Consider the barrel of a pump placed in a vacuum and closed by a piston at each end, and let us introduce between these a certain mass of air. The two pistons, through the elastic force of the gas, repel each other with a force which, according to the law of Mariotte, varies in inverse ratio to the distance. The method favoured by Ampère would first of all allow this law of repulsion between the two pistons to be discovered, even if the existence of a gas enclosed in the barrel of the pump were unsuspected; and it would then be natural to localize the potential energy of the system on the surface of the two pistons. But if the phenomenon is more carefully examined, we shall discover the presence of the air, and we shall understand that every part of the volume of this air could, if it were drawn off into a recipient of equal volume, carry away with it a fraction of the energy of the system, and that consequently this energy belongs really to the air and not to the pistons, which are there solely for the purpose of enabling this energy to manifest its existence. Faraday made, in some sort, an equivalent discovery when he perceived that the electrical energy belongs, not to the coatings of the condenser, but to the dielectric which separates them. His audacious views revealed to him a new world, but to explore this world a surer and more patient method was needed. Maxwell succeeded in stating with precision certain points of Faraday's ideas, and he gave them the mathematical form which, often wrongly, impresses physicists, but which when it exactly encloses a theory, is a certain proof that this theory is at least coherent and logical.[23] [Footnote 23: It will no doubt be a shock to those whom Professor Henry Armstrong has lately called the "mathematically-minded" to find a member of the Poincaré family speaking disrespectfully of the science they have done so much to illustrate. One may perhaps compare the expression in the text with M. Henri Poincaré's remark in his last allocution to the Académie des Sciences, that "Mathematics are sometimes a nuisance, and even a danger, when they induce us to affirm more than we know" (_Comptes-rendus_, 17th December 1906).] The work of Maxwell is over-elaborated, complex, difficult to read, and often ill-understood, even at the present day. Maxwell is more concerned in discovering whether it is possible to give an explanation of electrical and magnetic phenomena which shall be founded on the mechanical properties of a single medium, than in stating this explanation in precise terms. He is aware that if we could succeed in constructing such an interpretation, it would be easy to propose an infinity of others, entirely equivalent from the point of view of the experimentally verifiable consequences; and his especial ambition is therefore to extract from the premises a general view, and to place in evidence something which would remain the common property of all the theories.