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



quantity the magnitude of the periodic function. Thus, if v is a 

 varying function of the time, periodically repeating itself after every 



interval T, the magnitude of v 9 which we shall denote by v, is given 

 by the equation 



v = magnitude of v — ^/ ^ J" v z dt. 



When we are considering two such, functions, v v and v 2 , the mean 

 value of their product will be denoted by 



— 1 f T 



ViV 2 = — J VjVjdt. 



The mode of measuring electrical power known as the Three 

 Voltmeter Method,* is really dependent upon a curious property of 

 periodic functions which may be stated as follows : — 



If any two arbitrary periodic functions, v\ and v 2 , be represented in 

 magnitude by the lengths of two sides of a triangle, and if the third 

 side represents the magnitude of v (another periodic function equal 



to the sum, or difference, of the former two), then the mean value of 

 the product of any two of these functions will be represented by the 

 product of the lengths of the two corresponding sides and the cosine 

 of the angle between them. 



Thus, in the figure, if CA represents v h and BC (or CD) represents 

 r 2 , then BA will represent Vi-\-v 2 , and DA will represent v x — v 2 ; also 

 the mean product of (v 1 + v 2 )v 2 will be BAxBC cos ABC, the mean 

 product {Vi—v^Vi will be DA x CA cos CAD, and so on. Thus we may 

 regard v x and v 2 as vectors, the length of the vector representing the 

 magnitude of the periodic function, and the scalar product of tw r o 

 vectors the mean product of the two corresponding functions. Any 

 periodic function derivable from v x and v 2 , by a linear relation, can be 

 similarly represented in accordance with the ordinary properties of 

 vectors, and the scalar product of any two such vectors will be equal 

 to the mean product of the corresponding periodic functions. 



In order to establish this we may proceed as follows : — 



* See " The Measurement of the Power given by any Electric Current to any 

 Circuit," Professor W. E. Ayrton and Dr. W. E. Sumpner, ' Eoy. Soc. Proc./ 

 April 9, 1891. 



