some Problems of the Mass-Spectrograjrfi. 523 



If we take as axes of coordinates the tangent and normal 

 to the rays at their point of entry into the magnetic field, the 

 coordinates o£ their point oE emergence may be shown to be 



L sin </> _ L(l — cos (/>) 



and therefore by (9) 



J-Ja • i — — — — -L'O /-< i> — 



x= — sm<pe <P , y = — (1 — cos <p)e 4>o . 

 9o 9o 



The values of p can now be determined for general values 

 of cf> by the usual formula. The expression is complicated 

 and for our purpose it is sufficient to consider </> = $o an d 

 assume that <£ is moderately small. The leading term in 

 the expression for p is then 



The minus sign denotes that the concave side of the trailing 

 edge is outwards. In the existing instrument (£ = 4# =3« 

 To achieve the desired result we should require a concave 

 trailing edge in this neighbourhood of radius 3^L , where 

 L may be taken to be the diameter of the magnetic field. 

 This is far too severe to be of practical use. For (£ = 4# = !, 

 the radius of curvature would have to be -£L , which is 

 perhaps just practicable if L is large. Thus a serious 

 improvement in the position of the plate is barely feasible. 



§9. Second order focussing. 



In order to obtain the conditions for more exact focussing 

 we must start by examining the form of the beam for given 

 m after deflexion in the electric field. It is easily shown 

 that, when the electric field is a uniform field of intensity X 

 acting over a length I, the rays of various velocities all 

 diverge exactly from a virtual focus Z at the centre of the 

 field on the line of entry. For the path of the particle, 

 charge e, mass m, in the assumed field is a parabola, whose 

 equations referred to axes Oxy are (fig. 3 J : 



x = vQ,o$a.t, y=v sin u t~ ^X^ 2 , 



Xe 

 y — x tan a— — - 2 sec 2 a x 2 . 



