﻿Isotropic Spheres under Uniform Surface- Pressure. 179 



leaves the total volume of each material unchanged, and 

 does not disturb the material next the surface considered. 



Noticing the value of M in (55), we see from (57) that a 

 given alteration of rigidity throughout a given volume has a 

 greater effect the nearer the altered volume to the centre. 

 We have already come across this phenomenon in § 8. 



In the compound solid sphere the case when the layers, 

 though differing in rigidity, have all the same compressibility 

 is singularly simple, the same formula (60) supplying the 

 displacement throughout the different layers. 



Directions of no Extension. 

 § 13. In a simple solid sphere exposed to uniform surface- 

 pressure every unit element alters in length to the same 

 extent irrespective of its direction or position in the sphere. 

 In a simple shell (e . a . a) it is otherwise : there is in general 

 at any point a conical surface separating directions along 

 which elements lengthen from those along which they shorten. 

 This cone has the diameter through the point considered for 

 axis, and its semi-vertical angle 6 is given by the equation 



cos 2 = 



1 4n/r\* p*-p'a* m 



3 + 9AW p-p' V 



— 5 3 9k\aeJ p—p' 

 If p' = this becomes 



-^-i+S©'' (63) 



and, assuming as before 



Bk>2n, 



this gives a possible value for 6 for all values of r. The 

 directions along which elements shorten are included within 

 the conical surface. 



If p = then (62) becomes 



•""-i+fi©'- < 64 > 



which supplies a possible value for 6 only when 

 r/«<(3*/fcn)*. 



By means of the stress-strain relations the equation deter- 

 mining 6 may easily be thrown into the form 



oo*tf-5(*+{») + {«£_£}, . . (65) 



N2 



