o2S On Astigmatic Aberration. 



a € 

 since <^i= tan"^^ — , and cj)i is the eccentric angle o£ the 



point i in the ellipse on the receiving screen. 



A small point-like object placed in the beam at P will cast 

 a small point-like shadow on the screen, and the shadow with 

 the positions taken will be inverted in respect to top and 

 bottom, and will be unchanged in respect to side and side. 



A straioht wire placed say at 45° across the beam and 

 cutting the axis o£ the beam will depict on the screen a 

 straight-line shadow in ihe ellipse on the screen, and this 

 shadow is at a definite inclination. The inclination depends 

 on the relative distances and positions. 



If the wire is considered to intercept the bounding surface 

 of the beam in a point, the eccentric angle o£ which is </>, the 

 line shadow in the ellipse on the screen is at an inclination 

 defined by the eccentric angle <^. 



Generally, if the wire is inclined at 6 and is at a distance 

 z from the primary focal line, the value o£ ^ is obtained from 

 the equation 



bz 



tanO=.—TK X tan 6. 



a(o—z) ^ 



If an inclined straight wdre is placed across the jDencil or 

 beam and is moved, without varying its inclination, from the 

 primary focal line to the secondary focal line, the shadow 



rotates through an angle of ^, for (f) changes by ^ in 



passing from one focal line to the other. 



In non-homocentric pencils or beams that are not purely 

 astigmatic but are compounded of astigmatic and other 

 aberrations, the presence of astigmatism shows itself in or is 

 accountable for the twists or rotations that are discernible in 

 the shadows or images formed by objects placed in or moved 

 about in such pencils or beams. 



For instance, if in the experimental pencil or beam there 

 are astigmatic and spherical aberrations present, then a straight 

 wire placed across the beam and moved about does not cast 

 a straight-line shadow, but casts a distorted shadow^, such as 

 a figure of S, which is displaced through an angle or is 

 rotated, the displacement or rotation being due to the 

 astigmatic aberration present in the beam. 



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