(1)
where ν is the direction of the normal (drawn outwards) to the remaining face of the tetrahedron, and (x, ν) ... denote the angles which this normal makes with the axes. Hence Xν, ... for any direction ν are expressed in terms of Xx,.... When the above result is interpreted for a very small portion in the shape of a cube, having its edges parallel to the co-ordinate axes, it leads to the equations
Yz = Zy, Zx = Xz, Xy = Yx.
(2)
When we substitute in the general equations the particular results which are thus obtained, we find that the equations of motion take such forms as
| ρX + | ∂Xx | + | ∂Xy | + | ∂Zx | = ρƒx, |
| ∂x | ∂y | ∂z |
(3)
and the equations of moments are satisfied identically. The equations of equilibrium are obtained by replacing the right-hand members by zero.
| Fig. 1. |
| Fig. 2. |