Since the carbon rods in an arc are consumed during the process, the rods need to be moved continually closer to each other. Various clock-like mechanisms were developed, but a simpler mechanism used in the early days involved using the voltage to determine whether an adjustment was needed.

The relation between the current, the carbon P.D., and the length of arc in the case of the direct-current arc has been investigated by many observers with the object of giving it mathematical expression.

Let V stand for the potential difference of the carbons in volts, A for the current through the arc in amperes, L for the length of the arc in millimetres, R for the resistance of the arc; and let a, b, c. d, &c., be constants. Erik Edlund in 1867, and other workers after him, considered that their experiments showed that the relation between V and L could be expressed by a simple linear equation, V = a+bL.

Later researches by Mrs Ayrton (Electrician, 1898, 41, p. 720), however, showed that for a direct-current arc of given size with solid carbons, the observed values of V can be better represented as a function both of A and of L 01 the form c-FdL V =38.9+2.07L+. A There has been much debate as to the meaning to be given to the constant a in the above equation, which has a value apparently not far from forty volts for a direct-current arc with solid carbons. The suggestion made in 1867 by Edlund (Phil. Mag., 1868, 36, p. 35 8), that it implied the existence of a counter-electromotive force in the arc, was opposed by Luggin in 1889 (Wien. Ber. 98, p. 1198), Ernst Lecher in 1888 (Wied. Ann., 1888, 33, p. 609), and by Franz Stenger in 1892 (Id. 45, p. 33); whereas Victor von Lang and L. M. Arons in 1896 (Id. 30, p. 95), concluded that experiment indicated the presence of a counter-electromotive force of 20 volts. A. E. Blondel concludes, from experiments made by him in 1897 (The Electrician, 18 97, 39, p. 615), that there is no counter-electromotive force in the arc greater than a fraction of a volt. Subsequently W. Duddell (Proc. Roy. Soc., 1901, 68, p. 512) described experiments tending to prove the real existence of a counter-electromotive force in the arc, probably having a thermo-electric origin, residing near the positive electrode, and of an associated lesser adjuvant e.m.f. near the negative carbon.

This fall in voltage between the carbons and the arc is not uniformly distributed. In 1898 Mrs Ayrton described the results of experiments showing that if V 1 is the potential difference between the positive carbon and the arc, then V1=31.289 i 3.1L - A ' and if V2 is the potential difference between the arc and the negative carbon, then V2 -= = A where A is the current through the arc in amperes and L is the length of the arc in millimetres.

The total potential difference between the carbons, minus the fall in potential down the arc, is therefore equal to the sum of Vl+V2=V3.

V=a+bL+ A In the case of direct-current arcs formed with solid carbons, Edlund and other observers agree that the arc resistance R may be expressed by a simple straight line law, R=e+fL. If the arc is formed with cored carbons, Mrs Ayrton demonstrated that the lines expressing resistance as a function of arc length are no longer straight, but that there is a rather sudden dip down when the length of the arc is less than 3 mm.

The constants in the above equation for the potential difference of the carbons were determined by Mrs Ayrton in the case of solid carbons to be - Hence V3 = 38.88+ 22.6 A 3 I L The difference between this value and the value of V, the total potential difference between the carbons, gives the loss in potential due to the true arc. These laws are simple consequences of straightline laws connecting the work spent in the arc at the two electrodes with the other quantities. If W be the work spent in the arc on either carbon, measured by the product of the current and the potential drop in passing from the carbon to the arc, or vice versa, then for the positive carbon W = a + bA, if the length of arc is constant, W =c+dL, if the current through the arc is constant, and for the negative carbon W = e+f A. In the above experiments the potential difference between the carbons and the arc was measured by using a third exploring carbon as an electrode immersed in the arc. This method, adopted by Lecher, F. Uppenborn, S. P. Thompson, and J. A. Fleming, is open to the objection that the introduction of the third carbon may to a considerable extent disturb the distribution of potential.

The total work spent in the continuous-current arc with solid carbons may, according to Mrs Ayrton, be expressed by the equation W = I I.7+10.5L+(38.9+2 07L)A.