216 Mr M'Connel, On the effects of [Nov. 24, 



Integrating Ay + Bz + Lz = 0, 



for when t = 0, y = 0, s = and i = 0. 



Now when t = co , y = — GE 1 i = 0, 



.-. Bz = -ACE 1 . 



Thus the whole current through the galvanometer is indepen- 

 dent of L. It is clear that this result depends on the following 

 conditions : 



(1) The self-induction is in one branch only. 



(2) The currents in all the branches can be expressed as 

 linear functions of the current in that branch, of the difference 

 of potentials at its ends, and of certain other currents whose 

 integral value is fixed. 



Thus it would hold good if the galvanometer were linked with 

 any system of conductors, and condensers charged initially to any 

 potential, provided there were no electromotive forces in the 

 system and no appreciable self-induction except in the galvanome- 

 ter itself. 



The equation for the time constants is the quadratic in f. 



a + b + d — (a + d) —a 



— (a + d) a+ c + d + -^ a + c 



= 0, 



— a a + c a + c + g + %L 



or ?CL\bc+ (b + c)(a + d)} + %{L(a+b+d) + G(b+ c){ad + ag + dg) 

 + Cbc {d + g)} + {{a+c+g) (b + d) +a{c + g)} = 0....(8). 



What is practically required is to know that the smaller time 

 constant is so large that it may be safely treated as infinite ; so it 

 is convenient to have an inferior limit in a simple form. 



If we write equation (8) in the form P£ 2 + Qi; + R = 0, then if 

 the roots be real it may be easily shewn that the smaller time 

 constant is greater than R/Q ; if the roots be imaginary the real 

 time constant is Q/2P. 



But it is not likely that much error will arise in practice 

 through neglecting the self-induction in calculating the time con- 

 stant. For when L is large enough to govern the smaller time 

 constant, the latter approximates to the value (g + R) L, where R 

 is a positive constant depending on the resistances. Now the self- 

 induction of a galvanometer coil of given shape and size varies 

 approximately as the square of the number of turns, as does also 

 the resistance. Thus gjL will have much the same value in all 

 similar galvanometers. A short time ago I determined the self- 



