612 HARMON LEWIS 



compensation increased the size of the residuals. Is it not very 

 probable that the introduction of an error in depth actually dimin- 

 ished the residuals ? The most probable depth was calculated on 

 the assumption of completeness. If the assumption of complete- 

 ness was wrong, the depth of compensation which would appear 

 most probable would not be the true depth of compensation but a 

 depth which would counteract the effect of the wrong assump- 

 tion in regard to completeness. In other words the error in 

 the assumed mean depth of compensation would be such as to 

 decrease the residuals. Therefore the residuals which would have 

 been obtained had the correct depth been used would be larger than 

 the residuals actually obtained. The degree of incompleteness as 

 measured by Hayford's method would, therefore, be larger. 



If the depth of compensation were known independently, then 

 Hayford's method of finding the completeness would be legitimate. 

 To go back to the example cited before, suppose that it is known 

 independently that the depth of compensation is 25 miles and 

 suppose that the residual obtained on this basis and assuming 

 complete compensation is 15": this value would be the part of 

 the topographic deflection which had not been made up for by 

 compensation and therefore the ratio of 15 to 35.79 would be an 

 approximate measure of the incompleteness of compensation. 



The above argument will be made clear by a brief summary. The 

 depth and degree of completeness of compensation are unknowns 

 to be determined. It is claimed that these two unknowns can 

 not be determined by Hayford's method of assuming complete 

 compensation, calculating the most probable depth, and using 

 the residuals to tell the degree of incompleteness, because this 

 method would only be legitimate for the one case when compensa- 

 tion is actually complete. If compensation were not complete, 

 then Hayford's calculated depth would be wrong and would 

 furthermore be in error in such a direction as to at least partially make 

 up for the wrong assumption regarding degree of completeness. 

 The resulting residuals would therefore not furnish a maximum 

 measure of the degree of incompleteness; but the compensation 

 would appear to be more nearly complete than would be the fact. 

 We are forced to conclude that, from the geodetic evidence 



