6 R. F. GwYTHER, Specification of the elements of stress. 



It is not proposed to consider all these cases in this 

 paper, but only to set out general lines for investigation, 

 taking the case of Hooke's Law as the general case, but 

 having regard to departures from it. 



If Hooke's L.aw held universally, we should have a 

 second explicit expression for the stresses in terms of the 

 strains, and ultimately in terms of the three components 

 of the displacement. 



With two explicit forms for the elements of stress, 

 there are three modes of further investigation open : 



(i) To take the Hooke's Law specification as absolute 

 and to substitute in the statical equations. 

 This has been the practice. 



(2) To take the statical specification as absolute and 



to find the cases which satisfy Hooke's Law. 



(3) To combine the two specifications as may be 



found analytically most convenient in any case. 



This last method will doubtless be found convenient 

 in many cases, but it may be accompanied by the draw- 

 backs which are inherent in the combination of two 

 independent methods. 



I proceed with the second method with the object 

 of bringing into prominence the analytical connection 

 which is now to be demonstrated between the statical 

 expressions for the stress and the geometrical conditions 

 which are consequent on the fulfilment of Hooke's Law. 



The Geometrical conditiojis affecting elements of strain. 



Using the adopted notation for strains, the conditions 

 arc well known to be 



