Impulsive Motion of Electrified System. 



123 



the system. The formula (7) is, indeed, applicable to the 

 disturbance existing at any time after the impulsive change 

 of velocity, but the necessary integrations are difficult unless 

 we wait till the pulse has travelled out to a great distance. 



§ 7. If a sphere of radius vt be described about every 

 point on the surface of the system, in the position which 

 it occupies at £ = 0, the wave disturbance at time t, due to an 

 impulsive change of velocity of the system at £=0, will be 

 confined to the region contained by the envelopes of these 

 spheres. Whea vt is very large compared with the linear 

 dimensions of the system, this region will approximate to a 

 spherical shell of variable thickness p, this thickness never 

 exceeding the greatest distance between two points of the 

 svstem. When, as in the examples discussed in this paper, p 

 is infinitesimal compared with vt, the shell may be constructed 

 in the manner indicated in fig. 2. Any point 0, within or 



very close to the system, is 

 chosen as origin and a 

 sphere of radius vt is de- 

 scribed about it as centre^ 

 and P denotes any point 

 on this sphere. Two planes, 

 A A' and BB', normal to 

 OP, are drawn touching 

 the svstem at A' and B' 

 and cutting OP in A and B, 

 the plane AA' being the 

 nearer to P. If more than 

 two tangent planes can be 

 that is nearest to P and the 

 now, on the radius OP 



one 

 If. 



we 



drawn, we must select the 



one that is farthest from P. 



take Pa = OA and P£ = 0B, then p, the thickness of the shell 



or pulse in the neighbourhood of P, will be equal to ah. 



Let OX, drawn in the direction of n, cut the sphere in X, 

 and let OX = OP = rand POX = 0. Let 0' be a point on 

 OX at a distance at or nr from 0. 



Now take a point /on OP between a and h at a distance z 

 from P and draw a plane FF' normal to OP and at a distance 

 vt from /, so that it intersects the system and cuts OP in F. 

 Then, in the limit, any point of the section of the system by 

 this plane may be considered as being at a distance vt from 

 /and, further, the straight line joining it to P may be treated 

 as being parallel to OP. To this approximation, the wave 

 disturbance at/ at time t is due entirely to the charged particles 

 which lie in the plane FF', and the effect of these particles 

 is the same as if they were concentrated at the point F. 



