50 Geometrical Propositions applied 



right line Od will pass through N\ and the circle of contact, 

 described on the diameter di in a plane perpendicular to the 

 right line OdN, will be a section of the refracted cone. Now it 

 will be recollected* that, in general, the vibrations of a ray OT, 

 which goes to any point T of the wave surface, are parallel to 

 the line which joins the point T with the foot of the perpendi- 

 cular let fall from on the tangent plane at T. In the present 

 case, the perpendicular is the same for all the rays of the re- 

 fracted cone, and its extremity coincides with the point d : so 

 that the line dT, drawn from d to any point T of the circle of 

 contact, is parallel to the vibrations of the ray OT which passes 

 through T. Conceive, therefore, a plane perpendicular to ON 

 at the nodal point N. This plane will cut the refracted cone in 

 a circle whose circumference will pass through N', and a line 

 NT', drawn from the node to any other point T' of the circum- 

 ference, will be the direction of the vibrations in a ray OT' which 

 crosses the circle at this point. The plane of polarization is 

 perpendicular to the direction of the vibrations. 



57. The transverse section of the emergent cylinder is always 

 a very small ellipse, affording a hollow pencil of parallel rays in 

 complete accordance (55). If the crystal be thin, this ellipse 

 will be of evanescent magnitude. Hence the line 08 will be 

 the direction of a line drawn from the eye to the centre of the 

 rings commonly observed (50) with polarized light ; or it will 

 be what is called the apparent direction of one of the optic axes. 

 The diameter passing through N will be the direction of the 

 optic axis within the crystal. There are therefore two optic 

 axes, parallel to the two nodal diameters (19) of the surface of 

 refraction. 



As ON is equal to the mean semiaxis of the generating 

 ellipsoid, or to the mean index of refraction, when OS is unity, 

 it follows that the apparent direction of an optic axis is the 

 direction of an incident ray, which, if refracted in the ordinary 

 way, with an index equal to the mean index of refraction, would 

 pass along a nodal diameter of the surface of refraction. 



* Transactions of the Royal Irish Academy, VOL. xvi., p. 76 (supra, p. 12). 



