﻿282 Mr. R. F. Gwyther on an Analytical Discrimination 



on the left-hand side of (1) of terms representing the com- 

 ponents of the force per unit volume. 



These components can always be represented in a form 

 similar to that given for the displacement in (3), but I shall 

 suppose that the force per unit volume can be represented 

 by the simpler forms 



B(F+/x) 5(F+/,) ^(F+/ 3 ) 



P ^5i~' p ^i~' p ~^~' 



and I shall suppose that we have selected the axes and 

 that the yfrs are null. Then in (2) we must replace P 

 by P + ^F+Z,), Q by Q + p(F+/,;, R by R + p(F+/ 8 ), 

 leaving S, T, U unchanged. 



In the values found from the stress-strain relations there 

 are no such changes to be made. 



We shall thus obtain from (6) 



V*0i-~tfi = V 2 2 - P f 2 = V 2 3 -pfi 

 2n{3 /5 F4/>(/i+/>+y8)} + (3m + n)V!(^i + ^+.ft) 



= 6m {'S? + a? + a?}- ' (10) 



If we write 



^3 = </> + %3 + f + %/, 



where <£, % l9 ^ 2 , %3 have the values in (5) and (7) and may 

 be regarded as Complementary Functions, then we remain 

 with 



and 



2n/) F + (m + n>(/ 2 +/ 2 +/ s ) + (m + n) V 2 <£' 



which may be regarded as giving the Particular Integral 

 corresponding to the particular force acting. 



There are not many cases of interest. In the case of 

 gravity on the surface of the Earth, as under natural forces 

 generally, we have 



j\ =j 2 =fi = and (m + n) V 2 </>' + 2npF = 0. 



If we suppose ( — X, — ^, —v) to be the direction-cosines 

 of the attraction of gravitation, 



F ^^gfrt + fiy + pz) 



and ^ = _^ (x ^ + , + v ^. 



