28 
Proceedings of Royal Society of Edinburgh. [dec. 5 , 
provided none of tlie differences constituting tlie denominators is 
infinitely small. The case of any of these differences infinitely 
small, or zero, does not, as we shall see in the conclusion, require 
special treatment, though special treatment would he needed to 
interpret for any such case each step of the process. 
14. Substituting now for a, 6, e in (20) their values by 
(11) and (12) ; neglecting cr^ and higher powers ; and denoting by 
Sa, 8ft Sy the excesses of the three roots above a, /3, y respectively, 
we find 
.0. 
<5a = a|2cr^X +^2 + (Z/x + mX)2 
2o-m/x + (T2 ]|^ • . . . (21); 
^7 — 7 1 2crwy +cr^ |^i/2 - ^ ^ [mv + Ufif' — — J |- 
and using these in (19), we find 
^ E = o-( AZX + 'Qmix + Cnv) 
+^o-2{ AX2 + B^2^QJ;24.XJ(mv + ?^/x)2+M{wX + ^yp + N(Zyu + ?/^X)2} ^ , 
+ 2(72(0^2X2 + H??1-2^2 ^ I^2y2^ j 
where L = ?pf^; M = ^AzAl; N = ^ 
p - y y — a. a — p 
15. Now from (5) and (6), we find 
(mv + np)^==l-l^-\^ + 2{m-rnY-n^y^) .... (24); 
which, with the symmetrical expressions, reduces (22) to 
^E = cr(AZX + B??i/i + Cni;) ^ 
+ i(r2{L + M + N + (A-L)X2 + (B-MV2 + (C-N>2-LZ2-Mm2-Nw2 i ,(25) 
+ 2[(2G+L-M-N)Z2X2 + (2H + M-N-L)m2^2 + (2i + N-L-M)w2i;2]} j 
( 22 ); 
23). 
16. To interpret this result statically, imagine the solid to he 
given in the state of homogeneous strain (a, ft y) throughout, and 
let a finite plane plate of it, of thickness h, and of very large area 
Q, he displaced by a shearing motion according to the specification 
(3), (4), (5), (6) of § 8 ; the hounding planes of the plate being 
unmoved ; and all the solid exterior to the plate being therefore 
undisturbed, except by the slight distortion round the edge of the 
plate produced by the displacement of its substance. The analytical 
expression of this is 
<^=f(p) (26), 
where / denotes any. function of OP such that 
