narrow the avenue towards the west, where it was open to the light, and also towards the east, where 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 explanation, 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 transversely 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 perpendicular 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, while 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 versâ, also

refract the light at a definite angle of about 46°, and thus a second coloured ring outside the first is perceived; and in both, the refraction being produced directly without internal reflection, the colours are arranged in the same sequence, red in side and blue outside.

The third circle of white light sometimes observed outside the other two coloured rings is probably produced by reflections from the surfaces of the ice-crystals without refraction by the light passing through them.

In the bows seen at Invercargill the axes of the cones of coloured or white light producing the appearances were at right-angles to the parallel rays of moonlight—not coincident with them, as in ordinary halos. The phenomenon was therefore, I conceive, caused by reflection previous to refraction—reflection from the inclined surfaces of myriads of ice-crystals, which happened to be in the right position to reflect the incident rays outwards from the walls of frozen mist; their reflected rays passing through other crystals, and being refracted in their transit through them, gave rise to the divergent coloured rays, which produced in the eye of the spectator the arches seen both to the north and to the south. I think that in this way the effect was produced; for, although by an interior reflection in a prism coloured rays would be projected backward towards the source of light, I have not been able to find any position of the prisms that would account for the symmetrical arches on the theory of internal reflection.

The reflection from the outer surfaces of other crystals would doubtless weaken the light; but, as compared with ordinary halos, in which the light which reaches the eye has passed completely through a cloud, and has undergone very numerous reflections and refractions in its transit, the Invercargill appearance would have the advantage that the light would have been reflected and refracted at the outer surface of the cloud, and so would have lost less by absorption and dispersion than in the more ordinary appearance of the circular halo round the sun or moon; and this would be the more true if, as I suppose, the clear avenue in which the spectator stood was V-shaped—wider at the western end towards the moon and narrowing towards the east.

The straight ray of white light tangential to the northern white bow at its west end, which appeared only for a short time, may have been due to a local current of air which turned the majority of the ice-crystals there in a direction which reflected the moonlight; but it is very much more probable that it was the result of some of those intricate effects of refraction and reflection which produce the reversed circles sometimes seen in connection with halos, and those phenomena