Oscillations of various Periods. 373 



electrical language we may call it the resistance of the system 

 and denote it by the letter Bf. The other part of the ratio is 

 imaginary. If we denote it by ipU^r^ or 1/-^, 1/ will be 

 the moment of inertia, or self-induction of electrical theory. 

 We write therefore 



^(B' + ijpL')^; (4) 



and the values of W and 1/ are to be deduced by separation 

 of the real and imaginary parts of the right-hand member 

 of (3). 



Now the real part of i^±M! 



r ip a 22 + b 22 



_ b 12 2 b 22 +p 2 a n {2 b 12 a 22 —a 12 b 22 ) 

 b 22 +P 2 a 22 



_ftl2 _ j? Ol2ft22 — ^22 &12) : 



b 22 ~ b 22 (b 22 2 +fa 22 2 ) 



(5) 



so that 



T>/_A ^^12 2 , ^2< (<2l2 £22 — ^22 £12) (p\ 



E - Jn_ \ + ^w+A 1 r • • (6) 



This is the value of the resistance as determined by the 

 constitution of the system, and by the frequency of the im- 

 posed vibration. Each component of the latter series (which 

 alone involves jp) is of the form <*p 2 / ((3 + yp?) , where a, /3, 7 

 are all positive, and (as may be seen most easily by considering 

 its reciprocal) increases continuously as p 2 increases from zero 

 to infinity. We conclude that as the frequency of vibration 

 increases, the value of B/ increases continuously with it. At 

 the lower limit the motion is determined sensibly by the 

 quantities b (the resistances) only, and the corresponding 

 resultant resistance B/ is an absolute minimum, whose value is 



&u-S^ (7) 



#22 



At the upper limit the motion is determined by the inertia of 

 the component parts without regard to resistances, and the 

 value of B/ is 



o 22 o 22 a 22 



That the resistance in this case would exceed that expressed 

 by (7) might have been anticipated from the analogue of 

 Thomson's theorem ; but we now learn in addition that at 

 every stage of the transition, during which in general the 

 motions of the various parts disagree in phase, every incre- 



