Dr. Feussner gives a mathematical analysis of the paths taken by the two polarized rays within the Nicol prism, and finds that the emergent extraordinary ray can include an angular field of 29 deg., but that this extreme value holds good only for rays incident upon that portion of the end surface which is near to the obtuse corner, and that from thence it gradually decreases until the field includes an angle of only about half the previous amount. He finds, moreover, that, although of course the ray emerges parallel to its direction of incidence, yet that the zone of polarized light is shifted to one side of the central line. Also that the great length of the Nicol--3.28 times its breadth--is not only an inconvenience, but owing to the large pieces of spar thus required for its construction, prisms of any but small size become very expensive. To this it may be added that there is a considerable loss of light by reflection from the first surface, owing to its inclined position in regard to the long axis of the prism. [Illustration: Fig. 4., Fig. 5., and Fig. 6.] It is with the view of obviating these defects that the modifications represented in Figs. 2 to 6 have been devised. 2. _The Shortened Nicol Prism_.--This arrangement of the Nicol prism is constructed by Dr. Steeg and Reuter of Homburg v.d.H. For the sake of facility of manufacture, the end surfaces are cleavage planes, and the oblique cut, instead of being perpendicular, makes with these an angle of about 84 deg.. By this alteration the prism becomes shorter, and is now only 2.83 times its breadth; but if Canada balsam is still used as the cement, the field will occupy a very unsymmetrical position in regard to the long axis. If balsam of copaiba is made use of, the index of refraction of which is 1.50, a symmetrical field of about 24 deg. will be obtained. A prism of this kind has also been designed by Prof. B. Hasert of Eisenach (_Pogg. Ann._, cxiii., 189), but its performance appears to be inferior to the above. 3. _The Nicol Prism with Perpendicular Ends._--The terminal surfaces in this prism are perpendicular to the long axis, and the sectional cut makes with them an angle of about 75 deg.. The length of the prism is 3.75 times its breadth, and if the cement has an index of refraction of 1.525, the field is symmetrically disposed, and includes an angle of 27 deg.. Prisms of this kind have been manufactured by Dr. Steeg, Mr. C.D. Ahrens, and others.