72. The stress system considered in § 71 is equivalent, on the plane through the origin at right angles to the line of action of F, to a resultant pressure of magnitude ½F at the origin and a [1 − 2σ/2(1 − σ)] · F/4πr², and, by the application of this system of tractions to a solid bounded by a plane, the displacement just described would be produced. There is also another stress system for a solid so bounded which is equivalent, on the same plane, to a resultant pressure at the origin, and a radial traction proportional to 1/r², but these are in the ratio 2π : r−2, instead of being in the ratio 4π(1 − σ) : (1 − 2σ)r−2.
The second stress system (see fig. 31) consists of:
(1) radial pressure F′r−2,
(2) tension in the meridian plane across the radius vector of amount
F′r−2 cos θ / (1 + cos θ),
(3) tension across the meridian plane of amount
F′r−2 / (l + cos θ),
(4) shearing stress as in § 71 of amount
F′r−2 sin θ / (1 + cos θ),
and the stress across the plane boundary consists of a resultant pressure of magnitude 2πF′ and a radial traction of amount F′r−2. If then we superpose the component stresses of the last section multiplied by 4(1 − σ)W/F, and the component stresses here written down multiplied by −(1 − 2σ)W/2πF′, the stress on the plane boundary will reduce to a single pressure W at the origin. We shall thus obtain the stress system at any point due to such a force applied at one point of the boundary.