286 



Mr JR. F. Gwyther, On the solution of the [May 25, 



3. As we are to retain the squares and products of the 

 differential coefficients of £, rj, £, we must extend the expression 

 for the work (w) required to produce the change of state of strain 

 per unit volume. We therefore treat w as a cubic function of 

 the roots of 



K-A, c, b = 0, 



c, K— B, a 



b, a, K- C 



(Thomson and Tait, Vol. I. part 2. App. C. K.) 



where A . = 1 + 2e, etc., with the usual notation. 



We will therefore write (in order to obtain all the terms of the 

 second order) 



w = w 2 + co s + »,' (9), 



where 



2w 2 = - n {4fg + 4<ge + 4e/+ a 2 + b 2 - c 2 }, 



2co 3 = l [4<efg + abc - a 2 e - b 2 f- c 2 g], 



2^ = m(e+f+g)co 2 , 



neglecting higher powers of the components of the strains. 



From these we find the P's by the usual formulae. I shall write 

 down these formulae for P xx , P xy , and P xs , omitting the part of the 

 first degree, which has been already employed in forming the 

 equations and which being of the circular harmonic type con- 

 tributes nothing to the mean pressure. 



Then 



P = 



da> 2 dj[ dw a rf| da)? di; d_ 



dx db ' dz dc ' dy de 3 3 



xz dg 



de 



dco 2 dP, 

 df dy 



d% 



r *»- j*- j„+ da 'dz + dc "dx dc [ ' 3+ ah 



dc 



dz da ' dy db 



dP d . ,. 



and similar equations. 



Before simplifying these, I shall limit the cases to those in 

 which P xy = P yx , to which the special case I am considering 

 belongs. The general condition for this is that the complementary 



minors of the determinant , vs ' ' may separately vanish, from 



(jj \0C) (J y Z) 



which it follows that the leading minors vanish also. 



