56 Professor Thomson, Longitudinal Electric Waves, [Jan. 27, 



The solution of which is 



X=f{x-(p + aX)t] (2), 



where f(x) denotes an arbitrary function of X. 



Let us take a case where initially the volume density of the 

 electrification is represented by a harmonic function of x so that 



K dX AK . 



p = -. r~ = a — sin mx, 



r 4*tt dx 4nr 



so that when t = 0, 



A 



X = G H — cos mx, 

 m 



and hence by (2) 



thus -rr- = — 



X = G -\ — cos m (x — (p +■ aX) t), 

 dX _ A sin m [x — (p + aX) t] 



dx 1—Aat sin m {x — (p + aX ) t} ' 



We see from this equation that the denominator of this ex- 

 pression can vanish and thus p become infinite. The earliest time 

 at which this takes place is given by 



Aol' 



P + CtG , ._ 1 . 7T 



as -^— A h (2?i + \) — . 



Aa v 2/ m 



The value of dXjdx, which is proportional to the volume 

 density of the electrification, at the time t = 1/Aa is given by the 

 equation 



, . f / p + aX\ 



dx_ ABm \ m { x - !L Air) 



dx . ( ( p + aX\} ' 



comparing this with the initial value — A sin mX, we see that the 

 effect of the motion of the air in the neighbourhood of the positive 

 electrode has been to enormously increase the maximum value of 

 the density of the negative electrification, while it has diminished 

 the maximum value of the positive electrification. It has thus 

 accentuated the difference in the initial state of electrification in 

 the tube, for all the negative electrification is practically con- 

 centrated in a series of equidistant spots on the tube, while the 

 positive electricity is more evenly distributed than it was initially. 

 This effect, which is represented by equations similar to those 



