- Suppose that we propose to find a function F which satisfies Laplace’s equation inside the sphere S and which, at the different points M of the surface of this sphere, take given values V°.
The value of this function V at a point P inside the sphere will be the integral
∫(R2 – OP2)/4πr V°/MP3
extended to all the elements of the sphere.
- Let w and W be the smallest and the largest value that V° can take; they will also be, as we know, the smallest and the largest value that V can take.
If w is positive, so will be V° and V.
Besides, as MP is always included between R – OP and R + OP; we will have
V < (R + OP)/(R – OP)2∫V°dω/4πR
V > (R – OP)/(R + OP)2∫V°dω/4πR
Source: Henri Poincaré, Sur les Equations aux Dérivées Partielles de la Physique Mathématique, American Journal of Mathematics, Mar., 1890, Vol. 12, No. 3 (Mar., 1890), pp. 211-294. Translation by Nicolae Sfetcu. © 2023 Nicolae Sfetcu