62 Lord Kelvin. On the [June 15, 



generalised stress components, say p 6 , as uniform positive or negative 

 pressure in all directions. This makes s 6 uniform compression or 

 extension in all directions, and makes s^ . . . . , s 5 five distortional com. 

 ponents with no change of bulk. The condition that the solid shall 

 be incompressible is then simply that the coefficients of the six terms 

 involving p 6 are each of them zero. Thus, the expression for w 

 becomes merely a quadratic function of the five distortional stress- 

 components, j?i,. . . . , ^5, with fifteen independent coefficients : and 

 equations (3) of 3 above express the five distortional components as 

 linear functions of the five stress-components with these fifteen 

 independent coefficients. 



Added July 18, 1893. 



8. To demonstrate the propositions of 5, let OX, OY, OZ b& 

 three mutually perpendicular lines through any point of a homo- 

 geneous assemblage, and let x, y> z be the coordinates of any other 

 point P of the assemblage, in its unstrained condition. As it is a 

 homogeneous assemblage of single points that we are now considering, 

 there must be another point P', whose coordinates are x, y, z. 

 Let (x + Sx, y + Sy, z + Sz) be the coordinates of the altered position of 

 P in any condition of infinitesimal strain, specified by the six symbols 

 e > /j 9> j & c "> according to the notation of Thomson and Tait's 

 'Natural Philosophy,' Vol. I, Pt. II, 669. In this notation, c,/, g 

 denote simple infinitesimal elongations parallel to OX, OY, OZ re- 

 spectively; and a, 5, c infinitesimal changes from the right angles 

 between three pairs of planes of the substance, which, in the un- 

 strained condition, are parallel to (XOY, XOZ), (YOZ, YOX), 

 (ZOX, ZOY) respectively (all angles being measured in terms of the 

 radian). The definition of a, 6, c may be given, in other words, as 

 follows, with a taken as example : a denotes the difference of com- 

 ponent motions parallel to OY of two planes of the substance at unit 

 distance asunder, kept parallel to YOX during the displacement ; or, 

 which is the same thing, the difference of component motions parallel 

 to OZ of two planes at unit distance asunder kept parallel to ZOX 

 during the displacement. To avoid the unnecessary consideration of 

 rotational displacement, we shall suppose the displacement corre- 

 sponding to the strain-component a to consist of elongation perpen- 

 dicular to OX in the plane through OX bisecting YOZ, and shrinkage 

 perpendicular to OX in the plane through OX perpendicular to that 

 bisecting plane. This displacement gives no contribution to &B, and 

 contributes to Sy and %z respectively \az and \a,y. Hence, and deal- 

 ing similarly with 6 and c, and taking into account the contributions- 

 of e, /, g, we find 





