I50 



JOSEPH BARRELL 



The difference is the residual error due to the partial incorrectness 

 of a hypothesis. The exactly correct hypothesis would reduce all 

 residual errors to zero except for the errors of observation and 

 computation. A hypothesis which approximates to the truth will 

 give small residual errors. In a large mass of data the sum of the 

 squares of the residuals as derived from different hypotheses serves 

 as a test of the relative agreement of the hypotheses with nature, 

 that hypothesis applying best for which the sum of the squares is a 

 minimum. In all of the complete solutions a uniform distribution 

 of compensation was assumed to exist from the surface to the bottom 

 of the zone of isostatic compensation. That is, if the column under 

 a certain portion of land was 3 per cent lighter than under a certain 

 portion of water, then it was assumed that at any and every depth 

 the two columns differed in density by 3 per cent. The differences 

 abruptly terminate at the level where the two columns, the long but 

 light land column and the short but heavy sea column, become of 

 equal weight. At the level of this surface isostatic compensation is 

 complete and there is hydrostatic equilibrium. 



A tabulation of the probabilities of these hypotheses as applied 

 to the whole of the United States is as follows: 



TABLE III 



Hypothesis Sum of Squares of 765 Residuals 



Solution B (extreme rigidity; depth of compensation infinite) 107,385 



Solution E (depth of compensation 162. 2 km.) 10,297 



Solution H (depth of compensation 120. 9 km.) 10,063 



Solution G (depth of compensation 1 13 . 7 km.) 10,077 



Solution A (depth of compensation zero) 18,889 



The first investigation, that of 1906, favored Solution G, the 

 final, that of 1909, as shown in this table, favored H. The most 

 probable depth on the hypothesis of uniform compensation with 

 depth and of equal depth of compensation for the whole United 

 States was a little greater, being 122.2 km., 76 miles. It is seen, 

 however, that there is but little change in the sum of the squares for 

 a considerable range in the assumed depth. Further, Hayford 

 states that the hypothesis of all compensation being attained in a 

 lo-mile stratum whose bottom is at a depth of 35 miles is about as 

 probable as the solution which he adopted.^ Other variations in the 

 hypothesis are also possible with about the same probable error.^ 



»iQo6, p. 151. = 1906, p. 153. 



