With monochromatic light the rings would be simply bright and black--the bright rings occurring at those thicknesses of the spar which cause the rays to conspire; the black rings at those thicknesses which cause them to quench each other. Turning the analyzer 90° round, we obtain the complementary phenomena. The black cross gives place to a bright one, and every dark ring is supplanted also by a bright one (fig. 46). Here, as elsewhere, the different lengths of the light-waves give rise to iris-colours when white light is employed. [Illustration: Fig. 46.] [Illustration: Fig. 47.] Besides the _regular_ crystals which produce double refraction in no direction, and the _uniaxal_ crystals which produce it in all directions but one, Brewster discovered that in a large class of crystals there are _two_ directions in which double refraction does not take place. These are called _biaxal_ crystals. When plates of these crystals, suitably cut, are placed between the polarizer and analyzer, the axes (A A', fig. 47) are seen surrounded, not by circles, but by curves of another order and of a perfectly definite mathematical character. Each band, as proved experimentally by Herschel, forms a _lemniscata_; but the experimental proof was here, as in numberless other cases, preceded by the deduction which showed that, according to the undulatory theory, the bands must possess this special character. § 10. _Power of the Wave Theory_. I have taken this somewhat wide range over polarization itself, and over the phenomena exhibited by crystals in polarized light, in order to give you some notion of the firmness and completeness of the theory which grasps them all. Starting from the single assumption of transverse undulations, we first of all determine the wave-lengths, and find that on them all the phenomena of colour are dependent. The wavelengths may be determined in many independent ways. Newton virtually determined them when he measured the periods of his Fits: the length of a fit, in fact, is that of a quarter of an undulation. The wave-lengths may be determined by diffraction at the edges of a slit (as in the Appendix to these Lectures); they may be deduced from the interference fringes produced by reflection; from the fringes produced by refraction; also by lines drawn with a diamond upon glass at measured distances asunder. And when the length determined by these independent methods are compared together, the strictest agreement is found to exist between them.