The distance of the variable point P which has coordinates x, y, z from the fixed points (0, 0, 1) and (0, 0, -1) are denoted by u and v respectively. New variable ξ, η, ϕ are defined by \(ξ = \frac{1}{2}(u+v), η = \frac{1}{2}(u-v)\) and ϕ is the angle between the plane y = 0 and the plane containing the three points, i.e. \(ϕ = \tan^{-1}\left(\frac{y}{x}\right)\) over 1 ≤ ξ < ∞, -1 ≤ η < 1, 0 ≤ ϕ < 2π. The Jacobian of \(\frac{\partial (\xi, \eta, \phi)}{\partial (x, y, z)}\) has the value , then \(\int\int\int_{all\:space}\frac{u-v)^2}{uv}\:exp\left(-\frac{u+v}{2}\right)dxdydz=\)