双球座標系
∇ 2 f = ( cosh  τ − cos  σ ) 3 a 2 [ 1 sin  σ ∂ ∂ σ ( sin  σ cosh  τ − cos  σ ∂ f ∂ σ ) + ∂ ∂ τ ( 1 cosh  τ − cos  σ ∂ f ∂ τ ) + 1 sin 2  σ ( cosh  τ − cos  σ ) ∂ 2 f ∂ φ 2 ] {\displaystyle \nabla ^{2}f={\frac {(\cosh \tau -\cos \sigma )^{3}}{a^{2}}}\left[{\frac {1}{\sin \sigma }}{\frac {\partial }{\partial \sigma }}\left({\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}{\frac {\partial f}{\partial \sigma }}\right)+{\frac {\partial }{\partial \tau }}\left({\frac {1}{\cosh \tau -\cos \sigma }}{\frac {\partial f}{\partial \tau }}\right)+{\frac {1}{\sin ^{2}\sigma \left(\cosh \tau -\cos \sigma \right)}}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}\right]}