1887 .] Sir W. Thomson on Doctrine of Extraneous Pressure. 27 
2S/38y--(^-^-^j 
9S s 
P‘ 
8y2 
r 
ho? 
Y ■ 
(18). 
1/Sy2 Sa2 
r a- 
11. Now if E + 8E denote the energy per unit hulk of the solid 
in the condition 
(a + Sa, ^ + hp, 7 + 8y) ; 
we have, by Taylor’s theorem, 
8E = + H 2 + Hg + &c. 
where H 2 , &c. denote homogeneous functions of Sa, Sft Sy, of 
the 1®^ degree, 2"^ degree, &c. Hence omitting cubes, &c., and 
eliminating the products from H 2 , and taking Hj from (15), we find 
1/A, 
da' 
SE = ;f-8a + ?S^ + ^8y + G^ +1^^" 
t)' 
(19). 
where G, H, I denote three coefficients depending on the nature of 
the function ij/, (13), which expresses the energy. Thus in (19), with 
(14) taken into account, we have just five coefficients independently 
disposable. A, B, G, H, I ; which is the right number because, in 
virtue of a^y= 1, E is a function of just two independent variables. 
12. For the case of a=l, /S=l, y=l, we have A = B = C = 0; 
and G = H = I = G^, suppose ; which give 
8E = iGi(Sa2 + 8/32 + Sy2). 
From this we see that 2Gj is simply the rigidity modulus of the 
unstrained solid ; because if we make Sy = 0, we have 8a— -8/3 and 
the strain becomes an infinitesimal distortion in the plane (xy) 
which may be regarded in two ways as a simple shear, of which 
the magnitude is 8a * (this being twice the elongation in one of 
the normal axes). 
13. Going back to (10), (11), and (12), let o-beso small that errand 
higher powers can be neglected. To this degree of approximation, 
we neglect abc in. (10), and see that its three roots are respectively 
^ ^ ^ r-o) 
* Thomson and Tail's Natural Philosophy, § 175 ; or Elements, § 154. 
