696 BF.I.L SYSTEM TECHNICAL JOURXAL 



electromotive force is applied at time l = o, and that at this lime all 

 the currents in the inductances and all the charges on the condensers 

 are zero. 



So far wc ha\e been following the classical thcor\- of linear differ- 

 ential c(|ualions. We have seen that the forced exponential solution 

 and the derived steady state solution are extremely simple and are 

 mere matters of elementary algebra. The practical difficulties in the 

 classical method of solutions begin with the determination of the 

 constants C . . Cm of the complementary solution as well as the 

 roots Xi, . . \m of the equation D(X)=o. It is at this point that 

 Heaviside broke with classical methods, and by considering special 

 boundary conditions of great physical importance, and particular 

 l\pes of impressed forces, laid the foundations of original and powerful 

 methods of solution. We shall therefore at this point follow Heavi- 

 side's example and attack the problem from a different standpoint. 

 In doing this we shall not at once take up an exposition of Hea\iside's 

 own method of attack. We shall first establish some fundamental 

 thef)rems which are extremely powerful and will serve us as a guide 

 in interpreting and rationalizing the Heaviside Operational Calculus. 



CH.APTER II 



Thk Solution when .\.\ Aruitr.vrv Force is Applied to the 

 Network in a St.'lTE of Equilibrium 



In engineering applications of electric circuit theory there are 

 three outstanding problems: 



(1) The steady stale distribution of currents and potentials when 

 the network is energized by a sinusoidal electromotive force. This 

 problem is the subject of the theor>- of alternating currents which 

 forms the basis of our calculations of power lines and the more elabor- 

 ate networks of communication systems. 



(2) The distribution of currents and potentials in the network in 

 response to an arbitrary electromotive force applied to the network 

 in a state of equilibrium, i.e., applied when the currents and charges 

 in the network are identically zero. 



(3) The effect on the distribution of currents and potentials of 

 suddenly changing a circuit constant or connection, such as opening 

 or closing a switch, while the system is energized. 



We shall base our further anahsis of circuit thcor_\- on the solutions 

 of problem (2), for the following reasons: 



(A) It is essentially a generalization of the Heaviside problem and 

 its solution will furnish us a key to the correct understanding and 



