476 Dr. W. E. Sumpner. The Vector Properties of 



by assuming that the condenser is replaced by a conductor possessing 

 an arbitrary electromotive force E, related to the current passing 

 through the branch in a manner given by the equation 



E = Cdt. 



The vectors representing E and C are such that their scalar 

 product is zero, and the same remarks apply as in the preceding case. 



The case of a transformer can be similarly treated. The reaction 

 due to the varying induction of the core produced by a current C 

 through the primary, must be treated as an arbitrary electromotive 

 force E. The case will differ from that of constant self-induction, in 

 that the vectors representing E and C are at right angles in the latter 

 case, but not in the former, since, owing to hysteresis and eddy 

 currents, the core absorbs energy, and the scalar product of the 

 vectors E and C cannot be zero. 



It will thus be seen that the effect of inductance in a branch of the 

 network is the same as that of an arbitrary electromotive force placed 

 there, whether the inductance be due to constant self-induction, to 

 condenser action, or to transformer action. With this understanding 

 it follows that a vector figure can be draivn in space, completely repre- 

 senting the alternating currents and potentials in a network of conductors, 

 provided the number of independent alternating electromotive forces does 

 not exceed three. This figure can be drawn in a plane if the number 

 of such electromotive forces is not more than two. Of course, where 

 a linear relation, with scalar coefficients, connects three electromotive 

 forces, only two of these are to be regarded as independent. 



Now the vector method which has hitherto been so largely used in 

 explaining alternating current phenomena, will not be really service- 

 able in the new and extended application here suggested, except in 

 cases in which the vector figure is either confined to one plane, or 

 can, with all essential accuracy, be treated as if that were the case. 



Fortunately, severaJ circumstances conspire to simplify the appli- 

 cation of the theorem to alternate current problems. In the first 

 place the vectors which are needed are divisible naturally into two 

 sets : one representing the currents, and the other the voltages. It 

 is never necessary to find the product of two of these vectors unless 

 they belong to different sets. Although in some cases it may not be 

 possible to construct the complete vector figure in one plane, it may 

 still be possible to construct two figures — one for each set. In prac- 

 tical cases one of these figures (say the current vectors) will almost 

 certainly be confined to one plane, and if the vectors of the other (or 

 voltage) network be projected perpendicularly on to this plane, the 

 product of any projected voltage vector and the appropriate current 



