Alternating Currents and other Periodic Quantities. 469 



Whatever be the law of variation of v, v h and v 2 , if 



v = Vi + V 2 



V 2 = Vi+2V x V 2 -\-V 2 



(1): 



then 



v 



and, taking means, we have 



v 2 = v i 2 zk2viV-,-hv 2 2 , 



which equation is the same as 



02) 



But if the lengths of the three sides of a triangle respectively repre- 

 sent v, Vi, and 1% we have from geometry 



where is the angle between v x and v 2 . 



Comparing equations (2) and (3) with the figure, it will be seen 

 that for the case represented by the line AB the positive sign must 

 be taken in each equation, while for the other case, AD, the negative 



signs must be chosen. In either case we have v x v 2 = COS 0. 



Moreover, if in equation (1) Ave transpose either v x or v 2 , and square 

 and proceed as before, we find that whichever two sides of the tri- 

 angle ABC be chosen, the products of these sides into the cosine of 

 the angle between them is equal to the mean product of the corre- 

 sponding periodic functions. 



The foregoing argument shows that assuming it possible to form a 

 triangle with sides equal to the magnitudes of v, v u and v 2 , such 

 triangle has the properties stated. But we have yet to show that it 

 is always possible to construct such a triangle, or, in other words, to 

 show that any two of these quantities are together necessarily greater 

 than the third. 



Now if v = Vi + v 2 , 



we find by squaring and taking the mean that 



we shall therefore prove that 



V < V x + V 2 , 



provided we can establish the inequality 



(4); 



