It is proposed that the arrangements of the electron pairs in the valency shell of a central atom of a polyatomic molecule can be predicted by considering the equilibrium arrangements of similar particles on the surface of a sphere with an appropriate law of force between the particles. The arrangements resulting from an inverse square law of force, corresponding to electrostatic repulsions, and a force which is proportional to 1/rn where n is large, corresponding to Pauli forces, are considered specifically. It is shown that the arrangements predicted agree with those found experimentally for molecules containing only non-transitional elements. The possible arrangements for seven, eight, and nine pairs of electrons in a valency shell are discussed in detail. A method is suggested for predicting the arrangements of electron pairs in valency shells containing lone pairs which can occupy alternative non-equivalent positions. The effect of the interactions of electron pairs on bond lengths in certain molecules is discussed. The extension of the same principles to molecules containing transitional elements is briefly outlined.