744 SCIENCE AND HYPOTHESIS 



between Bertrand and Helmholtz. Helmholtz replaces Ampere's hypo- 

 thesis by the following : Two elements of current always admit of 

 an electro-dynamic potential, depending solely upon their position 

 and orientation; and the work of the forces that they exercise one on 

 the other is equal to the variation of this potential. Thus Helmholtz 

 can no more do without hypothesis than Ampere, but at least he does 

 not do so without explicitly announcing it. In the case of closed 

 currents, which alone are accessible to experiment, the two theories 

 agree; in all other cases they differ. In the first place, contrary to 

 what Ampere supposed, the force which seems to act on the movable 

 portion of a closed current is not the same as that acting on the 

 movable portion if it were isolated and if it constituted an open cur- 

 rent. Let us return to the circuit C', of which we spoke above, and 

 which was formed of a movable wire sliding on a fixed wire. In the 

 only experiment that can be made the movable portion a /3 is not 

 isolated, but is part of a closed circuit. When it passes from AB to 

 A'B', the total electro-dynamic potential varies for two reasons. First, 

 it has a slight increment because the potential of A'B' with respect to 

 the circuit C is not the same as that of AB ; secondly, it has a second 

 increment because it must be increased by the potentials of the ele- 

 ments AA' and B'B with respect to C. It is this double increment 

 which represents the work of the force acting upon the portion AB. 

 If, on the contrary, aft be isolated, the potential would only have the 

 first increment, and this first increment alone would measure the work 

 of the force acting on AB. In the second place, there could be no 

 continuous rotation without sliding contact, and in fact, that, as we 

 have seen in the case of closed currents, is an immediate consequence 

 of the existence of an electro-dynamic potential. In Faraday's experi- 

 ment, if the magnet is fixed, and if the part of the current external 

 to the magnet runs along a movable wire, that movable wire may 

 undergo continuous rotation. But it does not mean that, if the con- 

 tacts of the wire with the magnet were suppressed, and an open cur- 

 rent were to run along the wire, the wire would still have a movement 

 of continuous rotation. I have just said, in fact, that an isolated 

 element is not acted on in the same way as a movable element making 

 part of a closed circuit. But there is another difference. The action 

 of a solenoid on a closed current is zero according to experiment and 

 according to the two theories. Its action on an open current would be 

 zero according to Ampere, and it would not be zero according to 

 Helmholtz. From this follows an important consequence. We have 

 given above three definitions of the magnetic force. The third has 

 no meaning here, since an element of current is no longer acted upon 

 by a single force. Nor has the first any meaning. What, in fact, is 

 a magnetic pole ? It is the extremity of an indefinite linear magnet. 

 This magnet may be replaced by an indefinite solenoid. For 



