The Vector Properties of Alternating Currents, Sfc. 4G5 



it is seen that we have means at our disposal to separate any true 

 periodicity of a variable from among its irregular changes, provided 

 we can extend the time limits sufficiently. 



The proof of this proposition lies outside the limits of this paper. 



The application of the theory of probability to the investigation of 

 what may be called "hidden " periodicities, an instance of which has 

 here been given, may be further extended, and interesting results are 

 obtained when a number of periodicities, such as those supposed to 

 depend on the rotation of the sun about its axis, are critically 

 examined. A full treatment of the subject will shortly appear in 

 * Terrestrial Magnetism.' 



" The Vector Properties of Alternating Currents and other 

 Periodic Quantities." By W. E. Scjmpner, D.Sc. Commu- 

 nicated by 0. Henrici, F.R.S. Received May 28, — Read 

 June 17, 1897. 



It has been well known for many years that the variations of a 

 simple harmonic function, such as A cosp£, can be represented by 

 the projection, on a fixed line, of a vector of constant length A sup- 

 posed to revolve uniformly so as to complete one revolution in a 

 time T given by the relation == 2ir. The angle between the 

 revolving vector and the fixed line is pt at any instant t. In the first 

 edition of Thomson and Tait's ' Natural Philosophy ' (vol. 1, p. 38, 

 § 58), it is shown that any two simple harmonic functions of one 

 period can be compounded to a single simple harmonic function of 

 the same period, and that the vector, representing the compounded 

 function, is obtained from those representing the component func- 

 tions by the ordinary process of vector addition. 



This device has proved useful for many purposes, but it has 

 proved especially valuable in connexion with alternate current 

 problems. Its application to such cases was first clearly pointed 

 out by Mr. T. H. Blakesley more than twelve years since. The use 

 of it in alternate current work has gradually developed into what 

 may be described as a vector, or network, method of representing 

 alternate current quantities. In this method the length of the vector 

 denotes the magnitude of the current or voltage, while the angle 

 between any two vectors represents the time which elapses between 

 the instants at which the maximum values of the corresponding 

 quantities occur. In particular, if one line represents the voltage on 

 a conductor, and a second line denotes the strength of current 

 produced in it by this voltage, the power absorbed in the conductor 

 is measured by the product of the length of the two lines into the 

 cosine of the angle between them. It is to this possibility of 



