468 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. 



a certain finite or infinite number of geometrical parameters q s . 

 The electromagnetic forces due to the action of the currents may 

 be equilibrated by the action of certain impressed forces P S) and 

 these forces may be determined as partial derivatives with respect 

 to the parameters of the potential energy, or of the energy of the 

 field. These impressed forces we shall call the positional forces, and 

 since they are the negatives of the electromagnetic forces already 

 found, we have for any positional force P s , 



In order to specify the action of the system completely, we must 

 give, beside the values of the parameters q s , only the values of 

 the current-strength in every current-tube. If the currents are 

 distributed in three dimensions, this necessitates an infinite 

 number, but if there are a finite number of linear conductors, 

 only a finite number of electrical parameters I s . The energy of 

 the field is expressed as a homogeneous quadratic function of these 

 electrical parameters, the coefficients being functions of the posi- 

 tional parameters, whose velocities do not occur. If we consider 

 the negative energy of the field W as, instead of the negative 

 potential energy, the electrokinetic energy of the field, the current 

 strengths being considered as cyclic velocities, the analogy to a 

 mechanical cyclic system is complete. The cyclic coordinates q s} 

 being the time-integrals of the currents, represent the total 

 amounts of electricity that have traversed the respective circuits 

 since a fixed epoch. Since neither these coordinates, nor the 

 velocities of the positional coordinates occur in the expression for 

 the electrokinetic energy, all the conditions for a cyclic system are 

 fulfilled. A restriction must, however, be made, which is of no 

 importance in practice, namely that the velocities of the positional 

 coordinates must be small compared with a certain velocity, which 

 in this case is the velocity v, the ratio of the two units of elec- 

 tricity. For the case of all ordinary velocities, however, the 

 electrokinetic energy is accurately represented in the form already 

 found. 



If we have n linear currents, the electrokinetic energy is 

 T= IA/! 2 + M,JJ, ...... + M ln IJ n 



(2) + \LJ? + -23/2/3 



