R. Greenler and A. J. Mallmann, "Circumscribed halos," Science 176, 128–131, 1972.

R. Greenler, A. J. Mallmann, J. Mueller, and R. Romito, "Form and origin of the Parry Arcs," Science 195, 360–367, 1977.

W. J. Humphreys, Physics of the Air (McGraw-Hill, New York, 1929).

Precisely, *Y* is a *critical value* of *F* if for some *X*, *F*(*X*) = *Y* and the Jacobian matrix of *F* at *X* has less than maximal rank. Thus, at a critical value there has been an unusual amount of bunching, at least infinitesimally.This terminology is not restricted to halos resulting from two-dimensional sets of crystal orientations. For instance, a sun dog has a single critical value while a circumzenithal arc has none. The caustic of the 22° halo is the inner circular boundary of the halo.

If we replace P̂ by -P̂ then the resulting circle *C* is unchanged but the µ coordinate on C changes sign. The sense of µ is well-defined if P̂ is restricted to lie on one half of the equator.

For a given crystal and a given path of light through the crystal (see Fig. 13, for instance), there are two choices for P̂, one choice being the negative of the other. Throughout this section we assume that P̂ lies on the right half of the equator; that is, we assume that the y component of P̂ is positive. This convention insures that the sign of α is well-defined, once a crystal and light path are chosen.

To see analytically that for the upper and lower tangent arcs the caustic coincides with the minimum deviation locus, recall that *F*(θ,µ) = (θ,*D*(θ,µ)). So the Jacobian matrix of *F*, expressed in (θ,µ) coordinates, has determinant [Equation]Thus for the upper and lower tangent arcs, |*JF*| = 0 if and only if ∂*D*/∂µ = 0. [This argument breaks down at (cos Σ, 0, -sin Σ), which is a singularity of the (θ,µ) coordinate system.]

If θ and µ are coordinates for a point on the curve φ = 0°, then a computation shows cos µ = (sin Σ)/|sin θ|. For nonzero values of φ notice that µ(θ,φ) = µ(θ,0) + φ.

The (ξ,µ) coordinate system is defined analytically in the Appendix.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), has an enjoyable description of these and other halos.

J. M. Pernter and F. M. Exner, Meteorologische Optik (Wilhelm Baumuller, Vienna and Leipzig, 1910).

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).