454 Lord Rayleigh on a Statical Theorem . 



equations which determine the displacements yjr v i/r 2 when the 

 forces are given. The consequences which follow from the fact 

 that A 12 =A 21 may be exhibited in three ways. 



First Proposition. — Suppose ^ 2 =0. From the second equa- 

 tion, we see that v^2 = ^2i V ^i- Similarly if we had supposed 

 ^ = 0, we should get v^i = A ]2 "^ 2 , showing that the relation of 

 r \jr 1 to ^ in the second case is the same as the relation of ty 2 to M/\ 

 in the first. 



In order to fix our ideas we will take the case of a rod, not 

 necessarily uniform, supported in any manner in a horizontal 

 position — for example, with one end clamped and the other free. 

 Then, if P and Q be any two points of its length, we assert that 

 a pound weight hung on at P will give the same linear deflection 

 at Q as is observed at P when the weight is hung at Q ; and 

 the only thing on which our conclusion depends is the propor- 

 tionality of strains and stresses. If we take angular instead of 

 linear displacements, the theorem will run : — A given couple at 

 P will produce the same rotation at Q as the couple at Q would 

 give at P. Or if one displacement be linear and the other an- 

 gular, the result may be stated thus : — A couple at P would do 

 as much work in acting over the rotation at P due to a simple 

 force at Q, as the force at Q would do in acting over the linear 

 displacement at Q due to the couple at P. In the last case the 

 statement is more complicated, since the forces, being of different 

 kinds, cannot be made equal. 



Second Proposition. — Suppose that ^ 1 = 0. Then, from (C), 



From this we conclude that if yjr 2 is given, it requires the same 

 force ^ 1 to keep ^jr l = 0, as would be required in ^ 2 to keep ^ 2 = 0, 

 if ^i had the given value. 



Thus, if the rod be supported at P so that that point cannot 



fall, while Q is depressed one inch by a force there acting, the 



reaction on the support at P is the same as it would have been 



on a support at Q if P had been depressed one inch*. 



Third Proposition. — Suppose, first, that ^ r 1 = 0. Then, from (C), 



Secondly, suppose ^ 2 =0. Then 



Thus, when M/^ alone acts, the ratio of displacements ty 1 : ty 2 is the 



negative of the ratio of the forces ^ a : M/^ necessary to keep ty 2 = 0. 



If the rod is supported at P and bent by a force acting down- 



* The verification of these results with rods variously supported, or 

 more complicated structures, gives a very good experimental exercise. 



