Theory of Fog-Bows. 459 



the blurring due to the finite diameter of the sun into account, 

 it is clear that the violet would be far more diluted with red 

 and other colours than in the case illustrated. There is an 

 intermediate stage in which the green would reach its maxi- 

 mum purity, but the violet would be poor. 



This first case would be realized in a rainbow formed by 

 unusually fine drops. The other case might be described by 

 an observer as a double fog-bow, the outer bow being broad — 

 extending from 42-^° to 34J — and colourless except for a 

 reddish tinge on the outer edge ; and the inner bow — extend- 

 ing from 32° to 27-J° — being brightly coloured with the red 

 inside. This figure shows that the whiteness of the outer 

 bow is sufficiently explained (1) by the immense breadth of 

 the bow even when formed by homogeneous light, and (2) by 

 the approximation of the red to the violet maximum. In this 

 case they are about J° apart. The fact of the third bow being 

 so seldom observed can hardly be due to its faintness, so we 

 must attribute it to irregularity in the size of the drops. 



Thus far we have only treated of two special wave-lengths. 

 In fig. 3 may be seen the relative positions of all the different 

 colours in bows of various radii. The abscissae are wave- 

 lengths and the ordinates radii of the bows. The top curve 

 gives the geometrical bow, calculated by the formulae 



3 cos 2 cj) = iJi? — 1 , sin (f> = jju sin <£', radius = 4c$>' — 2$, 



from the values of (jl for water at 0° C, quoted by Landolt and 

 Bornstein. The indices for H«, H/3, H7 are from Wullner, the 

 other three from Riihlmann. I have drawn the curve on the 

 supposition that the small difference between the results of the 

 two sets is constant. The other curves are separated from this 

 by distances proportional to \* and to 3, 5, 6, 7, 10, and 12 

 respectively, and may serve to represent either the rainbow or 

 any of the supernumeraries. The two dotted vertical lines 

 indicate roughly the boundaries of the bright part of the 

 spectrum. The temperature-effects are trifling and may be 

 disregarded. At 60° F. all the radii are some M larger. 

 Below 32° F. the indices of refraction and, consequently, the 

 radii are almost constant. 



Nowhere is the superposition of colours perfect. But about 

 36° 45' they may be all contained within 15', so that any 

 supernumeraries near that radius must be colourless. At 32° 

 the spread of the colours is nearly as great as in the geometrical 

 bow, and in the reverse order. In practice, however, we 

 cannot expect the same vivid colouring under the most 

 favourable circumstances, as the first supernumerary, for 



