170 Mr. 0. Heaviside on the 



neglect of the battery-resistance in the one, and the conden- 

 ser's capacity in the other instance. All mathematical inves- 

 tigations of physical questions are approximative ; and being 

 such, impossible results arise in extreme cases. If R is the 

 battery-resistance, the current at x = when t = Q cannot be 



greater than =r 7 ; but since there is always self-induction, the 



current, when t = 0, is mathematically zero, rising in an ex- 



tremely short time to ^, and then falling to its final strength. 



The actual rise of the current is more complex, on account of 

 electromagnetic oscillations. Thus, from infinity we have got 

 down to zero for the current at x = when t = 0. 



22. When we introduce the coefficient s = T , calcula- 



tions become complicated by the presence of imaginary roots. 

 That there must be imaginary terms in the solutions will be 

 evident when it is considered that electromagnetic induction 

 imparts inertia to the electric current, thus causing oscillations, 

 and that 



~. . fax 7 V \ -£ 



cannot contain oscillatory terms with real values of a. When 

 there is a pair of terms in which A, a, and b are imaginary, their 

 addition causes the elimination of the imaginary parts, and the 

 result is real, as indeed it must be if the problem has physical 

 reality. It is also evident that if in a physically real problem we 

 we have a single imaginary root, it must be of the form 

 a=0±n V— 1, which makes a 2 real. 



Taking a simple example, let the line be to earth direct at 

 x = 0, and to earth through a coil of resistance rrikl and electro- 

 magnetic capacity L at x=l. Also let there be initially a 

 potential distribution 



eTi 



in the line, and a current 



it 



kl(l+m) 



through the whole circuit. This state would be produced finally 

 by E at #=0. At x = (),v = 0, and at x = I, 



= v + ml -T- + sP 



