Maintenance of Electric Currents by Mechanical Work. 475 



where x is the initial value of the current. This expression shows 

 that the current, if left to itself in a closed circuit, will gradually decay. 

 If we put 



L + 2M + N 



K + S r ' {S > 



then _£ 



x = x Q e r (4) 



The value of the time r depends on. the nature of the coils. In 

 coils of similar outward form, r varies as the square of the linear 

 dimension, and inversely as the resistance of unit of length of a wire 

 whose section is the sum of the sections of the wires passing through 

 unit of section of the coil. 



In the large experimental coil used in determining the B.A. unit 

 of resistance in 1864, r was about *01 second. In the coils of 

 electromagnets r is much greater, and when an iron core is inserted 

 there is a still greater increase. 



Let us next ascertain the effect of a sudden change of position in 

 the secondary coil, which alters the value of M from M, to M 2 in a 

 time t 2 — t lt during which the current changes from a?, to cc 2 . Inte- 

 grating equation (1) with respect to t, we get 



(R + S) r^+(L + 2M 2 + N> 2 -(L + 2M 1 + N> 1 = 0. (5) 



If we suppose the time so short that we may neglect the first term in 

 comparison with the others, we find, as the effect of a sudden change 

 of position, 



(L + 2M 2 + NK=(L + 21I 1 + N) l r 1 (6) 



This equation may be interpreted in the language of the dynamical 

 theory, by saying that the electromagnetic momentum of the circuit 

 remains the same after a sudden change of position. To ascertain the 

 effect of the commutator, let us suppose that, at a given instant, cur- 

 rents x and y exist in the two coils, that the two coils are then made 

 into one circuit, and that x' is the current in the circuit the instant 

 after completion; then the same equation (1) gives 



(L + 2M + N>' = (L + lI> + (N + M)y. ...(/) 



This equation shows that the electromagnetic momentum of the 

 completed circuit is equal to the sum of the electromagnetic momenta 

 of the separate coils just before completion. 



The commutator may belong to one of four different varieties, ac- 

 cording to the order in which the contacts are made and broken. 

 If A, B be the ends of the first coil, and C, D those of the second, 

 and if we enclose in brackets the parts in electric connexion, we 

 may express the four varieties as in the following Table : — 



(1) (2) (3) (4) 



(AC) (BD) (AC) (BD) (AC) (BD) (AC) (BD) 



(ABCD) (ABC) (D) (A) (BCD) (A) (B) (C) (D) 



(AD)(BC) (ABCD) (ABCD) (AD) (BC) 



(AD) (BC) (AD) (BC) 



