 not capable of complete theoretical justification, has more 

 theoretical justification than any other, and a great deal 

 more than that of the Method of Least Squares. 



The proposed procedure may, therefore, be called the 

 Method of Zero Sum (Z.S.) in contradistinction to the 

 Method of Least Squares (L.S.). But even if its theoretical 

 basis is accepted, two further objections may be urged 

 against it. The first arises in connexion with the second 

 problem of the adjustment of observations. We have 

 measured the three angles of a triangle and find that the 

 measured values do not add up to 180°. The method of 

 adjustment proposed is to choose true values such that the 

 sum of the errors is zero. But it is at once apparent that 

 it is impossible to choose such values which will at the 

 same time add up to 180° : for the sum of the true values 

 must be the same as the sum of the measured magnitudes : 

 in this example then the method will not work. 1 cannot 



