A Itemating Currents and other Periodic Quantities. 467 



Jo 



if we choose for a and (3 two perpendicular vectors of lengths numeri- 

 cally equal to 1/ the length of v = will be equal to 

 v/K^ 2 "*"/* 2 )' or *° ^ ne magnitude of F. 



If now we consider the value of the mean product of two functions 

 Fx x F 2 = 



(\i cos pt + jux sinpt) (\ 2 cos pt + fi 2 sin pi), 

 we see that this mean product is 



i (A-xX 2 + /tiyW 2 ), 



or the same as the scalar product of the corresponding vectors, 

 (X ia + ^) (\ 2 a + /i 2 /3). 



Every alternating current problem is reducible to the considera- 

 tion of the current, voltage, and electrical power spent in the 

 different conductors. Since all these quantities are varying periodi- 

 cally some average value has to be taken. For current and voltage 

 the mode of averaging selected is always that obtained by taking the 

 square root of the mean value of the square of the quantity during 

 the period. The electrical power is, however, the mean value 

 through the period of the product of the instantaneous values of 

 voltage and current. Hence, if the current and voltage are repre- 

 sented by vectors in the manner described, the scalar product of the 

 two vectors will represent the power. This vector method is 

 particularly suited to alternate current problems, and has been very 

 much used for such purposes by electricians, since it is simple to 

 understand and to apply, and it can be used for the simple solution 

 of many problems which would otherwise need the use of the 

 calculus. 



The method, however, assumes that all the voltages and currents 

 considered vary according to a simple sine law. In ordinary com- 

 mercial practice this is unfortunately not the case, owing chiefly to 

 saturation and hysteresis phenomena in the magnetic circuits, and 

 also to the fact that ordinary dynamos do not develop an electromotive 

 force varying according to a simple harmonic law. The object of 

 the present investigation is to show how far this method can be 

 modified, and used correctly, in cases where the currents, although 

 periodic, do not vary according to any simple law. 



Since we shall have frequently to refer to the value of the square 

 root of the mean square of a periodic function, we shall call such a 



