8 BELL SYSTEM TECHNICAL JOURNAL 



The solution of the partial differential equations for electron velocity 

 is thus reduced to the solution of a series of ordinary differential 

 equations, which in themselves contain no approximations. 



From equation 1, the inverse focal distance is now obtained by 

 dividing uhy r and w, which gives 



This is the general equation for focal distance as it is affected by 

 aberration. In using this equation, we are at liberty to set the higher 

 coefficients equal to zero at the incident side of the lens. This de- 

 termines the initial value of A in terms of the first conjugate focal 

 distance. The second conjugate focal distance is then determined by 

 solving for the coefficients at the exit side of the lens. Due to the 

 presence of the terms in r, this second focal distance varies slightly 

 with the radial distance at which an electron passes through the lens, 

 and the focus is therefore diffused along the axis. This diffusion of 

 the focus is called aberration. 



The coefficient A is of particular importance in the theory of 

 electron-optics. For paraxial rays, that is, rays near the axis, the 

 higher terms in the two series are negligibly small compared to their 

 first terms, and for such rays 



With the exception of aberration, the single coefficient A thus de- 

 termines the complete performance of a lens, and the principal con- 

 stants of a lens are determined by its differential equation alone. 

 In lenses where the rays are confined to a region near the axis with 

 proper diaphragms, the aberration terms are small and the coefficient 

 A describes the performance of a lens sufficiently well. 



The next section is devoted to the derivation of the principal lens 

 equations from this coefficient. The aberration terms are considered 

 only in the last section of the paper. 



Section II- — Rays Near the Axis 



For rays near the axis the optical characteristics of an electric field 

 are determined by the differential equation for A alone, 



^[2vA' + ^' = -y • (16) 



