1887.] Sir W. Thomson o)i Doctrine of Extraneous Pressure. 29 
r dpf{p) = f) (27). 
J 0 
If we denote by W the work required to produce the supposed 
displacement, we have 
W = Q^rfj5SE + W (28), 
8E being given by (25), with everything constant except o- a 
function of OP ; and W denoting the work done on the solid out- 
side the boundary of the plate. In this expression the first line of 
(25) disappears in virtue of (27) ; and we have 
w_ w , 
P — =1{L + M + N + (A-L)X2 + (B-M)/x2 + (C-N)i;2-LZ2-Mwi2-N7i2 
Q 
+ 2[(2G + L - M - N)Z2\2 + (2H + M - N - L)m2/x2 + (21 + N - L - ^i?o-2 . . (29) . 
When every diameter of the plate is infinitely great in comparison 
with its thickness, W/Q is infinitely small; and the second 
member of (29) expresses the work per unit of area of the plate, 
required to produce the supposed shearing motion. 
17. Solve now the problem of finding, subject to (5) and (6) 
of § 8, the values of /, n which make the factor { } of the 
second member of (29), a maximum or minimum. This is only 
the problem of finding the two principal diameters of the ellipse 
in which the ellipsoid 
[2(2G+L-M-N)A2-L]o;2+[2(2H+M-N-LV2-M]y2+[2(2H-N-L-M)i/2-N]^2::,const. . . (30) 
is cut by the plane 
'Kx + ixy + vz = 0 . . . . . . ( 31 ). 
If the displacement is in either of the two directions (Z, m, n) thus 
determined, the force required to maintain it is in the direction of 
the displacement ; and the magnitude of this force per unit bulk 
of the material of the plate at any point within it is easily proved 
to be 
{»•}$ (”). 
where {M} denotes the maximum or the minimum value of the 
bracketed factor of (29). 
18. Passing now from equilibrium to motion, we see at once 
that (the density being taken as unity) 
Y2={M} 
(33) 
