310 C. Bar-us — Flower-like Distortion of Coronas^ etc. 



the angle between the horizontal through o and the radius 

 vector, r, to a line of uniform color, ab, in the distorted corona, 

 and let A be the height of the extremity, <?, of the radius 

 vector above the datum line through o. If R be the distance 

 of r from the eye of the observer, 2r/B = s/B is the angu- 

 lar aperture of the corona. 



Let 8 be the diameter of the particles at the level passing 

 through c. Then if 8 and sj B be the corresponding quan- 

 tities (diameter and aperture) of particles in the datum level, 



3 5 = S s = -00144, (1) 



the number being found by experiment for normal coronas. 

 Hence r'd$+$'dr = (2) where r — s/2. Again if the angle 

 <f) increases counter-clockwise by dj>, 



rd<$> cos $ -f dr sin cf> = dh. (3) 



S^.i 



Let dh = adh (4) so that the diameter of the water particles 

 is supposed to decrease (a being negative) uniformly upward. 

 Other laws of distribution would merely complicate the prob- 

 lem without conducing to the present purposes, seeing that the 

 observed facts will be sufficiently interpreted by equation (4). 

 Combining (4), (3), (2), 



- -00072 dr/ar\ 

 by equation (1) 



r-cos 4>d4> + sin cf>dr = — (8 /a) (dr/r) 



00 



Put A — -00072 /a and integrate (5) whence sin cf> = C/r+A/r*. 

 To determine G, equation (1) is available, since for cj>=0,r=r . 



Therefore 



sin <£ 



{iA Is) (l Is -l Is). 



To construct these coronas distorted in consequence of the 

 linear distribution of size of particles, it will generally be more 

 convenient to express s in terms of $, so that finally, 



s = — (2A/s sm<f>) (1 — Vl + s\sm<j>/ A) = 



(S /asin<£) (1 — <\A + 2as sin <£/S ). (6) 



With this equation the following table has been computed 

 (a is for convenience entered positively). To have an average 

 case at the datum level, put B = -(J01 cm , as the diameter of 





