April 27, 1899] 



NATURE 



617 



That this is so, is shown in tables giving angles of incidence, 

 refraction, and the resulting final deviation, and is further ex- 

 plained by a simple trigonometrical deduction. The reflected 

 and refracted rays emerge, not parallel to one another, but 

 divergent or convergent. The limiting ray of minimum devi- 

 ation, and the rays in its immediate neighbourhood and ap- 

 pro.ximately parallel to it, would be the "efficient" rays of 

 Descartes. They have their .significance, though not that 

 which Descartes ascribed to them. For the C rays, this angle 

 (which corresponds to an angle of incidence of 59° 24') is 42° 4' ; 

 that is to say, the red-orange arc of the primary bow is seen 

 under that angle. Multiple reflection within the raindrops 

 renders an infinite number of other such limiting rays possible. 

 The emerging rays would emanate from various quadrants. We 

 should hence see bows, not only when standing with our backs 

 to the sun, but also when a cloud is between our eyes and the 

 sun (e.g. in the case of three and four internal reflections). 

 The direct sunlight would prevent our seeing those bows, but 

 they can be observed and shown in class-rooms when we let 

 the light fall on cylindrical glass rods after Babinet's fashion. 

 With cylinders instead of spherical drops, we see, of course, a 

 series of vertical coloured bands, arranged in a horizontal line, 

 instead of arcs. Miller, experimenting with water streams, 

 measured thirty such monochromatic bands. Pernter describes 

 a simpler arrangement, and calculates, in his popular treatise, 

 the angles of minimum deviation for fifteen bows, both for water 

 and glass. E.xperimenting with 

 white light and water streams, 

 I mm. and less in diameter, he 

 counted with I mm. drops one 

 bow and twenty-four secondaries 

 (supplementary bows) of beautiful 

 colours (white in the twelfth, after 

 which the sequence of the colours 

 is reversed), and with drops of 

 0"5 mm. eleven bows and second- 

 aries and some more bands of in- 

 distinct colour (white in the fifth). 



We recognise from Fig. i that 

 the emerging wave has not a 

 straight front like the entering 

 spherical wave AB, but a peculiarly 

 curved front, represented in ex- 

 aggerated curvature in Fig. \a. 

 Such a wave must give rise to in- 

 terference phenomena, and all the 

 rainbows, not only the supple- 

 mentary (or so-called spurious) 

 bows, are really diffraction phen- 

 omena of a peculiar kind. That 

 part of the wave-front which is 

 nearest to the ray of minimum 

 deviation might be called the 

 effective wave-front. In order to 

 arrive at an equation for that part, 

 Pernter starts from Wirtinger's 

 consideration that, if j,, s„, and 

 jj are the paths of a ray, in 



the air, in the water and again in the air, reckoned be- 

 tween the entering and the emerging wave-fronts, and Cj and 



< o the velocities of light in air and in water, then i -1- -? -t- — = 



fi ^2 ''i 

 const, for all rays of that wave. The constant can be chosen 



at will ; he takes the value -, in which a is the radius of 



the drop. Under the assumption that the curve consists of two 

 spherical arcs, one concave, the other convex, Pernter then 

 calculates the phase difference after Mascart. As regards the 

 amplitude, however, of his intensity equation, he has to refer 

 back to Airy; but he succeeds in showing that each colour 

 of the rainbow consists of an infinite number of coloured 

 rings of decreasing intensity, separated by rings of intensity o. 

 Pernter objects to the term ' ' spurious " rainbows, since they are 

 as much rainbows as the ordinary bows ; his own terms, 

 Hauptbogen, Nebenbogen, secundare Bogen (principal bow, 

 by-bow, secondary bows) are not suitable for literal translation. 

 When we replace the prism of a spectroscope by a glass rod, 

 2 mm. in diameter, and set the telescope under 22° 51' (principal 



bow for glass), we see a series of red bands as mentioned. If this 

 angle is not convenient, we adjust the instrument for one of the 

 other bows. The first is by far the brightest ; after the eighth 

 maximum the intensity diminishes very slowly ; Airy's original 

 curve brings this out very clearly. Replacing the rod by one, 

 less than i mm. in diameter, we notice that the colours are 

 different and less bright ; the blue, absent in the first experiment, 

 is prominent, and all the bands arc broader than before. 

 Smaller raindrops give broader bows, but, owing to their 

 diminished intensity, their number appears smaller. Fig. 2 is 

 Pernter's colour curve for raindrops 0'5 mm. in diameter. The 

 size of the actual raindrops lies, for our latitudes, between 0'05 

 and 2 or 3 mm. diameter. The fog-bow is produced by 

 the sun when shining on the water globules, 0^05 mm. and 

 less in diameter, of fine mists. We notice that we get real 

 white in the bow by superposition of the colours. This 

 is possible for drops of all sizes, and must occur with very 

 small drops. The sequence of the fog-bow colours is : very 

 faint yellow, whitish yellow, bright white, whitLsh violet ; 

 colourless gap ; then (secondary bows) faint whitish blue, 

 white, whitish red. To imitate these mists, Pernter fixed a glass 

 tube, o'5 mm. in diameter, in a lead pipe connected with the 

 high-pressure water mains, and directed the jet against a 

 metallic plate ; the mist thus produced consisted of drops 

 o'oio6 mm. in diameter. McConnel (Phil. Mag., 29, p. 453, 

 1S90), who made calculations for raindrops of certain sizes in 



iSgo, describes eighteen fog-bows, observed by Osmond in 

 1886-S7 on Ben Nevis; of these, ten were double. Exact 

 measurements of rainbows are exceedingly scarce. Pernter 

 differs from McConnel as regards the border colours of fog-bows ; 

 he also doubts that the pale colour of fog-bows can be due to 

 the uneven sizes of the drops, because the accompanying 

 phenomena, glories, require homogeneous conditions. But 

 dilution with white light, of course, m.akes all colours appear 

 whitish. 



Pernter's conclusions are interesting to meteorologists. The 

 greater the drops, the more secondary bows. Bright pink 

 and green, without blue, indicate drops from I to 2 mm. in 

 diameter ; intense red occurs with big drops only, but the max- 

 imum intensity is really in the violet. Drops of 0-5 mm. give 

 secondaries consisting of green and violet (and also blue, which, 

 however, is masked by contrast) immediately joining the 

 principal bow. Yellow in the secondary would mean drops of 

 o'3 mm. and, if there are separating gaps, of 02 mm. Drops of 

 diameters between o'i7 and 0'4 mm. show the greatest variety 

 of colours, also in the secondary bows; but real red is absent. 

 When we notice five and more secondaries of striking bre.adth 



NO. 1539, VOL. 59] 



