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



we shall then have 



Ml — = F a F M 

 M = = F a F„ 



uv = uXvX cos = F M x F c x cos = F«F ej 



or the three vectors a, v completely represent the series F a , 

 F«, F„. 



Since /3 and 7 are linear fnnctions of a., and r, while Fp, F y 

 are the same linear functions of F a , F«, and F p , it follows from the 

 foregoing reasoning that /3, 7 also represent the periodic functions 



F/3, Fy. 



It is thns established that : — 



Any three periodic functions can be represented by vectors in such a way 

 that the length of the vector represents the magnitude of the function, 

 while the scalar product of any tvjo vectors represents the mean value of 

 the product of the corresponding functions. Any other periodic function 

 derivable from the original three, in a linear manner, can be similarly 

 represented by a vector, derivable from the original three vectors by means 

 of the same linear relation. 



It will, however, be seen that it is impossible, in three-dimensioned 

 space, to find vectors fco represent four arbitrary periodic functions, 

 unless a linear relation connects them. For instance, the functions 

 sin pt, cos pt, sin2pt, cos 2pt cannot be so represented. The mean 

 product of any two of these is zero, and it is impossible in three- 

 dimensioned space to find four lines each of which is at right angles 

 to the other three. 



In discussing problems the solutions of which are expressible in 

 terms of the mean square of a Fourier function, or the mean product 

 •of two such functions, it may sometimes prove advantageous to make 

 use of the foregoing theorem. As a rule the complicated Fourier 

 series occurring in actual problems are obtained in an indirect and 

 somewhat artificial manner by expressing in precise mathematical 

 language the value of some periodic quantity which can be repre- 

 sented graphically in a very simple way. In some cases the curve, 

 though perfectly determinate, follows no recognised mathematical 

 law ; in other cases it consists of a series of discontinuous curves, 

 each of simple mathematical character. In all such instances the 

 mean square of the ordinate of such a curve, and the mean product 

 of the corresponding ordinates of two such curves, can more easily 

 and quickly be obtained than the Fourier expansions themselves. If 

 the problem only demands a knowledge of these mean squares and 

 mean products, the vector method will probably be the simplest one 



