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



The product of the magnitudes of two periodic functions is always 

 greater than the mean value of their product. 



From this and from the foregoing reasoning, it follows that : — 

 Any two periodic functions can he represented by vectors in such a way 

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

 and the scalar product of the vectors, the mean product of the functions. 

 Any other function derived from the first two by means of a linear rela- 

 tion can be represented in magnitude by a vector derived by means of the 

 same linear relation from the two original vectors. The scalar product 

 of any two such vectors will be equal to the mean product of the corre- 

 sponding functions. 



Now suppose a and ft are two vectors representing two arbitrary- 

 Fourier series, Fa and F/s. It follows from the law of vectors that 

 any vector in the plane aft will be a linear function of a and ft, and 

 must therefore represent a Fourier series derivable from F a and F/s 

 in a linear manner. If, therefore, we have a third arbitrary Fourier 

 series, F y , which cannot be expressed as a linear function of F a and 

 Fj3, it follows that this cannot be represented by a vector in the same 

 plane as a and ft. It remains to show that a vector, 7, out of this 

 plane can still be found to represent F Y . 



Let us first of all assume that three vectors a, ft, 7 (not in one 

 plane) have been found to represent three Fourier series F a , F/s, and 

 Fy. We can then show that any linear function of a, (3, 7, similarly 

 represents the Fourier series, which is the same linear function of F a , 



F|3, Fy. 



For if, in accordance with the notation already used, we denote by 

 the symbols 



Fa the square root of the mean square of the value^of F a , i.e., 



the magnitude of Fa, 

 FpFy the mean value of the product FpFy, 

 ex. the length of the vector a, 

 £7 the scalar product of the vectors ft, 7, 



If we substitute from the above relations, we get 



Vl v 2 = ai a. 2 + xy-^-y* + -x j = a x a 2 -—, 



where Z represents the value of a^x — a^y. 



It follows that v x v« is always less than a x a 2 , except in the limiting ease, in which 

 Z is zero at all parts of the period. This will only occur if 



ago = a x y 



at every instant, or when 



—June 11, 1897.] 4 



