198 



PIERCE. 



impressed sinusoidal e.m.f. of constant amplitude and frequency, a 

 and (f are given by expressions which factored may be written 



n = sinh-i ] + V' - ^' \ '^^ 1 + h- 



<P = sm- I ^ \' ^ ^' V - . - - J 



-1 J 



(5) 



+ 



V JV 1 + h^ =^ 1 - , 



in which T' and h have the value given in (2) and (3), and in which the 

 upper signs or the lower signs are to be tised together in each case and are 

 to be chosen so as to make a and <p both real quantities. For definitions 

 of a and ip see §3. 



6. Modified Form of Equations for a and (p. — Using the/- and 

 (/-functions given in (4), we may write (5) as follows: 



I. If r > 0, 



a = sinh"^ V2f' dW^ ^ — sin 



h = U/V. 



-1 i 



V2vm} 



(6) 



with 



II. ifr<o, 



a = sinh-i V-2VfUi), ^ = sin-^ !± V- 2V 5'W} 



with h = U/V. 



These equations are to be employed subject to the following rule 

 regarding the quadrant of cp: 



Rule Regarding (p. 



Equations (6) are the general expressions for the real attenuation con- 

 stant a and the real retardation angle (p per section in terms of U, V, f{h), 



