Adjustment of Observations. 191 



I maintain that, until it is shown to lead to misleading 

 results, the method of Z.S. holds the field against any other, 

 merely on the ground of simplicity. However, I am pre- 

 pared to admit that in certain cases it loses much or all o£ 

 its advantage, namely in those for which N is not very much 

 greater than ?i, and the number of observations not much 

 greater than the number of variables. It is then found — as 

 might be expected from the fact that p is little if at all greater 

 than 1— that values obtained for the constants vary greatly 

 with the precise grouping of the observations in the formation 

 of the normal equations, and the probable error of the result 

 is much greater than can be judged significant according to 

 the criterion which will be developed at, a later stage. Of 

 course the best method of dealing with such cases is to make 

 more observations and so cause N to be much greater than n ; 

 but if, for any reason, that course is impossible, and if some 

 single value must be obtained, then it is probably better to 

 employ L.S. unless N is at least as great as 3». But the use 

 of that method is a mere matter of practical convenience : 

 I deny altogether that, in general, the results obtained have 

 any greater theoretical significance than the widely differing 

 results obtained by the method of Z.S. There is simply no 

 theoretical ground for any single value whatever within very 

 wide limits. 



These considerations have a bearing on the second problem 

 of adjustment of observations, namely that in which it is 

 required to determine true values of the measured magnitudes 

 and not constants of an equation which they satisfy. The 

 true values are now values such that they satisfy some 

 equation of which the constants are definitely known. In 

 one form of the problem, this equation contains, besides the 

 variable magnitudes, a constant term to which a definite 

 numerical value is assigned. An example of this form is 

 the problem of the angles of a triangle, and we have already 

 noted that the method of Z.S. (and that of L.S.) must fail 

 when applied to that problem. But in a second form, the 

 " equation of condition " does not contain a constant term, 

 but relates only the measured variables. An example of 

 this form is the problem of a< I justing the results of a 

 levelling survey : here it is known (e. g.) that the height 

 of A above C must be the sum of the heights of A above B 

 and of B above C, but there is no numerical constant known 

 apart from the observations. 



In this problem it is possible to find true values such that 

 thev satisfy the equation of condition and make the sum of 

 the errors zero; and rules for applying the method of Z.S. 



