806 REPORT— 1898. 



Tolts, the current in amperes, and the length of the arc in millimetres may be 

 expressed by the equation : — 



V = ;]l-28 + ^ + ^"^'. . . ... , (1) 



The drop of potential between the arc and the iief/ative carbon, on the other hand, 

 is, I find, affected by the cvrrent only, and not by the length of the arc. The 

 equation expressing its conuection with the current is — 



V = 7-6 + l§^ (2) 



A. 



Thus this dio]) of potential at the negative carbon is by no means insignificant, nor 

 have I ever found its sign to change, either with silent or with hissing arcs, as it is 

 said to do by some observers. In all ray experiments this drop of potential has 

 been from arc to carbon, and its value has been about one-fourth of the value of 

 the corresponding fall of potential at the positive carbon. 



From equations (1) and (2) we can tind the equation for the drop of potential 

 at the positive carbon plus the drop of potential at the negative carbon ; i.e., the 

 whole fall of potential from carbon to carbon minus the fall of potential through 

 the arc itself. It is 



y = 38-88 + =^li-^i^ (3) 



A 



Now, the equation I found three years ago for the total P.U. between the main 

 carbons, which included, of course, the drop of P.I), in the arc itself, was 



V = 38-88 + 2-07/ + ^^-^^;i^"-/. . . . . (4) 



The coincidence between the first terms of equations (3) and (4) shows that 

 this constant quantity has at last been tracked home, and that it belongs net to 

 the positive carbon alone, as has hitherto been supposed, but to both the positive 

 and negative carbons in tbe proportions of about four-fifths to the former and one- 

 fiftli to the latter. 



It must be remembered that the experiments upon which equation (."3) is based 

 were made nearly two years after those from which equation (4) was obtained, and 

 that for the new equation (3) itself the experiments consisted of two entirely sepa- 

 rate sets ; the one made to find tlie drop of potential at the positive carbon and the 

 other to find the drop of potential at the negative carbon. 



As regards the accuracy with which equations (1), (-2), and (3) express the 

 results of the experiments, it may be mentioned that each of the 124 values from 

 which these equations for the fall of potential at the carbons alone were formed 

 was the mean of the results of from 6 to 12 experiments. Of these 124 values 06 

 differed by less than 1 volt from the numbers calculated from the equations, 25 

 diflfered by between 1 and 2 volts, and only 3 ditfered by more than 2 volts, and 

 these three all belonged to equation (1). 



This closeness of agreement between the observed and calculated values is the 

 result of no series of ingenious guesses, but of the algebraical expression of three 

 very simple straiglit line laws which, I find, exist between the power expended at 

 each of the CHrbons, the current Howing, and the length of tbe arc. These three 

 laws may be most simply expressed thus : — 



If W be the power in watts expended at either of the carbons (measured by 

 multiplyinff the current by the fall of potential at the carbon) and a, b, c, d, e, and 

 f he constants, then 



For the positive carbon {Z^""^^,^ ^''^ '' '°"'^^°* ^'"°^^' °*" '''^' 

 1 (^ W = e + rf< ., „ current. 



For the negative carbon W = <>+/A. 



A combination of the first two laws gives equation (1), and the third gives 

 equation (2). 



