Alternating Currents and other Periodic Quantities. 47S 



we then have a = F a , /3 = F/s, 7 = F y , 



£7" = F/sFy, 7a = Fyl\, ^= FJB>. 



If, now, we compare the value of the square of any vector 

 \a + w/? + f7 with the value of the mean square of the corresponding 

 Fourier series \F a + /uFp + vF y , we find, by expanding and using the 

 above relations, that the expressions are identical ; and, moreover, it 

 is clear that the scalar product of any two such vectors 



+ jitf + 1^7) (X 2 a + faft + v 27) 



is identical with the mean value of the product of the corresponding 

 pair of Fourier series 



(^Fa + fry? + * x F y ) (A 2 F« + fr-Fp + */ 2 Fy) , 



We have now, however, to show that whatever F a , F/3, F y may be, it 

 is always possible to find three vectors, a, /3, 7, to represent them. 



For this purpose let us consider two other Fourier functions, F« 

 and F„, derived in a linear manner from the given functions, F a , Fp, 

 Fy, in accordance with the equations 



Fa Fy 



a c, I 



where a, 0, and c are scalars determined from the mean products, 



a — FaFa, 



b = fje>, 



c = FaFy. 



It will then be seen that 



F«F tt = and F a F r = 0, 

 and whatever F« and F„ may be there is a real angle, 0, for which 



FJ?„ == Y u xY v x cos 0, 



since the product ¥ u ¥ v must be less than F„ x ¥ v . 



If, therefore, we take two planes inclined to each other at an angle* 

 0, and take two vectors, u, v, in these planes, drawn perpendicular to 

 the line of intersection, and of such length that 



u = F„ and v = F„, 



and if, in addition, we choose a vector, a, along the line of intersec- 

 tion of these planes, and of such length that 



F u = 



Fa Ff» 



a b 



and F„ = 



