RINGS SEEN IN CRYSTALS OF TWO AXES. 223 



whieli are normal to SQQ' and SRIl' will suffer no modification of their 

 molecular movements ; and as the analyzer AA is cro.sscd upon them all, there 

 will be seen in the field of view two dark lines, or bars, intersecting each other 

 at right angles in the point 0. 



Every other converging wave will, however, make a section of which the 

 axes are not in the planes SQQ' or SRR', Let, for example, the plane QIIQ'R' 

 turn around the line LL', and let KK' be at right angles to LL' in the turning 

 plane. Then CK gradually increases in length, while LL' remains constant. 

 When CK reaches the position SC, it becomes the major axis of the section. 

 The original position of the major axis being QC, it appears that, during the 

 turning, it changes its azimuth by the total amount QCK, while the minor axis 

 changes from CR to CL. There is therefore no position in which either of these 

 axes can be parallel to PP or RR'. It follows that every convergent wave not 

 normal to either of the two principal planes SQQ' or SRR' must undergo double 

 refraction ; and, therefore, in passing the analyzer A A, will exhibit chromatic 

 effects.- 



If now the direction of molecular movement in the incident wave be changed 

 to PP' or P''P", there will immediately be double refraction in both the prin- 

 ciprd planes, or the dark bars will disappear from them. But as, in the turning 

 of the plane QRQ'R' round the various diameters LL', the axes ot the section 

 made by the plane turn in azimuth, it is evident that some section can always 

 be f ;und which at some inclination will have one axis parallel to P'P'. 



To take an extreme case, let P'P' bo 46' distant from RC, when it will be 

 eqnidistant between the axes RC and Q'O. The optic axes of the crystal, 

 T/hich are in the plane SRR', will then be in azimuth 45° from the plane of 

 polarization. Now, since the axes of the section formed by the plane QRQ'R', as 

 it turns round LL' or KK', do not reach LL' or KK' until K or L reaches S, if 

 LL' be parallel to P'P', no light will come to the analyzer in the planes SLL' 

 SKK', without being or doubly refracted. The dark brushes will not therefore 

 appear in the central plane coinciding with, or normal to, the direction of inci- 

 dent molecular movement. 



There will be other sections, however, which will have an axis parallel to LL' 

 or P'P', or to KK' normal to LL'. To discover their positions, let us consider 

 for a moment the circular section QltQ'. If at N, in the plane SRR', there be 

 a plane TT', tangent to the surface of elasticity, and if, in this plane, the tangent 

 lines til and t'l'i be drawn — the first tangent to the elliptic section SNR, and the 

 second tangent to circular section QNQ' — then the angles made by the radius 

 CN, of the surface of elasticity, with the latter, will be right angles; but the 

 angle CxsV wiM be greater than a right angle, and the angle CN^j will be less 

 than a right angle. If the plane QNQ' turn about CN — say to the position 

 t"t"i — the angle CN^;" will be greater than a right angle, and the angle CN^", 

 will be less than a right angle. The minor axis of the elliptic section 

 made by the plane m this position will therefore fall toward <''i, from N. So, 

 if the plane turn toward the position t'"t"'i, the minor axis of the section it 

 makes will fall toward t'", from N ; that is, beloAV the circular section in each 

 case. 



Now, LL' being supposed parallel to P'P', and KK' normal to it, let CII CI 

 be the intersections of the planes SLL' and SKK' with the circular section. If 

 the plane QNQ' turn about CII, so that Q' approaches L, the minor axis of the 

 elliptic section it makes will faH to the right of 11. But if another line, to the 

 left of CH, as CO, be made the turnhig line, a position may be found for it in 

 which, ior a given amount of turning, the minor axis of the section, which will 

 be to the right of CO, may fall in the plane SHL. The nearer is to H, the 

 kss the plane will be required to turn to produce this effect. Accordingly there 



