NOTE ox KOHLRAUSCH'S DETERMINATION 83 



in Kohlrausch's instrument. Kohlrausch's principal error lies in the 

 omission of the coefficient of self-induction from his equations. 



For the sake of clearness, and because the subject is quite often 

 misapprehended, I shall commence at the beginning and deduce nearly 

 all equations. 



Let us proceed at first in the method of Helmholtz, using the nota- 

 tion of Maxwell's ' Electricity.' 



Let a current of strength / be passing in a circuit whose resistance 

 is 7?, and coefficient of self-induction L. Also let a magnet be near the 

 circuit whose potential energy with respect to the circuit is IV. Let A 

 be the electromotive force of the battery in the circuit. 



The work done by the battery in the time dt is equal to the sum of 

 the work done in heating the wire, in moving the magnet, and in 

 increasing the mutual potential of the circuit on itself. 2 Hence we have 



AUt = PRdt + l~dt + -L j 

 dt 2 



and if A is equal to zero, we find 



/=_.7r + L*L\ 



If we apply this to the case of a magnet swinging within a coil the 

 angle of the magnet from a fixed position being x, we have since -j- 



&3s 



is the moment of the force acting on the magnet with unit current and 

 may be denoted by q, 



dx , r 



where my R is Kohlrausch's w. 



This expression differs from that used by Kohlrausch in the addition 

 of the last term, which is the correction due to self-induction. The 

 last term vanishes whenever the magnet moves with such velocity as 

 to keep the induced current constant ; but in the swinging of a galvano- 

 meter-needle it has a value. 



To form the equation of motion of the needle, we can proceed the 

 rest of the way as Maxwell has done (Electricity, art. 762). Assuming 

 that all frictional resistances to the needle are proportional to the 

 velocity of the needle, we have 



B< S + c w + l)x = r ' ....... ^ 



where B, C, and D are constants. 



2 See remarks in Maxwell's ' Electricity,' art. 544, near bottom of page. 



