where a1, a2, a = a1a2/√ (a12 + a22) is the radius of the spheres and their circle of intersection, and r1, r2, r the distances of a point from their centres.

The corresponding expression for two orthogonal cylinders will be

ψ′ = Uy ( 1 − a12− a22+ a2).
r12 r22r2

(8)

With a2 = ∞, these reduce to

ψ′ = ½Uy2 ( 1 − a5) x, or Uy ( 1 − a4) x,
r5 ar4 a

(10)

for a sphere or cylinder, and a diametral plane.

Two equal spheres, intersecting at 120°, will require

ψ′ = ½Uy2 [ x− a3+ a4 (a − 2x)+ a3− a4 (a + 2x)],
a 2r132r15 2r232r25