116 DOUBLE EEFEACTION. 



Any ;;7an/' parallel to tlie axis of the crystal necessarily coincides with the 

 optic axis, and every such plane is called ^principal plane ov principal section. 

 This term is one of very convenient use. Any plane at right angles to the optic 

 axis (and therefore to all the principal planes) may be called a rc/yw^a/e plane or 

 section, which term will be also found to have its convenience. In every such 

 conjugate plane the separation of the two rays by double refraction is at its 

 m iximum ; and, what is also important, the extraordinary ray, as well as the 

 ordinary, obeys in this plane the law of Snellius. The indexes of refraction for 

 thi- two are, however, necessarily different ; that of the ordinary ray being 

 1.6543, and that of the extraordinary, i.4833. In directions which do not 

 correspond with either a principal or a conjugate plane, the index of refrac- 

 tion of the ordinary ray will be found to be invariably the same, but that of 

 thp extraordinary ray will gradually increase from the direction perpendicular 

 to thf^ axis to that which coincides with the axis. In this last case the two in- 

 dexes become equal, and double refraction disappears. The index of t\w ex- 

 traordinary ray at any inclination (denoted by «) with the optic axis, may be 

 found from the following formula, in which ?i = 1.6543, %' = 1.4833, and iV^is 

 the index sought : 



iV= ^7?—{n^—n^) siu'^a= 72^7367^ oTsses'shi^'a. 

 We see now why it is that one of the images seen through the crystal is 

 apparently nearer than the other. The general effect of refraction by a single 

 plane surface of a body denser than air, is, as has been already illustrated, to 

 bring the object apparently nearer to the surftice. This effect must depend for 

 its degree upon the refracting power, and this power is a direct function of the 

 index of refraction. The indexes of the two rays are different; and therefore 

 the apparent distances of their images are different likewise. 



One of the most remarkable facts connected with the refraction of the ex- 

 traordinary ray is that, unless the incidence is in the plane of a principal section, 

 or of a conjugate section, the refracted ray is not in the plane of incidence. And 

 if the refracting surface, whether the natural surface of the crystal or one artifi- 

 cially prepared, be not coincident with a principal or a conjugate plane, the 

 extraordinary ray is bent at the surfxce, even when the incidence is perpendicular. 

 In observing through the crystal prepared by truncating its vertices by 

 conjugate planes, in which case we have the advantage of having both refracted 

 rav^ in all positions in the plane of incidence, we shall see that the extraordinary 

 ray is always the most distant from the normal to the surface. But this normal 

 represents the direction of the optic axis. The extraordinary ray, therefore, 

 has the semblance of being repelled from this axis. As there are crystal- in 

 which the" -ipparent effects are reversed, that is, in which the extraordinary ray 

 is nearer to the optic axis than the ordinary, as if it were attracted, these two 

 classes have been distinguished by the terms negativp and positive. In the 

 negative the extraordinary index is less than the ordinary ; in the positive, 

 greater. 



A curious observation in regard to the paths of the two rays through a cry.-tal 

 of Iceland spar, by which an object at a little distance beyond it is seen, origi- 

 nally made by Monge, may be mentioned here. The object 

 being at 0, and the eye being at E, the ordinary image 

 will appear above the extraordinary and nearer, as at 0', 

 0'' being the extraordinary image. The emergent rays 

 are therefore P'E and Q E. But the rays incident on the 

 under side of the crystal from the object must be respec- 

 tively parallel to these. Draw then OP parallel to P'E, 

 pj,^ g and OQ parallel to Q'E, and join PP', QQ'. The en- 



"' tire path of the ordinary ray is then OPP'E, and that of 



the extraordinary is OQQ'E, which lines, when the plane of incidence is a 

 principal plane, necessarily cross each other in the crystal. 



