OF ARTS AND SCIENCES: MAY 10, 1870. 217 



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limiting angle of transmission is reached. There is an interior as well 

 as a superior tangent arc to the halo of 22°. Their figures are complex, 

 and they join in a single curve, circumscribing the halo itself, when the 

 altitude of the sun exceeds 29° 15'. The inferior arc is rarely visible, 

 unless the sun is more than 22° high. The halo of 46° is less bright 

 than that of 22°, because it is larger and broader; and more light is 

 reflected by the prisms under the larger incidences. The tangent curve 

 to this halo is a circular arc having the zenith for its centre. It cannot 

 be formed if the sun's altitude exceed 32° 12'. The semi-amplitude 

 increases from 57° 48' to 90°. But when it is 90°, its height is also 90° 

 and its radius is reduced to zero. The maximum brightness is in the 

 middle of the whole arc. This arc actually touches the halo of 46° only 

 when the altitude of the sun is 22° 8'. It sensibly touches between the 

 altitudes of 15° and 28°. If the sun were in the horizon, the tangent 

 arc would be 12° 4' above the summit of the halo. If the sun were 30° 

 high, the tangent arc would be 3° 39' above the halo. The altitude of 

 22° 8' is most favorable, because, in this case, the middle of the arc is 

 formed by rays which have suffered a minimum deviation. A tan- 

 gent arc to the lowermost point of this halo is not impossible, but rare. 

 In this event, the light must enter a vertical face and emerge at the base. 

 The limits of altitude are complementary to those which the superior 

 tangent requires ; that is, the sun's altitude must be between 57° 48' 

 and 90°, the arc actually touching the halo at the special altitude of 67° 

 52'. If the axes of the prisms are shifted from a vertical to a horizon- 

 tal position, the inferior and superior tangent arcs are changed to what 

 are called infra-lateral and supra-lateral. 



I have taken renewed interest in this theory of halos, which has been 

 admirably developed by Bravais,* on account of the halo seen at Cam- 

 bridge, January 6, 1870. This halo was seen about two o'clock, when the 

 altitude of the sun was not far from 25°. The principal feature of the 

 phenomenon, on that occasion, was the tangent curve to the halo of 46°, 

 though the halo itself was not visible. At Waltham, the mock suns were 

 seen, but not the tangent curve. The tangent curve seemed to be a com- 

 plete circle, and the colors were very vivid, the red being the outermost 

 color, or nearest to the sun. I have stated that, theoretically, the maxi- 

 mum amplitude of this curve is 180°, and, if the sun had an altitude of 

 25°, the amplitude would be only about 138°. The history of halos fur- 



* Journ. de l'^cole Poly technique. Cahier 31. Tome xviii. 

 VOL. VIII. 28 



