Alternating Currents and other Periodic Quantities, 477 



vector will accurately represent the corresponding quantity of power, 

 since it is a well-known property of vectors that the scalar product 

 of any two vectors is the same as the scalar product of one of them 

 and the perpendicular projection of the other on any plane containing 

 the first. 



It will only rarely happen that a plane figure will fail to give a 

 sufficiently accurate representation of the relations between the volts 

 and amperes in alternate current circuits. These circuits are gener- 

 ally simple in character. Complication is not caused by a multitude 

 of conductors unless some of these are inductive. A network of 

 non-inductive resistances may involve a large number of currents 

 and potentials, but they are all connected by linear relations, and 

 the construction of the vector network presents no difficulty. Wher- 

 ever a transformer is used, the primary and secondary circuits can 

 be treated as quite distinct. The vector figures representing them 

 can indeed be drawn to different scales, the only necessary connexion 

 between them being the relation which connects the power put into 

 the primary core of the transformer with that taken from the secon- 

 dary. As a type of the most complicated circuit likely to occur, we 

 may take the case of an alternator feeding the primary coil of a 

 transformer, the primary terminals of which are also connected 

 through a condenser. Here we have three inductive circuits in 

 parallel. There are only three currents to consider. One of these 

 can be expressed as the sum of the other two, and hence the current 

 vectors lie in a plane. There is only one voltage. This may not lie 

 in the plane of the currents, but its projection on this plane will 

 give all that is needed. Let E be the electromotive force of the 

 dynamo, B, the resistance of the armature and leads up to the 

 terminals of the transformer, kept at voltage Y, and let C be the 

 armature current, we then have 



E = RC + V. 



The vectors representing E and V may not be in the plane of the 

 currents, but their projections on this plane are vectors which are 

 connected with each other by the same linear relation as in the above 

 equation, and in nearly all practical cases the projections may be 

 taken to represent the voltages. 



The chief purpose for which this vector method has hitherto been 

 used is to explain or to predict the nature of the changes caused in 

 alternate current circuits by variations in resistance, self-induction, 

 or capacity, and to deduce the relations which must be fulfilled in 

 order that certain results may be obtained. It has been possible to 

 do this because from the network representing one particular case it 

 has been easy to draw the figures representing other cases by merely 

 increasing the length of certain of the lines in the proper proportion. 



