228 VARIABLE STATE. 



at this point will only depend on the source found there. The 

 portion OE of the unlimited wire is therefore in the same condition 

 as if it were alone. 



The current at the point P, on the wire OE at a distance x from 

 the origin O, is the algebraical sum of the currents which would be 

 produced at this point if we suppose that all the sources were on 

 an unlimited wire. 



If all the sources are raised to the constant potential V , then, 

 for a portion of the potential due to the source O, we shall have 

 (228) 



V a _a"* V a 

 * = 4=* ir 

 pjirt 



For sources on the left it will be sufficient if we successively 

 assign to x the values x + 2/, x + 4!, . . . x + 2#/, . . . The sources 

 O', O", O"' will produce currents in opposite directions to the 

 preceding if they were at the potential V ; but, as their sign has 

 been changed, the flows of electricity which they produce are still in 

 the same direction; we ought accordingly to replace x by 2/-^, 

 4/- #,..., znl-x, . . . , which gives for the current I, 



When we have made x = /, that is to say, when we consider the 

 phenomenon at the point E, at the end of the wire in connection 

 with the earth, the expression becomes simplified, and we see 

 directly that the intensity is double that which all the sources on 

 the left would give. We have then 



2V, a F 2 ' 2 a2 ( 3? > 2 2 (50 2 -I 



Putting e u = v, we get 



The current is at first zero, since v is zero when t is equal to 



zero ; it then increases towards the limiting value . 



pl 



