924 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



The permanent terms, for z<z', are the above with signs all changed. 



To deduce the transitory modes for 2;<z' from those for z>z\ note that S^'\ Sl'\ and 



S^ for 2<z' are got from their values for z<z' simply by changing e~~~^ into e~^ ! 

 the same therefore holds for Si? and S^^ but in S^^^'^ we require in addition to change the 

 signs of ^, 0, and •^. 



Thus, for P and force, ii^ and u^ are even functions of z-z\ but u^ is an odd 

 function oi z — z! \ for Z force u^ and u^ are odd, but u^ even in z — 2'. 



As for the stresses, it follows from their expressions in terms of the displacements 

 as given in Art. 1 (4) that, for P or ^ force, jop, pw, ww, and zz are even functions of 

 z — z', but pi and ooz odd ; for Z force fp, /ow, cow, and 22 are odd, but fz and oJi even 

 in 2 — 2'. 



If we compare the permanent terms here with those already found for a unit 

 element of traction (Arts. 12 and 13), we see at a glance that the latter are just those 

 here with p' put equal to a. 



The same is true, though by no means obvious on a simple inspection of the 

 analytical formulge, for the transitory terms also. The result is only what might be 

 expected from physical considerations, and will be demonstrated independently in next 

 Article with the help of a fundamental theorem. 



20. Betti's Theorem. The solutions for a unit element of surface traction special 

 cases of those for a unit intei'nal force. 



We shall in the succeeding pages make considerable use of the fundamental 

 reciprocal theorem due to Betti.* According to this theorem, if two deformations of 

 a body he taken, with the coi'respondiiig body and surface forces, then the ivhole ivork 

 done by the forces of the first acting over the displacements of the second is equal 

 to the ivhole work done by the forces of the second acting over the displacements 

 of the first. 



It is known that the solutions of Art. 14 (divided by ^-wiJiv), or any solutions having 

 the same singularities as those at {x' , y' , z'), can be used in applications of the theorem 

 on the understanding that a unit X, Y, or Z force is included among the forces belonging 

 to the three solutions respectively. 



The same holds good of the solutions for a unit element of surface traction (Art. 5 

 at (8), (9)), as we see by applying the theorem first to the case of traction over a finite 

 area, and proceeding to the limit as in Art. 5. 



We can easily deduce that the solutions given above for unit elements of traction 

 at («, «', z') are the limits for p' = a of the solutions given for unit forces at (/>', w', 2'). 



To prove this, make two applications of Betti's Theorem, the first deformation in 

 each being the solution given for a force (P, ii, Z) at {p" , «", z"), and the second 

 deformation being in the first case the solution given for a force (Pq, ^\^, Zu) at 



* LovK, ElusHcily, Arts. 121, 1G9. 



