32 MR CLERK MAXWELL ON 



If we assume three functions ABC, such that 

 and put 



d 2 A 

 dydz 



T == - — 



dzdx 



u - d2C 



dxdy 



X == — 

 dx 



dy 



z== — 



dz 



(2), 



then a sufficiently general solution of the equations of equilibrium is given by 

 putting 



p = (PB d*C _ y 



dz 2 dy 2 



(IH) 



d-r 2 



q = v; - v: - v 



d*A 

 dz 2 



> ■ 



(3), 



rf?/ 2 da a 



I am not aware of any method of finding other relations between the com- 

 ponents of stress without making further assumptions. The most natural 

 assumption to make is that the stress arises from elasticity in the body. I 

 shall confine myself to the case of an isotropic body, such that it can be deprived 

 of all stress and strain by a removal of the applied forces. In this case, if 

 a (3 y are the components of displacement, and n the co-efficient of rigidity, the 

 equations of tangential elasticity are, by equation (6) §§ 670 and 694 of Thoms< >n 

 and Tait, 



dz dy n n dydz 



(4), 



with similar equations for b and c. A sufficiently general solution of these equa- 

 tions is given by putting 



a = ± 4-(a-b^g) ^ 



In d:c \ / 



^ = ^|( B - C - A ) I • • • W 



2n dz \ J J 



The equations of longitudinal elasticity are of the form given in § 693, 



*-(»*J-)£+(»-S-)(f*3) • • • «, 



where k is the co-efficient of cubical elasticity, with similar equations for Q and 

 R. Substituting for P, a, /3 and y in equation (6) their values from (3) and (5), 



3\ / 2 \/d 2 B_dK>_d 2 A dK_dPA_d^B\ 

 ?) + \ 3 n J\dy 2 dy 2 dy 2 +dz 2 dz 2 dz 2 ) ' 



. /d 2 B d 2 G „\ /, ,4 \(d 2 A 



d 2 A d?B_d 2 C 

 dx 2 dx 



