ARC IN AIR BETWEEN CARBON ELECTRODES 19 



represents a point of unstable equilibrium. If all the con- 

 ditions were absolutely constant, a small current repre- 

 sented by the abscissa of P" could exist, but any increase 

 in the current would cause E to decrease faster than E" 

 would increase, so that the sum of the two would be less 

 than the E.M.F. supplied by the dynamo. This would 

 cause a further increase in the current and it would con- 

 tinue to increase until it became equal to 7, the amount 

 corresponding to the point P. 



In a similar way it can be shown that any decrease from 

 the amount corresponding to the point P" would cause the 

 arc to be extinguished, so that P" represents a point of 

 unstable equilibrium. 



As the resistance in series with the arc is varied the slope 

 of the line AP will change, but there will be a point where 

 there is stable equilibrium as long as this line intersects 

 the curve PP'P". The limiting case will occur when AP 

 is tangent to PP'P", 



To state mathematically what has been given above, the 

 condition for stable equilibrium is that 



dE ^ dE" . u dE" D dE ^ D 



- < > but -TT = R\ /. - -77 < R, 



dl dl dl dl 



where R is the resistance in series with the arc. 



For many of the arcs it is approximately correct to say 



that E = a + j, where a and b are constants for any 



given length. In such cases it would follow that 

 dE b b t> ^ VT 



-Ti=p> *' P <R > or i <RI - 



But E' = E + RI; :. E' > E + |- 

 That is, the E.M.F. of the dynamo must be greater 



