% 

 214 GENERAL THEOREMS 



=l dx...\ 



Jo J 



and therefore 



I rh rh' r r 00 



1= x- / <bu du I d>u du I dot. I da H. 



iTrJ J J-oo J-oo ' 



Let H= K cos (a.u -f a,M ; ) + L sin 

 Then it will easily "be seen that 



N N ' 



where, if we take the case of three variables, 



/ am + OL t m^ an + a l n l am + a t m l cup + a,/?, 



wL\ - ftOC I 1 -- T 



\ a o a c 



a b 



ap + ajt? y am + a ,m / aw + a / w / cup + & t p\ 



D={a*+ (am + a m,) 2 } {J 2 + (aw + a,^) 2 } }c 2 + (op + aj?,) 2 }. 



(Precisely the same law of formation of these quantities 

 would obtain if we were to take any number of variables. I 

 have taken the case of three merely for distinctness of repre- 

 sentation.) 



Putting for cos (aw + a,w,) and sin (OLU + a t u t ) their expo- 

 nential values, we find that 



= a... cjl - V(- 



and as 



/ \ 2 2 f 1 // -vaw + c^wil f 



(am + a < m / ) 2 = a 2 |l - V(-l) -- ~JT \ j 



