RINGS IN CRYSTALS OP TWO AXES. 213 



and before analysis, therefore, the two rays may be i-epresentccl by Roe and 

 lleo> which symbols show that each has been equally modified in its passage 

 through the system, and hence, that they reach IIk:' analyzer without any differ- 

 ence of path. In the foregoing formulae, accordingly, 7(i,=0 for this case, and 

 the chromatic term disappears. 



We also obtain an explanation of the effects produced .by inclining the lamina 

 to the incident light. In general, the increased thickness of the crystal which 

 the rays will have to traverse at an oblique incidence, will have the effect to 

 increase the value of h, and the colors will descend, or take the tints belonging 

 to thicker plates. But the new direction of the rays within th<; crystal may be 

 one in which their difference of velocity is greater or less than that which 

 belongs to the direction of perpendicular incidence. In this case the tints will 

 descend more rapidly by inclining in one direction than they do when the in- 

 clination is opposite ; or they may possibly remain stationary, or rise on one 

 side and descend on the other. We suppose here as before that the analyzer is 

 crossed upon the polarizer 



Crystals of two axes, cut at right angles to either axis, will exhibit elliptical 

 rings, the variations of velocity of the two rays being subject to different laws 

 in the principal plane which contains the two axes, and in two other principal 

 planes co-ordinate to that. When the axes are not largely inclined to each other, 

 a lamina of the crystal taken perpendicularly to the line bisecting the angle 

 between them will exhibit both systems at once. In these crystals neither ray 

 obeys, in general, the law of Snellius. But there are thi-ee planes — those just 

 mentioned — in which one of the rays obeys this law. These three planes are, 

 in the first place, that which passes through both axes ; and, secondly, both the 

 planes normal to the first, which bisect the angles between the axes. The terms 

 " ordinary ray " and " extraordinary ray." in the sense in which those words 

 have been used, in speaking of crystals of one axis, are inapplicable in the 

 present case. 



The following equation, deduced by Fresnel from the general theory of double 

 refraction, expresses the relation between the velocities of two rays travei-sing 

 the crystal in the same direction, but possessing the diflering polarizations 

 produced by its double refraction : 



1 1 /I 1\ . . , , , 



-J^~-.= [^--:)^^^9^^^9'^ [41.] 



In this formula, v and v' are the two variable velocities; a and c are the Snel- 

 Uan velocities (constant) in the two principal planes co-ordinate to that which 

 contains the axes ; and <p and (p' are the angles made by the common direction 

 of the two rays with the axes themselves. 



It may be remarked that the rays whose velocities are here denoted by v and 

 •i/, cannot be the two rays which proceed from one incident ray, since these two 

 rays do not pursue the same course within the crystal. This consideration is 

 not important, when the divergence of the rays produced by double refraction 

 is small, (which is the case with most crystals of two axes, and with all for rays 

 in the vicinity of the axes themselves ;) and therefore we may employ this law 

 for the purpose of determining the forms of the colored rings, in plates cut 

 so as to make it possible to observe both axes at once. Putting vv'=^ac and 



v-\-v'=^2^ ac, suppositions which arc sensibly true near the axes, the formula 

 gives — 



r— .'=-_p^, (-,^-j sm^^smco'-g-^-smsrsms.', (very nearly.) 



As we propose to confine the inquiry to the immediate vicinity of the axes, 

 where f and ^' are small, we may take the angles themselves, or their chords. 



