some Problems of the Mass-Spectrograph. 525 



the equation, valid when 6 is small, for the electric deflexion 



(0 O + a)v 2 = const. 



The focus of the emergent beam is the point where the 

 line (11) touches its caustic (i. e. the envelope of the family 

 (11) when the parameter a is varied). This point can be 

 determined (by the usual rule) by differentiating with 

 respect to «, and determining d<j)/da and dv/da. from the 

 deflexion equations. This leads to a position of the focus 

 agreeing when <f> is small with the position already deter- 

 mined (loc. cit.) by simpler arguments. In order to obtain 



second-order focussing the coordinates of this point on the- 

 caustic must also satisfy the equation obtained by differen- 

 tiating (11) twice with respect to a. 



The exact condition thus obtained is an equation for 

 d' 2 ~L/da 2 and is somewhat complicated. In order to appre- 

 ciate its meaning, we shall assume that dL/dx=0 and retain 

 only the lowest powers of </>. It then reduces to the relation 



L^ 1_ b' + j'L 



L " 46V J '&' + L(l-</>/46> )* 



Near <£ = 4# , the important neighbourhood, and for 6 Q =^ 

 we have L"/L somewhat greater than 36 — for <9 = -J some- 

 what greater than 9. If we work out the radius of curvature 

 p of the trailing edge of the field we find, with dL/du = 0- 



