344 G. F. Becker — Proof of the Earth's Rigidity. 



by F r . This force vanishes when 3 cos 2 #=l or when #=54° 44 ; , 



nearly. The lines of no compressive force form a cone of two 



sheets, the axis of which coincides with the line connecting the 



two halves of the moon. Within this cone the forces F r are 



directed from the center and tend to produce dilatation of the 



mass, and F r is a maximum on the axis where it has the value 



Mr 

 2-T^r. Between the sheets of the cone is an annular mass 

 D 



bounded on the surface by a zone. Within this mass the 



forces are directed inward, and tend to produce compression.. 



The compressive force is maximum when #=90 and thus has 



the value Mr/D 3 . 



This distribution is illustrated in the diagram on page 343, 

 where an arbitrary length is assumed as equal to the value of 

 the force on the a?-axis and the other forces are exhibited in 

 the true proportion to it.* 



When strains are small, they may be supposed to take place 

 successively. Thus one may arrive at the distortion of a com- 

 pressible sphere by supposing it first deformed as if it were in- 

 compressible, and that it then becomes compressible. If this 

 course is adopted in the present case, the incompressible sphere 

 would be distorted to an ellipsoid of the same volume, in which 

 the compressive stresses exhibited in the diagram would exist, 

 but would be inoperative. If compressibility now supervenes, 

 it is clear that dilatation will take place along the major axis 

 and compression at right angles to it. 



General result for elastic spheroid. — Hence compressibility 

 must increase the ellipticity which an elastic sphere assumes 

 under a tide-generating stress. Hence also the ellipticity is 

 always greater than rwa^/^n, and any conclusions which can be 



* It is a noteworthy fact that all ellipsoids of infinitesimal ellipticity and constant 

 volume intersect the surface of the unstrained sphere at the circles of no force. 

 One way of proving this is as follows: Let the semi-axes of the ellipse be 

 a{\ + 6) and a(\ — 6/2) where 6 is so small that its square can be neglected. Them 

 the volume being constant, 



a 3 = a(l + rf).a 2 (l - 6/2)- 

 and the equation of the ellipse is 



( ^ + (T^fe-)5 = a9 = a;! ( I - M ) +! ' s<l + <!) 



when this ellipse cuts the circle, a 2 = x- + ?/ 2 , and 



2a; 2 = y' 2 or tan 2 ■& = 2 and cos 2 ■& = — 



irrespective of the spherical value of 6. 



The equipotential surfaces are hyperboloids of revolution asymptotic to the cone 

 mentioned in the text, and the equation of the potential is simply 



/ 



Mr 2 

 V rfr = V=— (3cos 2 #-l) 



