Theory of Fog-Boies. 



457 



The intensity of the light in any direction is proportional 

 to the square of the expression 



w 



= 1 cos ■= (w d —mw) 



div. 



(i) 



where m is connected with the angnlar separation % of that 

 direction from the geometrical bow by the equation 



X = qm 



(2) 



as 



q was left by Airy an undetermined constant. It has recently 

 been shown by Boitel* and Larmorf that, for the primary 



bow. 



<f= 



9 4-m 2 

 4096 (jf-iy 



(3) 



Thus q, which may be treated as constant for different colours, 

 = 0*465 nearly. 



Everything turns on the value of W 2 . Airy, calculating by 

 quadratures, found that, for negative values of m, W 2 , starting 

 from a small value at m=0, rapidly and continuously de- 

 creases and becomes very soon insensible ; but on the positive 

 side W 2 has an infinite number of fluctuations, vanishing 



between the successive maxima, 

 the following table :— 



Selected values are given in 





m. 



W 2 . 





-4-00 



0-000009 





-1-00 



0-0745 





o-oo 



0357 





0-80 



0-940 



1st maximum ... 



1-08 



1-006 





1-20 



0-995 





2-50 



0000 



2nd maximum ... 



3-47 



0-615 





4-36 



0-000 



3rd maximum . . . 



5-14 



0-510 





5-89 



0-000 



4th maximum . . . 



6-58 



0-450 





724 



o-ooo 



5th maximum . . . 



7-87 



0-412 





8-48 



0-000 



* Comptes Bendus x 1888 ; or Phil. Mag. August 1888. Boitel poirt3 

 out that Airy's equation to the wave-surface is only a first approximation, 

 and has found experimentally some small discrepancy between theory and 

 observation which he attributes to this. But Larmor's analysis of Miller's 

 observations on fine jets of water shows no discrepancies greater than 4', 

 up to x = 10°. 



t Proc. Camb. Phil. Soc. vi. p. 283 (1888). 







