三次元弾性論における変位の方程式の解として1885年Boussinesqは変位を三つの調和関数\({\varphi _0},{\varphi _3},{\lambda _3}\)で示した．直角座標\(\left( {x,y,z} \right)\)で\[\begin{array}{l} 2G{u_x} = \frac{\partial }{{\partial x}}\left( {{\varphi _0} + z{\varphi _3}} \right) + 2\frac{{\partial {\lambda _3}}}{{\partial y}}\\ 2G{u_y} = \frac{\partial }{{\partial y}}\left( {{\varphi _0} + z{\varphi _3}} \right) - 2\frac{{\partial {\lambda _3}}}{{\partial x}}\\ 2G{u_z} = \frac{\partial }{{\partial z}}\left( {{\varphi _0} + z{\varphi _3}} \right) - 4\left( {1 - \nu } \right){\varphi _3} \end{array}\]ここで，\[\begin{array}{l} {\nabla ^2}{\varphi _0} = {\nabla ^2}{\varphi _3} = {\nabla ^2}{\lambda _3} = 0\\ {\nabla ^2} = \frac{\partial }{{\partial {x^2}}} + \frac{{{\partial ^2}}}{{\partial {y^2}}} + \frac{{{\partial ^2}}}{{\partial {z^2}}} \end{array}\]Gは横弾性係数，νはポアソン比である．\({\varphi _0},{\lambda _3},{\varphi _3}\)はそれぞれ第1，2，3基本解と呼ばれる．ベクトル表示は\[2G\boldsymbol{u} = {\rm{grad}}\left( {{\varphi _0} + z{\varphi _3}} \right) - 4\left( {1 - \nu } \right)\left[ {0,0,{\varphi _3}} \right] + 2\,{\rm{rot}}\left[ {0,0,{\lambda _3}} \right]\]