192 



BELL SYSTEM TECHNICAL JOURNAL 



Instead of the frequency, a more general symbol p is used and may be 

 thought of as the differential operator d/dt which occurs in the fundamental 

 linear differential equations taken as describing the oscillatory system. 

 That is 



d , . 



P = -■ = a -\- iw 

 dl 



(12.53) 



(12.54) 



(12.55) 



The function ju^ may have the form 



The result of taking a general variation b of (12.54) is then 



^ + m = 



A 



Since (12.54) is a function of the three variables p, a, and V the variational 

 equation (12.55) may be expressed in terms of partial derivatives with 

 respect to these three variables. That is 



dd 



„^ , „„ , hV \ -\- 1. \ ~ ()f} -+- ^ da -\- 



A\_dp "^ da dV 



dp ^ da dV 



-- 

 (12.56) 



The solution of (12.56) for the variation in p is 



bp =^ - 



57 4- ^ 5c 

 A\dV ^ da 



dv'' ^da ^ 



-h i 



dd 



(12.57) 



.1 Yp ■ dp 



It is a property of functions of complex variables that, provided they 

 possess derivatives at all, then the value of the derivative is the same regard- 

 less of the direction in which the limiting point is approached. This fact is 

 expressed by 



dA 



dp 



dd _ dd _ . dd (12.58) 



dp ^ ^ 



and bp = ba -{- iSco 



and provides means by which the real and imaginary parts of (12.57) may 

 be separated to yield the two equations 



ba = 



\_A da \dV i'a j 



d(^\A dV 



, 1 5-1 . 

 A da 



v. 4 6co/ \5co/ 



(12.59) 



