## Abstract

A display of halos of unusual radii occurred on 14 April 1974, and was visible in southern England and in Holland. Photographs of the display were taken by Professor Scorer and the present author was privileged to examine them. They showed not only circular halos but also arcs tangential to some of them. The investigation described in the following paper was undertaken to try to elucidate the mechanisms by which these arcs might be formed. It led to a new theory of the formation of circular halos and also of the tangential or quasitangential arcs associated with them. It also led to a picture of the shape of the ice crystals giving rise to this particular display.

© 1979 Optical Society of America

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### Equations (15)

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(1)
$${\mu}^{\prime}=\frac{\mu {(1-{cos}^{2}\theta /{\mu}^{2})}^{1/2}}{sin\theta}$$
(2)
$$cos{N}_{1}{N}_{2}=sin{\mu}_{1}sin{\mu}_{2}+cos{\mu}_{1}cos{\mu}_{2}cos({\mathrm{\lambda}}_{2}-{\mathrm{\lambda}}_{1}).$$
(3)
$$A=180\xb0-{N}_{1}{N}_{2}.$$
(4)
$$cosZ{N}_{1}{N}_{2}=\frac{sin{\mu}_{2}-sin{\mu}_{1}cos{N}_{1}{N}_{2}}{cos{\mu}_{1}sin{N}_{1}{N}_{2}},$$
(5)
$$cosP=-cos{\mu}_{1}sinZ{N}_{1}{N}_{2}.$$
(6)
$$cosL=\frac{-sin{\mu}_{1}cosP}{cos{\mu}_{1}sinP},$$
(7)
$${B}^{\prime}=RQ={\mathrm{\lambda}}_{1}-L.$$
(8)
$$cos\nu =sin{\mu}_{1}/sinP.$$
(9)
$$cos\theta =sin\text{\u2211}cosP+cos\text{\u2211}sinPcos({B}^{\prime}-\beta ).$$
(10)
$$cos\delta =\frac{sin\text{\u2211}-cos\theta cosP}{sin\theta sinP}.$$
(11)
$${i}_{1}=\nu +\delta .$$
(12)
$$sin\overline{\mu}=cos\theta cosP+sin\theta sinPcos(\delta -D).$$
(13)
$$cos{S}^{\prime}ZK=\frac{cos\theta -sin\overline{\mu}cosP}{cos\overline{\mu}sinP},$$
(14)
$$\mathrm{\lambda}={B}^{\prime}-\beta -{S}^{\prime}ZK.$$
(15)
$${p}_{1}{p}_{1}^{\prime}$$