MRS. II. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC. 
319 
How to Ascertain with Certainty whether there is a Constant or a Variable Back 
E.M.F. in the Arc or None , and how to Jincl the True Back E.M.F. if there 
is One. 
Returning to the equation 
we have 
SA — SA 
V = E + A r, 
SV SE , . . Sr . , ,, , 
— = — + r + A — when noth R and r vary, 
X A X A oA. 
and hence 
TT A A SE . , Sr 
v - a sa= e - a Ia- a 'sa- 
If the alternating current with which SV/SA is measured is of such high frequency 
that it does not alter the resistance of the arc, and if, also, the back E.M.F. is 
constant, or, being variable, the alternating current is too small to affect it, then 
\ — A SY/SA = E. 
To see whether the arc has any back E.M.F. at all, therefore, it is only necessary 
to measure SV/SA with a superimposed alternating current of a frequency that has 
been found not to affect its resistance and to subtract A SV/SA from Y. If the result 
is zero, the arc has no back E. M. F. If it is not zero, SV/SA must be measured in the 
same way for other arcs differing widely in current and length. If all the values of 
Y — A SV/SA thus obtained are equal, or nearly so, the arc has a constant back 
E.M.F. which is equal to this value. If A' — A SY/SA is not the same for all the arcs, 
but varies according to some definite law, then there is a variable back E.M.F. which 
may or may not be affected by the alternating current used to measure SV/SA. 
Suppose, for instance, that two measurements of SY/SA were made, using the same 
direct current and length of arc, but different alternating currents. If one of the 
alternating currents had a root mean square value equal to one per cent, of the direct 
current, and the other a value equal to five per cent., one would be five times as great 
as the other, and yet both would be small compared with the direct current. It 
would, of course, be possible to make the frequency of each of these currents so great 
that the resistances of the arcs to which they were applied were not altered by them. 
Yet it would not necessarily follow that when this had been done the two values ot 
SV/SA thus obtained would be equal. For the back E.M.F. might vary not with the 
frequency of the alternating current, but with its magnitude. If, therefore, it were 
found that E was variable, it would be necessary to measure SY/SA with smaller and 
smaller alternating currents, till two were found which, while differing considerably 
from one another, both gave the same value of SY/SA. Only a value obtained in this 
way could be accepted as measuring the true resistance of the arc, and A r — A SY/SA 
would be the true back E.M.F. of the same arc. 
