86 



When a stress and a strain are of the same type, they are said to 

 be concurrent ; or, if directly opposed, they are said to be negatively 

 concurrent. When a stress and a strain are of any different types, 

 the degree of their concurrence, or simply " their concurrence," is 

 measured by the work done by the stress applied to a body of unit 

 volume acquiring the strain, divided by the product of the magnitude 

 of the stress into the magnitude of the strain. The measure of per- 

 fect concurrence is therefore + 1, and that of perfect opposition 1. 

 When work is neither spent nor gained in the application of a cer- 

 tain stress to a body while acquiring a certain strain, that stress and 

 that strain, or any stresses or strains of the same types respectively, 

 are said to be orthogonal to one another. The measure of their con- 

 currence is zero. 



A system of stress or strain coordinates involving symmetrically 

 six independent variables, perfectly analogous to the system of triple 

 coordinates for specifying the position of a point in space, is laid 

 down. The concurrence of a stress or strain with six orthogonal 

 types of reference being denoted by I, m,n,\, p., v, it is demonstrated 

 that 



and it is proved that if cos 6 denote the mutual concurrence between 

 two stress or strain types, whose concurrences with six orthogonal 

 types of reference are respectively (I, m, n, X, p, v) and (I 1 , m', n', 

 \', p.', v'), we have 



cos = IV + mm' + nn' + XX' + pp.' + rv' . 



The treatment of the subject in the text of the paper is quite 

 abstract, but along with it a series of examples are given, illustrating 

 the statements by applications to familiar types of stresses and 

 strains. 



Part II. commences with an interpretation of the Differential 

 Equation of the potential energy of Elasticity of a solid, in terms of the 

 mode of specification of stresses and strains laid down in Part I. 

 The Quadratic Function expressing the potential energv of an elastic 

 solid when strained to an infinitely small amount, is next considered ; 

 and its simplest possible form, that of six squares with coefficients, is 

 interpreted. Hence it is proved that an infinite number of systems 

 of six types of strains or stresses exist in any given elastic solid, 



