of a Crystal according to Boscovich. 417 



express the five distortional components as linear functions of 

 the five stress-components with these fifteen independent 

 coefficients. 



§ 8. To demonstrate the propositions of § 5, let OX, OY, 

 Z be three mutually perpendicular lines through any point 

 of a homogeneous assemblage, and let x, y, z be the co- 

 ordinates 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 + 8x, y + 8y, z + Sz) be the coordinates of the altered position 

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

 six symbols e, f, g, a, b, c, according to the notation of Thomson 

 and Tait's < Natural Philosophy,' vol. i. pt. 2, § 669. In this 

 notation, e,fg denote simple infinitesimal elongations parallel 

 to OX, Y, OZ respectively ; and a, b, c infinitesimal changes 

 from the right angles between three pairs of planes of the 

 substance, which, in the unstrained condition, are parallel to 

 (X Y, X Z), (Y Z, Y X), (ZOX, ZO Y) respectively 

 (all angles being measured in terms of the radian). The 

 definition of a, b, c may be given, in other words, as follows, 

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

 ponent motions parallel to Y of two planes of the substance 

 at unit distance asunder, kept parallel to Y X during the 

 displacement ; or, which is the same thing, the difference of 

 component motions parallel to Z of two planes at unit dis- 

 tance asunder kept parallel to Z X during the displacement. 

 To avoid the unnecessary consideration of rotational displace- 

 ment, we shall suppose the displacement corresponding to the 

 strain-component a to consist of elongation perpendicular to 

 X in the plane through O X bisecting Y Z, and shrinkage 

 perpendicular to X in the plane through X perpendicular 

 to that bisecting plane. This displacement gives no contri- 

 bution to 8x, and contributes to Sy and Sz respectively \az 

 and \ay. Hence, and dealing similarly with b and c, and 

 taking into account the contributions of <?,/, g, we find 



8% = ex -\~-%(bz + cy) \ 



Sy=fy + i(cx + az) I . . . . (4). 



Sz =gz + i(ay'+ba) J 



§ 9. In our dynamical treatment below, the folio wing- 

 formulas, in which powers higher than squares or products 

 of the infinitesimal ratios 8x/r, Sy/r, Sz/r (r denoting OP) 

 are neglected, will be found useful : — 



Sr _ xSx +ySy + zSz , , Sx 2 + Sif + Sz 2 x f xSx+ySy +z8z\ 2 



r ■ ' r * + 2 r 2 2 ^ r2 J . (5). 



Phil. Mag. S. 5. Vol. 36. No. 222. Nov. 1893. 2 F 



