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 V1 is the potential difference between the positive carbon and the arc, then
| V1 = 31.28 + | 9 + 3.1L | ; |
| A |
and if V2 is the potential difference between the arc and the negative carbon, then
| V2 = 7.6 + | 13.6 | , |
| 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 V1 + V2 = V3.
| Hence V3 = 38.88 + | 22.6 + 3.1L | . |
| A |
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 straight-line 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 + fA.
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