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



We have now to see what is the meaning of G + L-y as an 

 operator in a determinant. 



If we consider the formation of a current in a circuit of 

 resistance E and coefficient of self-induction L by an electro- 

 motive force E, we have the equation for the current i 



L$ + Ei=E. 

 at 



Write thus 



or, by notation of the calculus of operations, 

 f=E(l4 + R)-'. 



But now the solution of the above differential equation 



E 



under the conditions t = 0, i = 0, and t=. oo, i=— , is 



Comparing these two expressions for i together, we have 



(1-.-E-) 



Hence we may substitute in the expression for the combined 

 resistance of galvanometer and shunt for Lj+G, 



(XL 



G 



and we have as a result, 



R= 



G+s(l-e-l') 



OS 

 We see that when / = 0, R=S, and when t= oo, R=- 



G+S' 



Hence the result shows that at the first instant of starting a 

 current through a shunted galvanometer, when the shunt 

 has no self-induction and the galvanometer a considerable 

 one, the galvanometer behaves as if it had a high spurious 

 resistance, which in time dies away, allowing the total current, 

 after an infinite time, to be divided between the galvanometer 



and the shunt in the ratio of ~ — m to ^ — -r. 



lT+ b <JT-|-b 





