241, 242] INDUCTION OF CURRENTS. 499 



ones if the R's are zero. Calling the modulus of a numerator B r) 

 and its argument r , 



(25) 2 s D rs (ia>)E s = r e ie r, 



6 r is a small angle if the resistances are small. We thus have 



(26) " = oir]: 



where 



(27) A.= {/vM 



Retaining now only the real parts, we have for the solution 



(28) q r > 



Thus if the resistances are small, all the oscillations are in 

 nearly the same phase. If the frequency of the impressed force 

 coincides with that of any one of the free oscillations, o> v s 0, 

 and one factor of the denominator reduces to /^, so that if the 

 damping of that oscillation is small, the amplitude is very large, 

 or infinite if there is no damping. This is the case of resonance. 

 (Resonance may also be defined in a slightly different manner as 

 occurring when ico is one of the roots of the equation D (\) = in 

 which all the R's have been put equal to zero. This corresponds 

 with our example in 240. In practical cases the difference is very 

 small.) 



If now we have a system acted on by electromotive forces each 

 one of which is the sum of any number of harmonic components of 

 different periods, any component may cause resonance with any 

 free oscillation of the system, so that resonance may occur in a 

 large number of ways. 



242. Examples. Two Circuits. We shall illustrate the 

 principles of the preceding section, aside from the examples that 

 have already been given in 239, 240, involving one degree of 

 freedom, by an example of two circuits. Consider an induction 

 coil in which both the primary and secondary contain a condenser 

 in series. This is the case of the so-called Tesla high-frequency 

 coil, in which a Leyden jar produces an oscillatory discharge 

 through the primary, while the ends of the secondary are usually 

 connected with a small capacity, say a pair of knobs. We shall 



322 



