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



so that if we can show for any two periodic functions that the pro- 

 duct of their magnitudes is necessarily greater than the mean value 

 of their product, we shall have shown that it is always possible to 

 construct a triangle the sides of which represent the magnitudes of 

 any three periodic functions related as in equation (1). 



The meaning of (4) can be illustrated by supposing that vi repre- 

 sents an alternating voltage applied to a conductor, arid producing a 

 current, v 2 , through it. The right-hand element of (4) then repre- 

 sents the power absorbed, while the left-hand element is the product 

 of volts and amperes. The fact that the power absorbed in an 

 inductive circuit is always found to be less than the product of volts 

 and amperes, is usually stated by asserting that the " power factor " 

 is less than unity for such circuits. Experience has not shown any 

 case in which the power factor of a circuit is greater than unity, but 

 this of course is no evidence of the truth of (4), which must be 

 proved for the general case in which the two periodic functions are 

 perfectly arbitrary Fourier series.* 



Let us therefore put 



Vi — 2 (A m sin mpt + Bim cos nipt) (5). 



m 



v 2 = % (A 2m sin mpt -f B 2m cos nipt) (6) . 



m 



By taking the mean value of twice the square of each of these 

 equations, we get 



2(53* = S (A^+B^), 



m 



2(7 2 ) 2 =2(A 2rrt 2 -i-E 2m 2 ), 



m 



and by taking four times the value of the mean product we get 



m 



* In the proof given, the two functions v\ and v 2 are assumed to hare funda- 

 mental periods that are commensurable with each other. This will always be so in 

 practical cases, and what happens with functions of incommensurable periods is of 

 but theoretical interest. It is, however, worth notice that no limitation is thereby 

 imposed on the application of the theorems given in this paper. These theorems 

 deal with mean products, and the truth of them is directly deducible from the 

 fundamental inequality given in (4). Now when the periods are incommensurable 

 the product v x v 2 consists of terms such as sin mpt sinnpt, where m and n are incom- 

 mensurable, and as each of those can be expressed as the sum of two sinuous func- 

 tions of angles denoted by (m±n)pt, the mean value is zero, the vectors representing 

 v x and v 2 are perpendicular, and the truth of (4) is at once evident. But inasmuch 

 as the value of (m — n) may be infinitesimally small, it may be necessary to take an 

 infinite time to determine the mean values. 



