122 Sir W. Thomson on Cauchy's and Green's 



their cubes and corresponding products, and all higher 

 products. We have 



^ + *§ + B ? + a8/38y + /38 y 8a + ry8a8j3 = . (17); 

 a P 7 

 whence 



whence, and by the symmetrical expressions, 



11. Now, if E +SE denote the energy per unit bulk of the 

 solid in the condition 



(a + 8a, /3 + S/3, 7 + Sy), 



we have, by Taylor's theorem, 



SE=H 1 +H 2 +H 3 +&c., 



where H 1? H 2 , &c. denote homogeneous functions of 8a, 8/3, 87 

 of the 1st degree, 2nd degree, &c. Hence, omitting cubes 

 &c, and eliminating the products from H 2 , and taking H, 

 from (15), we find 



SE: 



!(^ + > + 5^ + Hf + i?0.a9), 



where G, H, I denote three coefficients depending on the 

 nature of the function yjr (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 aj3y—l, E is a 

 function of just two independent variables. 



12. For the case of a = l, |8 = 1, y = l, we have 



A = B = C = and G = H = I=G 1 , suppose ; 



which give 



8E = ±Gc 1 (8a 2 + 8p i + 8y 2 ). 



From this we see that 2G! is simply the rigidity-modulus of 

 the unstrained solid; because if we make 8y = 0, we have 

 8a=— 8/3, and the strain becomes an infinitesimal distortion 



