Theory of Elasticity. 277 



definite stresses and strains, or else that under certain applied 

 forces a certain condition of stress and strain exists. Or it 

 may be that we merely wish to investigate the consequences 

 of some supposed state of stress and strain. Whatever the 

 basis, we must eventually have as the starting point for our 

 strictly mathematical investigations a definitely given state of 

 stress and strain under certain definite conditions as to temper- 

 ature and applied forces. Thence, if we know the stress- 

 strain relations that are true for the body and material under 

 consideration, by the ordinary procedure of the theory of elas- 

 ticity we may (theoretically) follow all subsequent elastic 

 changes in the body's condition consequent on known changes 

 in the applied forces and temperature, with perfect certainty. 

 Here we are in the domain of exact analysis ; cause and effect 

 follow in unquestionable sequence. But the foundation on 

 which the results are thus built, it must always be remembered, 

 was one furnished by imperfect observations or mere assump- 

 tions, and' we must never permit ourselves to fall into the error 

 of attributing to our structure a security and certainty which 

 its base never offered. We should be sure that oar subsequent 

 labors have introduced no further uncertainties, but never for- 

 get that those primarily existing must persist through our 

 entire work. 



Let us now consider some simple illustrations of the applica- 

 tion of the principles of absolute and primary stresses and 

 strains. 



First let us consider how a simple and fairly definite state 

 of primary stress and strain might be created. Imagine three 

 bars of the same material, relatively broad for their thickness, 

 relatively long for their breadth, practically the same in thick- 

 ness and in breadth, but not quite of equal length, one being a 

 trifle longer than the other two. Suppose the three bars were 

 placed side by side, and lightly clamped together, their larger 

 sides in contact and the longer bar in the middle ; suppose 

 there was applied to the ends of the longer bar such a com- 

 pressive force that it was shortened to just the same length as 

 that of the other two bars, and then that the three bars were 

 firmly united by forcing some binding material between their 

 adjacent faces. ' Now suppose the compressive force removed 

 from the middle bar ; it would expand, carrying with it the two 

 adjacent bars, supposing the binding material sufficiently strong, 

 and as a result we would have a composite bar built up of 

 three layers of equal thickness, the two outer layers subject to 

 tension, the middle layer to compression. It is evident that 

 the force of compression of the middle layer would equal the 

 sura of the tensions on the two side layers, that is, would be of 

 intensity double that in the side layers. That is, the center 



