458 Mr. James C. M c Connel on the 



Airy's calculation extended from m= — 4 to ?n= +4. Out- 

 side these limits the calculation by quadratures becomes al- 

 most impracticable, but Sir George Stokes * has found a 

 rapidly convergent series for W in descending powers of m. 

 From this follow simple approximations. For the dark bands 

 m=3 (i— i)fj where i is given the values 1, 2, 3 . . . successively. 



2 

 For the maxima, m=3(i— £.)* and W 2 = 7=. These are 



accurate enough for most purposes when i>l. 



The variations of W 2 with m are exhibited graphically in 

 PI. X. fig. 1. It will be observed that the first band is much 

 broader as well as higher than the others, and that the intensity 

 at m=% = is by no means negligible. Thus, putting the 

 question of colour on one side, the effect of diminishing the 

 size of the drops is to spread out the principal bow in an 

 inward direction, leaving its outer edge but little affected. 

 After the third band the brightness diminishes very slowly. 

 Even the 30th maximum at m = 28*5 has the value W 2 = 0'217. 

 Of course the irregularity of the drops would prevent any 

 colour-effects or distinct bands as far out as this, and we should 

 only look for white light ; and this is exactly what is observed. 

 The space within the primary rainbow is strikingly brighter 

 than that outside ; and Mr. Backhouse describes the glare 

 within a fog-bow he witnessed as being not much inferior in 

 brightness to the bow itself. 



Some idea of the colour-effects in two cases may be gleaned 

 from fig. 2. I have taken as the representative red and violet 

 of the spectrum, light of wave-lengths 0'000615 millim. and 

 0*000455 millim. It happens that the radii of the geometrical 

 bows for these colours are 42° and 41°. The curve represent- 

 ing the brightness of the red is set out in accordance with 

 equation (2), 42° being taken as equivalent to m = x=0. 

 In the curve for violet, which is dotted in the figure, 

 41° is taken as equivalent to % = 0. The upper pair of 

 cmves represent the case when 2a, the diameter of the 

 drops, is 0*3 millim., the lower pair the case when 2a = '024 

 millim. In the former the drops are of about the size 

 adapted to give the most vivid colours possible in the 

 principal bow and the first two supernumeraries. When the 

 drops are very large, say 5 millim. in diameter, it is true 

 that the principal maxima for red and violet are better sepa- 

 rated, but the principal violet maximum gets entangled with 

 a number of the smaller red maxima. Thus, when we take 



* Trans. Camb. Phil. Soc. ix. 1850, or Collected Papers, ii. p. 329. 



