﻿506 Mr. R. F. Grwyther on Conditions for 



To complete the equations, we substitute for {e,f,g, a, b, c} 

 from (2), equate to the mechanical elements of stress given 

 by (1), and substitute for {<j> u </> 2 , <£ 3 , Xi, % 2 , %3} to find the 

 six equations for condition in {0 1 6 2 , #3, ^1, ^ 2 , ^3}* This 

 general process will be now simplified. 



In order to carry out the scheme of this paper it has been 

 necessary that the axes should be arbitrary and the stress- 

 elements should be written in full. But this stage having 

 been completed we may now select a special set of axes, and 

 also simplify the arbitrary stress-elements. 



Thus, {e,f, g, a, b, c} act on transformation of axes as 

 {■x 2 , y 2 , z 2 , 2yz, 2xz, 2xy\ act, as do also {a 2 — kfg, b 2 —^eg y 

 c 2 — ief, 4:ae — 2bc, 4:bf—2ac, 4=cg — 2ab\, and since 



• \dv 2 OvJ d^ 0^5 0^6 



we find that 



v\0t? + v 2 y 2 + v 3 z 2 + 2v±yz + 2v h xz + 2v 6 xy — 1 



is an invariantal ellipsoid, and the principal planes of thi& 

 ellipsoid are planes of elastic symmetry in the body. 



I shall take these principal planes as the coordinate planes, 

 and consequently v 4 = 0, v 5 = 0, v G = 0, while v 1? v 2 , v 3 now 

 stand for the roots of the Discriminating Cubic. 



The equations (11), (12), and (13), become for these axes, 



01 = "i(#2 + As), 2xi = — O2 + ^3)^1, 



02 = ^2(^3 + #i), ^%2 = — (v 3 + Vi)^ 2 , 



03 = V'i(0 1 + 6 2 ), 2X3 = — (Vl + V 2 )^3, 



P = X e + (\-2v 3 )f+(\-2v 2 )g, 

 Q = (X-2v s )e + \f+(\-2v 1 )g, 

 R = (\-2v 2 )e + (\-2v 1 )f+\g, 

 S = Vl a, T = v 2 b, U = vtp. . • • (14) 

 It will now suffice to put ^ = 0, i/r 2 = 0, ^3 = 0, so that 



a^i 30 2 303 

 a^ oy o^ 



and it follows, from above, that %i = 0, X^ — ^-> %3 = ^ a ^ ^ n ^ 

 same time. We shall therefore remain with only three 

 equations in U 2 , #3, when these simplifications are made. 



