Alternating Currents and other Periodic Quantities. 475 



to apply, and possibly many problems exist which may be solved in 

 this manner. This seems the more likely when it is remembered 

 that, in problems involving periodic forces and fluxes, the energy 

 depends npon the mean product of the force and its corresponding flux, 

 or, if the forces and fluxes are proportional, upon the mean square of 

 the value of either. The theorem, however, seems most immediately 

 applicable to the case of alternate current problems. 



To apply the theorem to alternating currents, let us first of all 

 consider the case of a network of conductors in which there are 

 only three arbitrary periodic electromotive forces, Ei, B 2 , and E 3f 

 acting in the branches. If the resistances are all non-inductive, it 

 will always then be possible to express the current in each branch, 

 and the potential between any two points, as a linear function of E lt 

 E 2 , and E 3 , the coefficients only involving the resistances of the net- 

 work branches. If x x , 02, and a 3 are vectors representing the periodic 

 functions Ej, E 2 , E 3 , it will be possible to represent all the potentials 

 and all the currents involved by means of a network of vectors. 



A case in which the resistance of one of the branches is inductive, 

 owing to its possession of a constant coefficient of self-induction, can 

 best be treated by assuming that such a branch possesses an arbitrary 

 electromotive force E, related to the current C, passing through the 

 branch in a manner given by the equation 



dt 



We know nothing of the relation between the vectors representing 

 E and C, except that their scalar product is zero, and they are in 

 consequence perpendicular to each other. No energy is wasted, due 

 to self-induction, since 



CL-dt = o. 



This relation gives a certain connexion between the coefficients 

 which express C as a linear function of the vectors denoting the 

 electromotive forces, but does not make it possible to express E as a 

 linear function of the other electromotive forces E x and E 2 . For 

 although we may put E = X^Ex + X 2 E 2 , the coefficients Xi and \ 2 

 will not be scalar quantities, since they will involve not only the 

 resistances of the branches of the network, but also the operator 

 d\dt. It will always be possible to find a vector to represent E, but 

 only in special cases will this vector be in the same plane as the 

 vectors denoting E x and E 2 . 



Similarly a case in which two points of the network are connected 

 with the terminals of a condenser of capacity K can best be treated 



