July 7, 1911] 



SCIENCE 



27 



red and blue rays, then the foci for these rays 

 ■with respect to the curvature of the vertical 

 meridian will fall, say, at r and h, while the 

 foci with respect to the curvature of the hori- 

 zontal meridian will fall at B and B respect- 

 ively. The relative distances are exaggerated 

 for the sake of clearness. Helmholtz gives 

 the distance between the focal planes for red 

 and violet {BB or rh in the diagram) as 

 about 0.6 mm. "Whether r falls to the right or 

 left of B depends on the degree of astigma- 



tism. In an otherwise emmetropic eye 

 focused on infinity the retina will be some- 

 where between the points B and r. If the ac- 

 commodation is such that the point B falls 

 on the retina, the eye perceives instead of a 

 point a vertical red band. The direction of 

 the bands is indicated by short lines under the 

 letters in the diagram. A horizontal blue 

 band is perceived if the point h falls on the 

 retina, in each case the band being surrounded 

 by an indistinct halo due to the other images. 

 If B and r fall close enough to the retina, B 

 and h will be out of focus and practically in- 

 distinguishable, so that only a horizonal red 

 band crossing a vertical band of blue wiU be 

 seen. While the writer normally sees these 

 bands crossing, he is able, by changing the ac- 

 commodation of the eye, to observe the vertical 

 red band (surrounded by' a bluish halo) or the 

 horizontal blue band. To see the two bands 

 crossing, the degree of astigmatism must, of 

 course, fall between certain limits, but these 

 limits turn out to be surprisingly wide, partly 

 because the appearance of crossed bands is 

 heightened by the effects of contrast, partly 

 because the bands become longer with in- 

 creasing astigmatism. As would be expected, 

 the red band loohs nearer than the blue. 



As for the quantitative relations, it is easy 

 to show that there is a degree of astigmatism. 



well within the limits conmionly found, for 

 which the crossed red and blue bands are in 

 focus simultaneously. For Helmholtz, as 

 stated above, shows that the distance BB be- 

 tween the focal lengths for red and blue due 

 to dispersion is about 0.6 mm. On the other 

 hand, in an astigmatic eye the difference be- 

 tween the focal lengths due to the two curva- 

 tures at right angles {Br or BV) may be any- 

 thing from zero to 2 mm. or more. Thus an 

 eye need be only slightly astigmatic (correc- 

 tion about 1.5 diopters) in order to bring the 

 points B and r into coincidence. 



If the eye has an astigmatism so complex 

 that several astigmatic axes in different planes 

 have to be dealt with, the red and blue bands 

 are correspondingly complex. Indeed, this 

 method might perhaps prove useful in the 

 examination of astigmatic eyes. By placing 

 cylindrical lenses of varying focal lengths be- 

 fore the eye, it is possible to tell by the disap- 

 pearance of the bands when the optimum 

 correction has been attained. It may be re- 

 marked in passing that the phenomenon of 

 color dispersion furnishes an interesting dem- 

 onstration of the manner in which ametropia 

 is corrected. For, as is well known, far- 

 sighted and near-sighted eyes perceive purple 

 sources as surrounded by a red or a blue halo 

 respectively, and this halo can be increased, 

 or reduced to a well-marked minimum, by the 

 use of suitable spherical lenses. 



Even a small source of white light looks a 

 little ruddier on two opposite sides if the eye 

 is astigmatic. 



Very peculiar effects are produced in an 

 astigmatic eye by repeating the experiments 

 described above with a purple or " blue " light, 

 using in place of a round hole a row or group 

 of pinholes, or a slit, or a maltese cross set at 

 varying angles. 



To show the crossed red and blue bands ob- 

 jectively, an astigmatic lens of high disper- 

 sion was made in the following manner: the 

 surface of a large incandescent bulb was 

 tested at different points, until a region was 

 found where the ratio of the curvatures in 

 planes parallel and at right angles to the axis 

 of the bulb was of the desired value (a 60- 



