Manchester Memoirs, Vol. /xii, (1917)7^. 1 3 



A Theory of the Displacements of the Material Bodies 

 as a Consequence of Stress. 



In this final portion of the paper I hope to be able to 

 explain the proposal I wish' to put forward of a method of 

 treating questions of the stress and displacement in an elastic 

 body. It has been with the object of justifying a method of 

 this kind that I have written the several parts of the paper, 

 but the details of the scheme and its practical application have 

 developed themselves in the course of the work, and it would 

 have been better if this latter portion had been ready first. 



To explain my proposals, I shall commence with the remark 

 which is, I think, obvious : That a material body can only be 

 free from stress between its component particles when each such 

 particle is moving freely under such system of forces as the 

 particles are subject to; and that this is the case whether the 

 body be rigid or yielding. 



If a biearn is at rest, supported in any mode under gravity, 

 the material of the beam is in a state of stress, and if the beam 

 is (Swinging about an axis under gravity, the material is in a 

 state of stress, which in this case varies not only with the position 

 of the particle considered, but also with the time. 



My proposal is intended to be applicable to cases of motion 

 as iwell as to cases of rest. Wfe are to deal first' with the hypo- 

 thesis of rigidity, and accordingly I shall assume that questions 

 of the Statics and Dynamics of the rigid body do not enter into 

 the present enquiry. In fact, I shall proceed not only as if such 

 questions were solvable, but as if they had been actually solved. 



We will treat the number of elements of a stress as six, 

 and not nine, and this is undoubtedly the case in the material 

 stresses of a rigid body. But if the stress is definite, and as 

 the number of conditions from which it can be deduced are Only 

 three in number, we are entitled to assume that there is some 

 condition generally affecting the elements of a material stress. 

 Any such hypothesis must be reasonable, and must find justifi- 

 cation both on mathematical and physical grounds. The condi- 

 tion which I shall assume is : That the elements of a material 

 stress are functions of Ithe first differential co-efficients of some 

 vector. The physical justification of this hypothesis lies in the 

 superstructure of analysis of stress and strain which has been 

 developed out of Hooke's Law, and the general acceptance of 

 the doctrine by engineers and physicists. The mathematical 

 justification is put forward below. (Appendix A.) 



On this hypothesis it has been shown in Part IV. that six 

 equations are to be found, giving at any rate to a first approxi- 

 mation the six elements of Stress, and in the earlier parts of this 

 paper it is shown in general terms how they are to be solved. 



