THE STRENGTH OF THE EARTH'S CRUST 443 



isostatic balance, and supported permanently and rigidly by the 

 strength of the crust ? 



These problems have been studied in the last three parts of this 

 article by means of the evidence presented by Hay ford and Bowie, 

 but that evidence has not been used to solve the depth of the out- 

 standing masses. Yet it is seen that all of the aspects just enu- 

 merated are bound up in that factor. It is especially this problem 

 of the depth and the consequent areal extent and mass of the units 

 which produce the residuals of the deflection and gravity observa- 

 tions which is attacked in this part. It is necessary for this investi- 

 gation to enter into a study of the complex relations between the 

 anomalies and residuals which depend upon the depth and form 

 of masses. It is a subject upon which, so far as the writer is 

 aware, but little has been done, so that about half of this chapter, 

 published as Section A, consists of a study of these relations pre- 

 liminary to their application. 



For facility of rhathematical treatment the individual outstand- 

 ing masses must be regarded as equivalent to spheres, spheroids, or 

 cylindrical disks, either as units or as aggregates. If only the 

 epicenter (the point on the surface vertically above the center) of 

 the disturbing mass is determinable and the deflections at two or 

 three points on one side of it, then the mass may be most simply 

 interpreted as a sphere; since the mass of a sphere acts as if con- 

 centrated at its center. With fuller observations a close approxi- 

 mation to the depth of the mass and a less close approximation 

 to its form and density may be made. The first problem then is 

 to determine the epicenter of the outstanding mass and its depth, 

 using for this purpose the nature of the anomalies and residuals. 

 For a sphere beneath a plane surface it is shown that the value of 

 the maximum gravity anomaly at the surface is 2.6 times the 

 value of the maximum deflection residual, both being measured in 

 the same units of force. The former occurs vertically over the 

 abnormal spherical mass, that is, at the epicenter. The latter 

 occurs at a horizontal distance from the epicenter equal to 70 per 

 cent of the vertical depth to the center of mass. Oblate spheroids 

 and broad cylindrical disks with vertical axes and the same depth 

 of center as the sphere give maximum deflections at greater 



