Mat 10, 1918] 



SCIENCE 



459 



Fleure, who was appointed professor of zoology 

 at the college ten years ago, will now devote 

 all his energies to the department of geography. 

 Dn. R. H. Jesse, Jr., head of the depart- 

 ment of chemistry at the Montana State Uni- 

 versity at Missoula, has been appointed dean 

 of men for the institution. 



DISCUSSION AND CORRESPONDENCE 



ASTIGMATISM AND COMA 



Perhaps the clearest statement of the pre- 

 vailing theoretical distinction between the 

 five spherical aberrations is that given in the 

 last edition of the Encyclopedia Britanniea by 

 Dr. Eppenstein of the Zeiss factory. 



The differentiation there made between 

 astigmatism and coma is not, however, in 

 strict conformity with the facts. The term 

 " astigmatism " as applied to lenses has al- 

 ways referred to the increasing lack of sharp- 

 ness in the image towards the edge of the 

 field in an uncorrected or poorly corrected 

 lens system and " coma " to the peculiar rad- 

 ial flare sometimes very evident in the outer 

 portions of the field. 



The explanation given in the article just 

 referred to is that astigmatism is the aberra- 

 tion due to obliquity and is therefore fully 

 shown by very narrow bundles of rays, while 

 coma can not be shown at all except with a 

 wide bundle. 



This explanation is the result of reasoning 

 from the theory of astigmatism devised by 

 Sturm, who assumed a behavior of oblique 

 rays completely at variance with the facts. 

 By the use of a method developed by the 

 writer it is possible to calculate with strict 

 accuracy the path and focal point of any ray 

 through a lens surface from any point of the 

 field by the use of which it became at onoe 

 evident that the two foci calculated by Sturm's 

 method locating the position of the two astig- 

 matic surfaces are pure fictions, though this 

 calculation is nevertheless a rough numerical 

 approximation of this aberration. The detail 

 of the new method of calculation will be pre- 

 sented elsewhere. 



As a matter of fact only distortion and 

 curvature are independent of the bundle width, 



and both coma and astigmatism are increased 

 witli increase in the width of the ray bundles, 

 and it is not true, as stated in this article, 

 that coma alone is the result of the width of 

 the ray bundle. This can be very easily 

 proven without recourse to mathematical cal- 

 culations by the use of a poorly corrected 

 photographic lens, examining the images on 

 the ground glass or making photographs of a 

 grating, using a wide and a narrow stop. 



The best known test for astigmatism is the 

 fact that where this aberration is uncorrected 

 one of two crossing lines may be very vague, 

 while the other is sharp and distinct. This is 

 best seen with the wide stop. The effect is 

 usually explained according to the Sturm 

 theory by saying that only one of these lines 

 can be in focus at a time and that either may 

 be brought into focus. If one will shift the 

 ground glass he can easily prove that only 

 radial linos can be sharply focused by an un- 

 corrected lens, and that towards the edge of 

 the field lines at right angles to these radial 

 lines can not be brought into focus at all and 

 are in fact most nearly in focus on the same 

 plane as the radial lines. 



When the grating is rotated 90° the lines 

 that were vague may become sharp but only 

 when a line is approximately radial is the 

 effect of astigmatism nullified. 



Both astigmatism and coma consist in a 

 longitudinal spreading out of the image pro- 

 duced by the zones of the lens. The radial 

 lines remain sharp because the shifting is 

 radial and the shifted images of a radial line 

 are superimposed, the line remaining sharp 

 because its width is not increased to an ap- 

 preciable extent. 



Instead therefore of making the distinction 

 expressed by Dr. Eppenstein that the features 

 of lateral aberration due to obliquity consti- 

 tute astigmatism, and that those dependent on 

 difference of zones produce coma, the writer 

 would suggest that the former be defined as the 

 difference of focus produced by the median 

 region of the lens and that of the most dis- 

 tant marginal point while the latter repre- 

 sents the focal difference of the nearest mar- 

 ginal point of the lens. 



