228 



NA TURE 



[January 3, 1901 



" looked about fruitlessly for a guide " ; he found 

 nothing to help him, and the volume now under review, 

 which has been translated and amplified by Prof. 

 Thompson, is the result. 



Let us consider the problem ; the photographer needs 

 to produce on a flat surface an exact picture of a distant 

 object. To take in a wide field of view the plate must 

 subtend a considerable angle at the centre of the lens ; 

 the rays of light traverse the lens in very various 

 directions, some are parallel to its axis, others are 

 oblique, and the angle between them and the axis may 

 reach 30° to 40°. 



It is impossible to attain this result with any single 

 lens ; we must examine, then, in what respects an actual 

 lens differs from the perfect instrument of Gauss's 

 theory, what are its defects, and how they are to be 

 remedied. 



Now this theory is an approximation for real lenses 

 depending on the assumption that the atigle between 

 any ray and the axis of the system is so small that 

 all its powers above the first may be neglected. 



We wish to inquire, then, what are the conditions 

 which must hold in order that a refracting system may 

 produce an image coinciding with that given by Gauss's 

 theory, even when we retain in our mathematical theory 

 powers of the obliquity above the first. This was done by L, 

 von Seidel in the years 1856, 1857 {Astronomische Nach- 

 richten, 835, 871, 1027-1029) for the case in which powers 

 of the obliquity up to the third are retained. His work is 

 hardly known in England. He found that to this degree 

 of accuracy a perfect image could be obtained provided 

 five separate equations of condition between the curva- 

 tures and refractive indices of the lenses and their dis- 

 tances apart were satisfied. These five conditions, which 

 we may denote by Sj = o . . . 85 = 0, are sufficient for 

 all cases in which the light employed is of definite re- 

 frangibility ; to correct, however, for dispersion, two other 

 conditions are required, and the modern photographic 

 objective, possible only in consequence of the discovery 

 by the Jena factory of special " anomalous " glasses, is 

 the outcome of the attempt to satisfy these conditions. 



In the work before us no attempt is made to prove the 

 above equations ; this is not part of the scheme of the 

 book, but it is shown that to each condition a distinctive 

 physical meaning can be attached, and this physical 

 meaning is brought into very clear light. 



The first equation, Si = o, is the condition that the 

 image of a point on the axis may be a point, free, there- 

 fore, from aberration, even when the full aperture of the 

 lens is used. 



Let this be satisfied, and suppose the point source to 

 move away at right angles to the axis to some neigh- 

 bouring position, let the refracted beam be received on a 

 screen perpendicular to the axis ; in general, a blurr of 

 light, more or less pear-shaped, will be formed at the 

 screen ; in some positions the narrow end of the pear is 

 towards the axis, in others it is removed from it. The 

 shape varies as the screen is moved, but unless the lens 

 is corrected a point image is never formed. This is the 

 defect or aberration called coma, so beautifully shown by 

 Prof. Thompson to the Physical Society last session. 



Now we know that when a small pencil of rays is 

 obliquely refracted, the refracted rays diverge from two 

 NO. 1627, VOL. 63] 



focal lines which lie in perpendicular planes ; and we 

 may look upon the finite pencil as composed of a series 

 of small pencils incident at different parts of the lens. To 

 each of these corresponds two focal lines, a primary and 

 a secondary. But the refraction produced by the lens is 

 such that the primary line belonging to a small pencil 

 traversing the lens near its edge does not coincide with 

 that of the central pencil. The primary focal line has a 

 different position for each of the small pencils into which 

 the finite incident pencil has been resolved ; if, for a 

 moment, we look upon the primary line as an image of 

 the source, the position of this image depends on the 

 portion of the lens by which it is formed. The next step, 

 then, is to superpose these partial focal lines so as to form 

 two single, primary and secondary, focal lines for the whole 

 pencil, and this is the meaning of Seidel's second condition, 

 82 = 0. This condition, which was known to Frauenhofer, 

 is shown to be identical with a law distinguished as the 

 8ine-law, which states that if P be a point on the axis 

 of the system, and Q its image, and if a, a be the angles 

 which an incident ray through P and the corresponding 

 refracted ray through Q make with the axis, then 

 sin a/sin a is a constant for all rays. If a small object 

 be placed at P, perpendicular to the axis, and a perfect 

 image of this be formed at Q, then, as von Helmholtz 

 showed, the ratio sin a/sin a measures the linear mag- 

 nification of the image. The connection between this 

 and the physical meaning of the condition 82 = is an 

 obvious one. 



But if this condition be satisfied, we are far from 

 having a point image of our source ; we have, instead, 

 two linear images — one lying at right-angles to the axial 

 plane through the point, the other, the secondary image, 

 lying in that plane. If, instead of considering the image 

 of a point, we deal with a plane cutting the axis at right- 

 angles in P, we get, as the image of this plane, two curved 

 surfaces, the one made up of primary lines, the other of 

 secondary lines ; these both cut the axis at right-angles 

 in Q, the image of P. If by any means we can make the 

 primary and secondary lines coincide, we shall have a 

 point image of our point source ; and this is done if the 

 condition 83 is satisfied. 8uch an image is said to be 

 stigmatic, or sometimes an-astigmatic. If this be done 

 for the whole field, the curved surfaces move up to co- 

 incidence ; the image of the plane is a single surface, not 

 two ; in general, however, this image surface will be 

 curved, and as we cannot dish out our photographic 

 plate to get it, we must, if possible, make it plane. The 

 equation 84 = expresses the condition for this. 



Thus, if these four conditi ons are satisfied, we obtain 

 a plane stigmatic image of any object lying in a plane 

 normal to the axis of the lens. 



But this alone is not sufficient to give us a perfect 

 image ; it must be similar to the object, there must be 

 no distortion. It is found that the equation 85 = expresses 

 the condition for this. 



In some such manner as the above. Dr. Lummer and 

 Prof. Thompson explain the meaning of the five con- 

 ditions cf no aberration, and then proceed to show how 

 each is satisfied. 



For the details of this the reader must refer to the 

 book ; there is one other point, however, which it is 

 desirable to follow up more completely here. The 



