FEBRUARY 1937 MONTHLY WEATHER REVIEW THE GEOMETRICAL THEORY OF HALOS-I11 By EDGAR W. WOOLARD [Weather Bureau, Washington, D. C., February 19371 REFLECTION The calculation of the image produced by external reflection is easily accomplished as indicated in figure 6.' Internal reflection (which occurs whenever an internal ray meets an interface at an angle with the normal greater than y=arc sin -7 p>1) conforms to the same law, but may be associated with refraction a t some other point in the complete path of the ray; when the ray is incident 1 P FIGURE fi.-Simplc Reflection: 6 , luminous source: SC, iucideut ray; C!N, normal to inter- face nt point of incidence; i. angle of incideuce; (261, reflected ray; R, angle of reflection; S', virtual image; D, deviation. on, and also emerges from, a face parallel to that from which it is int,ernally reflected, the emergent ray is parallel to the course it would have ta.ken if merely reflected a t the point of inc.idence. The geometrical relntions on a sphere of indefiilitely great ra.dius, figure 7, readily lead to approprin te forinulne for computation, Formulae 11. It is a ~orollary of the Law of Reflection, that the source and the image lie a t equal distances from the geometric pole of every great circle normal to the reflecting plane (any such-pole is on the great circle PP). Furtherinore, the projectioiis of the incident and the reflected rays on any plane through the normal to the reflecting surface make equal angles with the normal, just as if they were nc.tua1 rays. These principles constitute Bravais' Laws of Reflection. The deviation (angular displacement of the image from the true position of an indefinitely distant source) is away from the normal N. I n the case of prismatic refraction accompanied by an internal reflection from a principal plane, figure 8, the projection of the ray on the principal plane is not altered by the reflection; nnd hence the positions of the images produced with and without suc,h a reflection are syni- metric with respect to the princ,ipal plane. The calcu- 1a.tion of the ima.ge, figure 9, is the,refore acconiplished with Formulae I except t1in.t the altitude relnt,ive to the principal pla.ne is --h and (6*) COS D=COS* h(l+cos D')-l (I*) cos h sin D' (7*) sin A= sin by the Law of Cosines and t,he Law of Sines, respect,ively. The deviation is toward V and away froin P'. Prismatic refraction accompanied by an internal reflec- tion from m y arbitrarily given plane, figure 10, requires the more complicated Formulae 111; see figure 11. When I The figure3 am numbered consecutively with those In paper 11. 55 the reflecting plane coincides with a principal plane, p=O" and the formulae reduce to the previous ones; when the reflecting plane coincides with a lateral face, p=90°. Multiple internal reflection will, of course, take place if the once-reflected ray meets the next interface a t too great an angle with the normal. The positions of the images formed by light from the sun or the nioon incident on a particular face of an ice crystal of specified form in a given orientttion may now be calculated by superposing the appropriate figures on the celestial sphere, with S in the position of the luminary: The preceding formulae give the. position of an imnge relative to the source; and from t h s , the position relative to the horizon is readily found for any given altitucle of the sun or moon, by solving the spherical triangles involved. The formulae required will be given in the next section. I f I \- \ . FIGUhE 'I.-Calculalion of the Image Produced by Simple Refldion: See Formulae 11, PPP reflecting plane CN, normal at point of incidence; N. pole of reflecting plane. on sahm side of plane as source 6; S', image; I), deviation; C , observer. The deviation, D, is away from N. Formulae I1 CALCULATION OF IMAGE PRODUCED BY SIMPLE REFLECTION Given To Find Angle of incidence, i Coordinates of S relative to Relative to reflecting plane: reflecting plane: Altitude 90'- i Relative Azimuth 0' Relative Azimuth Oo Coorainates of S': Altitude Relative to S Deviation, D Positiou Angle 180' a Computation (1) D=18Oo-2i oo--un principal plane; D, diviation; A, position angle. IC, observer at center of sphere IS omitted from the diagram; SIC, Y’IC would correspond to CO. C’O, respectlvely, in h u r e 10.1 Formulae I11 Computatio I I }Table 3 CALCULATION OF IMAGE PRODUCED BY PRISMATIC REFRAC- (1) sit1 =p sin k TION COMBINED WITH INTERNAL REFLECTION (2 ) A’= X+z1--r, To Find (3) cos w=sin k cos O+cos k sin COS X’ Given . sin p sin X‘ 11. a; h; 8, x; i 1 Coordinates of S‘ (4) sin y= Coordinates of S: Relative to Principal Plane sin w Relative to Principal Plane Altitude -h’ (5) sin k’=sin k cos 2 wfcos k sin 2 w cos y Altitude h Relative Azimuth f D’ (6) sin h’=p sin k’ Relative Azimuth 0’ Deviation u Position Angle f A Relative to S sin 2 w sin y cos k’ (7) sin z= (8) T~’=LT+z--T~ (Internal reflection occurs if r:>$ for h’) (9) D’=il+il’--a Table 3 with h=h’ (10) cos D=cos h cos h’ cos D’-sin h sin h’ \I cos h’ sin D’ sin L) (11) sin A=- [See figure 11. The t.rigonometrica1 relations on the sphere follow from t.he Law of Cosiiies nnd the Law of Sines.]