1887.] Sir W. Thomson on Doctrine of Extraneous Pressure. 25 
x = i Ja + apt ] 
y = r] Jf3 + (rpm (3), 
2 C x/y + o-jm J 
where p = 0~P = Ja + prj J/S + v^ Jy (4), 
with + -\-7i^ —1 ; \^ + + v'^ — 1 (5) , 
and l\ + 7nfjL-{-nv = 0 (6); 
X, fjL, V denoting the direction-cosines of OP, the normal to the 
shearing planes; and I, w, n the direction-cosines of shearing dis- 
placement. The principal axes of the resultant strains are the direc- 
tions of OM in which it is maximum or minimum, subject to the 
condition 
+ + (7). 
and its maximum, minimax, and minimum values are the three 
required strain-ratios. Now we have 
OM^ = x‘^-\-y^ + z^ 
= + f/3 + -1- 2cr{li Ja + mrj 4- Jy)p + . . . (8) : 
and to make this maximum or minimum subject to (7), we have 
ciaOM^) . c^(iOM2) c?(iOM2) . ... 
where in virtue of (7), and because OM^ is a homogeneous quadratic 
function of rj, 
p = OM2 (10). 
The determinantal cubic, being 
-p)- -p)- i w- p) - p) + = 0 , 
where 
<^=a(l-l-2o-ZA.-l-(r2A.2) ; J^=/3(1 -f2o-m/i + o-V^) ; <^=y(l + 2<r?ti' + o-i;2) . . (11), 
and 
V{(3y)[a(mv + niuL) + (T'^luLv]; h=V{ya)[(T{n\-\-lv) + g^vX\", c=\Z(a/3)[<r(Z/>t-l-mA.)4-o-2X^] . .(12), 
gives three real positive values for p, the square roots of which are 
the required principal strain-ratios. 
9. Entering now on the dynamics of our subject, remark that the 
isotropy (§ 7) implies that the work required of the extraneous 
pressure, to change the solid from its unstrained condition (1, 1, 1) 
to the strain (a, /?, y), is independent of the direction of the normal 
axes of the strain, and depends solely on the magnitudes of a, /3, y. 
