284 Nature and Bearings of Isostasy. 



tude depend upon very precise measurements of the point 

 in which the projected vertical at any station pierces 

 the celestial sphere, measured by reference to the stars. 

 The geodetic determinations, on the other hand, are made 

 by very precise triangulations from a network of other 

 stations, knowing the size and shape of the geoid. The 

 geodetic and astronomic coordinates of the same station 

 will differ because of the deflections of the vertical pro- 

 duced by the gravitative effect of the topographic relief 

 and the heterogeneities in density. Let it be assumed 

 for purposes of calculation that every detail in topog- 

 raphy is balanced or " compensated' ' by a correspond- 

 ing variation in density. Suppose these variations are 

 uniformly distributed down to a certain datum plane, 

 at which level they abruptly cease. There is postulated 

 then a condition of perfect isostasy depending upon a 

 uniform density compensation which extends to a certain 

 depth of complete compensation. This is the form of 

 hypothesis which Hayford adopted, necessarily artificial, 

 but one which lends itself readily to mathematical cal- 

 culation. On this basis let the gravitative influence be 

 calculated of the surrounding topography and its as- 

 sumed compensation acting upon the vertical at each 

 station. This gives a correction to the deflections of the 

 vertical which modifies the geodetic coordinates as ob- 

 tained without the aid of the theory of isostasy. Note 

 the remaining unexplained divergence between the astro- 

 nomic and these modified geodetic calculations of the 

 directions of the vertical or plumb line. They are found 

 to possess an average value of about three seconds of 

 arc. These differences in the meridian and also in the 

 prime vertical for each station are the de-flection resid- 

 uals. They rest upon the imperfections of the meas- 

 urements plus the imperfections of that hypothesis of 

 isostasy which was adopted. The errors of measure- 

 ment are small in comparison, and in a large number of 

 observations would tend to cancel out. A perfect theory 

 of isostasy (not a theory of perfect isostasy) would 

 reduce to zero that part of the residuals due to the 

 hypothesis. Hayford 's hypothesis of uniform compen- 

 sation to a depth of 122 kilometers reduced the sum of 

 the squares of the residuals to a quantity less than one- 

 tenth of the sum of the squares which is obtained under 

 the hypothesis that there is no isostatic compensation, 

 or that the compensation was uniformly distributed 



