THE STRENGTH OF THE EARTH'S CRUST 453 



In this (/= density, c = constant of gravitation, i? = radius, D = 

 depth. Solving this equation for the values chosen gives 



F= .0853 dyne 



Take the earth's surface as a plane and any point on it as located 

 by the dip angle 6, made by a line from the point to the center of 

 the sphere of outstanding mass. Then the vertical component 

 Fv and the horizontal component Fh are given by the following 



equations : 



Fv (dynes) = 0.0853 sin^' 6 



Fh (dynes) = o . 0853 sin^ 6 cos 



To convert Fh into seconds of arc divide by 0.00475 and 

 Fh (seconds) = 17 . 94 sin^ 6 cos 9 



The maximum value of Fh occurs for ^ = 55° and is o. 0328 dyne 

 or 6.9 seconds. The curves for Fv and Fh are shown in the 

 unbroken lines of Figs. 9 and 10. They are seen to be quite close 

 in character to the curves of Fig. 8. Changes in the mass or depth 

 of the sphere will serve to change only the scales of forces and 

 distances so that these curves may be adapted readily to apply to 

 all spherical masses situated within the lithosphere. 



Influence of sum of intersecting spheres approximately equivalent 

 to spheroids. — The analysis of the gravitative forces which a sphere 

 exerts upon points in an external plane serves as a starting-point 

 for the consideration of the problem of the influence of those unit 

 masses of excess or defect of density which exist in the crust. As 

 a further step, any one mass may be considered as approximating 

 in form either to some oblate or prolate spheroid or to some ellip- 

 soid of three unequal axes. But the equations for the forces exerted 

 by spheroids upon an external plane are complicated and laborious 

 to solve. A sufficient approximation to the influence of a spheroid 

 may be made, however, by employing several intersecting spheres 

 which together give an approximation to the right quantity and 

 distribution of mass. The influence of the composite mass is 

 readily attained by summing up the curves given by the modifica- 

 tions for the several spheres. 



In Fig. 9 the unbroken lines, as previously noted, are the curves 

 of force due to the single unit sphere. The broken lines show the 



