(7) 



522 Dr. F. W. Aston and Mr. R. H. Fowler on 



either or both of the objects might be achieved. It turns 

 out that it is unlikely that any serious improvement is 

 feasible in the position of the plate, but that improvements 

 should be possible in v the focussing. The arguments by 

 which one can obtain these results are as follows. 



Consider, first, the problem of trying to achieve focussing at 

 a (roughly) constant distance from the centre of the magnetic 

 field. The equations governing the deflexions are (loc. cit.) 



v 2 = const., i'<£/L = const., 



where L is the length of the path in the magnetic field and 

 is now no longer assumed constant. On differentiating we 

 have 



dO , dv _ d(f> dv _ dh 



u v <p v 1j 



and on eliminating v, 



2dcf> dO _ 2dL 

 <j> ~ L 



We shall, for example, achieve focussing at a constant dis- 

 tance a from the magnetic field (fig. 1) if we make dcf> = 2dd 

 for all values of (f> by proper adjustment of dh. That is to 

 say, we must have by (7), 



deb d6 dh /ox 



J-^ =L' (8) 



for 6 is a constant Q the same for all rays. Equation (8) 

 can be integrated and we find that, putting <p o = A0 , 



~T e <?o W 



This gives the necessary length of path L as a function of (/>, 

 and thus determines the shape of the trailing edge of the 

 magnetic field. It is clear that it is the shape of the trailing 

 edge only which is relevant here, for the rays for various 

 values of m all enter at the same point on the leading- 

 edge. 



So far all is satisfactory, but the result will only be of 

 practical value if the curvature of the trailing edge is not 

 too large compared with the size of the field. If the edge 

 is very sharply curved, the stray field will be large and may 

 be expected to spoil the effect which we desire to produce. 

 The calculation of this radius of curvature p is straightforward. 



