250 Dr. J. A. Fleming on the Distribution of 



and accordingly we have the following equation for the value 

 of y at any instant 



^=(l 1 |+g)" 1 (l 1 l 2 c 2 ^+(l 1 s+l 2 g)o 2 ^+c 2 gs|+c(g+s))' 



which may be written 



+ (GTSCLi + C 2 G 2 S) j t + GCS+s)~0. 



This linear differential equation is solved when we know 

 the roots of the auxiliary biquadratic; and according as they 

 are all real or partly imaginary, so will be the nature of the 

 solution. 



If the roots are all real the solution is a sum of exponentials, 

 whose total value first increases and then dies away as t 

 increases, indicating that the discharge produces a wave of 

 electricity through the galvanometer always in one direction; 

 but if two or all of the roots of the auxiliary biquadratic are 

 unreal, it indicates as the form of solution a function of sines 

 and cosines which will have periodic values, and points to the 

 fact that fhe discharge is a series of alternations. The general 

 case, when both the galvanometer and shunt have coefficients 

 of self-induction, when treated to determine the conditions for 

 an oscillating discharge, leads to an expression of considerable 

 complexity and not much practical use. The reduced case, in 

 which the galvanometer is wound to have self-induction and 

 the shunt so as to have no coefficient of self-induction, is, 

 however, a practical case, and can be treated without much 

 difficulty. 



Taking the differential equation for q, equation (iv.), and 

 writing in it L 2 =0, we have 



CLxS^+OGS^+CG+SJ^O, 



or 



_JLA, GS dq q 

 G+ S c^ + L^G + S) dt + CL, 



The discharge will be oscillatory if the auxiliary quadratic 



S o , GS 1 ~ 



G + S ^(G+Sr CI* 

 has unreal or imaginary roots 



