Oscillations of various Periods. 379 



so that 



1 _ g 1 Q6) 



R' + i>L' *R + vpL v J 



Now 



^ 1 _^ R— iph 



^R + s^L^R'^+p 2 !; 2 ' 

 or, if we write 



%+7L 2 = A? t W+fU =B ' ' ' ' ( 17) 



Hence 



A ^ B 



E ' = A 2 +y 2 E*' L ' = A 2 +/B 2, ' " (18) 



Equations (17) and (18) contain the solution of the problem. 

 Whenp = 0, 



tv- l L/ _ S(LR-^) 



When ^? = co 7 



B'=f£?, L' = ^. . . (20) 



These examples will suffice. 



The relation between "SP^ and ^i expressed in (4) may be 

 exhibited in another way in terms of the phase difference (e) 

 and the ratio of maxima. Thus if 



we have j - 



P=i/(R' 8 +j? 2 L' 2 ), tan e=^. . . (21) 



As p increases from to <x> , € usually ranges from to \ it. 

 At first sight it might appear probable that every increment 

 of ip would involve an increment of e, but this seems not to be 

 generally true. For consider a case in which 



a n = 0, a 12 = 0, a l3 = 0, .... 

 so that by (10) 



L , =2 oM> P 



Here pJJ begins (as usual) at zero and ends at zero. During 

 part of the range, therefore, it falls ; and thus since R/ rises 

 throughout, it follows that e does not rise throughout. 



