546 Transactions. — Miscellaneous. 



narrow the avenue towards the west, where it was open to the 

 light, and also towards the east, w^here it was closed in, and 

 so the arches would seem to spring from about the same 

 points in the horizon. The order of the colours in the inner 

 and second arches being the same points also to this explana- 

 tion, as this is also the case in halos. To account more in 

 detail for the mode in which the appearances were produced 

 under this very unusual combination of circumstances we 

 must go back to the more ordinary phenomenon of halos, or 

 circles of coloured and white light seen round the sun or 

 moon when shining through thin frozen clouds. 



The basis of the explanation is the fact that ice-crystals 

 are hexagonal prisms, and that when light passes trans- 

 versely through such a prism it is refracted, producing a 

 spectrum of coloured rays, the mean angle of divergence of 

 the emergent rays (i.e., the emergent angle of the yellow ray) 

 being about 22° from the entering ray, if that entering ray 

 strikes a face at angles between 35° and 55° from the per- 

 pendicular to the face. If the crystal be so turned that the 

 entering ray is similarly inclined to the face on the other side 

 of the perpendicular it emerges at the same mean angle of 

 22° in the opposite direction. As the successive perpendiculars 

 to the faces of the crystal are inclined to one another at 

 angles of 60°, it follows that, in a semi-revolution of a crystal 

 on its major axis, during 100° of the revolution the parallel 

 rays of light will have been refracted at this definite angle of 

 22° to one side or the other, w^hile 80° will have been non- 

 effective ; or, which comes to the same thing, if a multitude of 

 ice-crystals be placed at random in all possible positions in 

 the path of the light, of those through which the light passes 

 transversely, 55-5 per cent, will be effective in refracting the 

 light in all directions, inclined at 22° to the incident ray. 

 Hence, from each unit-volume of crystallized vapour (say, 1 

 cubic foot, or 1 cubic yard) a series of concentric cones of 

 coloured light issue, the angle at the apex of the cone being 

 always the same. If the light be proceeding from its source 

 to the eye, passing through such a frozen cloud, it is evident 

 that the conditions exist for the formation, in the eye of the 

 observer, of a circular halo around the source of light, whether 

 sun or moon ; for such of those cones of coloured light as 

 conform to the surfaces of similar imaginary concentric cones 

 in the opposite direction, having their common apex in the 

 observer's eye, are effective in producing in that eye the sense 

 of concentric coloured rings of light at about 22° from the sun 

 or moon. 



Similarly, it may be shown that those ice-crystals which 

 are so placed that the light passes through them obliquely, 

 entering at a side and issuing at an end, or vice versa, also 



