120 Sir W. Thomson on Cauchy's and Green's 



For brevity, we shall designate as («, fi, 7) the strained con- 

 dition thus defined. 



8. As a purely kinematic preliminary, let it be required to 

 find the principal strain-ratios when the solid, already strained 

 according to (1), (2), is further strained by a uniform shear. 

 er, specified as follows; in terms of x, y, z, the coordinates of 

 still the same particle, M, of the solid and other notation, as 

 explained below : — 



X = %\/oi + apt \ 



y=zf}y/P + (rpm> (3), 



z — $*/<y + crpn J 

 where 



p = OP = \Z\/u+fjL V S/3 + vWY • • • (4), 

 with 



P+m 2 + ra a = l, \ 2 +p- + v 2 =l . . . (5), 

 and 



I\ + nip + nv = (6); 



\, fi, v denoting the direction-cosines of OP, the normal to 

 the shearing planes, and I, m, n the direction-cosines of 

 shearing displacement. The principal axes of the resultant 

 strains are the directions of OM in which it is maximum or 

 minimum, subject to the condition 



F4V+P=1 (7); 



and its maximum, minimax, and minimum values are the 

 three required strain-ratios. Now we have 



OM 2 =.* 2 +2/ 2 + * 2 



= ? a + v '>/3 + ¥ry + 2a(l^* + m V Sfi + n^y)p + ay (8), 



and to make this maximum or minimum subject to (7), we 

 have 



dt p S' d v PVi at -n - • ( 9 ); 



where in virtue of (7), and because OM 2 is a homogeneous 

 quadratic function of £, ij, £, 



p = OM 2 . 



The determinantal cubic, being 



{S4-p){@-p)(£-p)-a 2 {S4-p)-b 2 (3-p)-c 2 (£-p) + 2abc = 

 where 



£4=*(1 + 2<tI\ + <t 2 \ 2 ); &=/3(l + 2<rmfi+o*ft*) ; 



^=y(l + 2o-ni/ + o-V) (11) 



