192 Dr. Norman Campbell on the 



can easily be devised. But since the equations of conditions 

 are never much more numerous than the observations (they 

 are usually much less numerous), N is never large compared 

 with n. Accordingly Z.S. gives results varying widely as 

 different methods of grouping are adopted ; and though 

 there is no reason to believe that all these results are not 

 admissible, ifc is difficult to fix precisely one method of 

 grouping, so that the first requisite in problems of surveying, 

 namely that a definitely unique set of values shall be obtained, 

 is fulfilled. It is certainly better to adjust by the method 

 of L.S. And it is fortunate here that the method loses most 

 of its disadvantages. The coefficients of the equations of 

 condition are usually small integers, so that the calculation 

 is easy. Moreover it. will appear in the sequel that this is 

 the one form of problem to which L.S. is strictly applicable 

 on theoretical grounds. It is here that there is most evidence 

 that the Gaussian law of error is true, and here that the 

 method is an adequate expression of the Gaussian theory. 

 I believe indeed that the Gaussian method was originally 

 elaborated to deal with just this problem : if so, it was 

 completely justified. It is only its extension to the other 

 form of the second problem (where there is a constant 

 term in the equation of condition), and to the third 

 problem, that is both theoretically illegitimate and prac- 

 tically inconvenient. 



It is admitted then that, in this direction, room still remains 

 for the method of L.S. This admission may seem to weaken 

 somewhat the case for its replacement elsewhere by Z.S. 

 Accordingly, before proceeding (in a subsequent paper) to a 

 discussion of the validity of the two methods according to 

 the theory of errors, it may be well to point out that there 

 are examples to which L.S. is as clearly inapplicable as Z.S. 

 is to that which has just been discussed. These examples 

 occur when the equation (1) reduces to the simple form x = by, 

 as happens when we have to determine a density (b) from 

 measurements of a mass (x) and a volume (y). . 



An elementary student, when he had measured several sets 

 of associated values of x and y, would doubtless take the ratio 

 cejy in each set, and take for b the mean of these ratios. A 

 more experienced worker w r ould realize at once that such a 

 procedure gives undue w 7 eight to the raiios derived from the 

 smaller values of x,y, which are likely to be less (relatively) 

 accurate. He would probably add all the #'s and all the ?/'s 

 and take the ratio of the sums. This is exactly the procedure 

 of the method of Z.S. But nobody, I believe, would adopt the 



J 



