6 GwVTlIER, Conditions of Sticsscs in a Heavy Body. 



body may yield but it must not collapse or visibly suffer 

 change of shape. A portion of the solution will indicate 

 definite gravitational structural stresses, and a further por- 

 tion will indicate ' complementary ' stresses, conditioned 

 but not determined. 



Then, as an assumption, these stresses may be equated 

 to the corresponding expressions in terms of strain for 

 the purpose of determining the conditions under which 

 the hypothesis is admissible, if admissible at all, and 

 finally of determining the displacements when the stress 

 is really elastic. 



If we eliminate the conditioned functions we should 

 obtain the elastic displacement equations. This may be 

 convenient, but is not necessary. The conditions which 

 it is proposed to find are to be obtained by eliminating 

 the components of the displacement in every possible 

 way, or if it is found more convenient in any case, the 

 displacement equations may be solved first, provided it is 

 recognised that the conditions of possibility still require 

 to be investigated. 



If a body is rectangular, the difficulty of finding a 

 formal solution of the statical equation will be great, but 

 this case offers the best opportunity of estimating the 

 number of conditions. 



The elements of a stress, generally, are subject to three 

 statical conditions. If the stress is a purely elastic stress 

 its elements are subject to six further differential con- 

 ditions. Hence the assumption that the stress in a heavy 

 body is a purely elastic stress involves a considerable 

 assumption, but allows of estimation by known formulae. 



Stresses in a Spherical Shell. 

 Taking the polar system of co-ordinates (/', y, ^), I shall 

 write the elements of the stress {P, Q,R, S, 7'sin^, f/sin0), 



