Alternating Currents and other Periodic Quantities. 471 

 and we shall have proved (4) if we can show that 



or that 



{2 (A 1;/i 2 + B m 2 )} x {2 (A 2M HB 2>S 2 )}>{S (A m A 2m + B lm B 2wt )} 2 . 



m m m 



This will be proved if we can establish the algebraic inequality — 



(A, 2 + A 2 2 +.... + A» a ) (Bf + B 2 2 + . . . . + B„ 2 ) 



> (AxBiH- A 2 B 2 + .... -\-A n B n y .... (7). 



Now it will be found quite easy to establish this inequality by an 

 inductive method, but there is no need to do so here, since it is proved 

 in the theory of determinants that 



A 1 2 + A 2 2 + .... + A n 3 AjBi + A 2 Bo + .... +A„B W 

 A^+AoBo^- .... -f A»B M Bf + B 2 2 + .... + B„ 2 



|A X A 2 | 2 

 ~~ | B , B 2 | ' 



and is necessarily positive. 



It follows that the inequality (7) is true. In order that the 

 inequality may become an equality the necessary and sufficient condi- 

 tions are 



Aj/Bi = A 2 /B 2 = . . . . = A,/B w . 



In other words, the two Fourier series must bear a constant ratio 

 to each other at every instant of time, and must be multiples of the 

 same Fourier function. This is a limiting case which is not con- 

 templated in (4), and with this exception only, we have shown* 

 that : — 



* [The following much simpler proof has recently been noticed : — 

 Whatever Vi and v 2 may be, we can always put 



v i — a>\+®, v 2 = a 2 + y y 



where a x and a 2 are scalar constants given by the equations 



«i = v u a 2 = v 2 . 



By squaring each expression and taking means we get 



= 2a^ + r, = 2a 2 y + x 2 + y' 2 , 



where by x is meant the arithmetical mean of x, &c. 

 Also, we obtain by multiplication and taking means, 



V\V 2 = a x a 2 + xy + a x y + a 2 x. 

 VOL. LXI. 2 L 



