182 Dr. Norman Campbell on the 



help thinking that it is the failure of the method in this 

 important application which has prevented its serious con- 

 sideration. I shall argue later that in this example the 

 Gaussian method also will not work, and that its application 

 in the ordinary manner leads to a result which is directly 

 contradictory to the assumptions involved in it. The 

 Gaussian method, like that of Z.S., involves the assumption 

 that there is no systematic error, and in this case the 

 assumption cannot be maintained. We must base our 

 procedure on a theory which recognizes systematic error. 



However, at present we are concerned only with the third 

 problem, and in connexion with this an objection may be 

 urged against the method of Z.S. It may be said — and of 

 course the statement of fact cannot be disputed — that the 

 result which will be obtained will depend on the manner in 

 which the observational equations are grouped to obtain 

 the normal equations : if one grouping is adopted, one 

 set of values will be obtained ; if another grouping is 

 adopted, another : Z.S. does not, like L.S., lead to an 

 unique solution. Herein lies, to my mind, one of the 

 chief advantages of Z.S. For the apparent uniqueness 

 of the solution by L.S. is altogether misleading. It is 

 true that the introduction of " probable error " admits 

 implicitly that solutions other than that given by the 

 normal equations are admissible ; but that admission is 

 so important that it ought to be stated explicitly. When 

 we discuss in detail the theory of the matter we shall see 

 that there is not the slightest reason of any kind for 

 selecting one of the admissible solutions rather than 

 another. However, for the purposes of practical con- 

 venience, it is certainly desirable to have some standard 

 method of selecting a single value to represent the obser- 

 vations, even if that which is selected is not really different 

 in importance from many others, simply in order that no 

 scope may be left for personal choice and that all persons 

 who consider the same observations may arrive at the same 

 single value for expressing them. But if it is admitted 

 that the choice of that standard method is to a large extent 

 arbitrary, there is no difficulty in devising one that is 

 suitable. 



Accordingly in the* application of the method of Z.S. 

 it is proposed that the groups should be selected in the 

 following manner. There is always at least one of tho 

 measured magnitudes (x, y, z, . . .) which may be assumed 

 to be free from error : let it be y. (This assumption is also 

 involved in L.S.) The observational equations are to be 



