An attempt is put forward to formulate as exactly as possible the basic assumption proposed by Pais in his ω-space theory, and a wider framework for the description of elementary particles is obtained which involves that of Pais as a special case. It is found necessary to introduce some field which characterizes the structure of the total manifold. The b field equations introduced by Yang and Mills give one of the admissible forms of the field equations for this new field. Finally the b field is put in comparison with the electromagnetic and the gravitational fields, and similarities and dissimilarities between them are pointed out.