∫ ∫ lpdS = ∫ ∫ ∫ρX dx dy dz,
(1)
where l, m, n denote the direction cosines of the normal drawn outward of the surface S.
But by Green’s transformation
| ∫ ∫ lp dS = ∫ ∫ ∫ | dp | dx dy dz, |
| dx |
(2)
thus leading to the differential relation at every point
| dp | = ρX, | dp | = ρY, | dp | = ρZ. |
| dx | dy | dz |
(3)
The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically.