388 Prof. G-. G. Stokes on the Foci of Lines 



and by similar triangles 



Yq : PxM^OT : OQ = OB 2 : PM . OQ, 



since 



OB 2 =0T.PM; 



also 



OP=OA 



ultimately. Hence as 



OA:OB:OQ=a:c:l, 



we have ultimately 



where u, p! denote the ordinary and the principal extraordinary indices of 

 refraction, which are the reciprocals of a, c. 



In this particular case the ordinary and extraordinary images cannot 

 be distinguished directly by their polarization, since each consists of rays 

 polarized in all azimuths. But if the objective of the microscope be 

 limited by a narrow aperture, so as to give a predominance to rays hying 

 in one plane, there will in the ordinary image be a predominance of 

 polarization in a plane parallel to the length of the aperture, and in the 

 extraordinary image of polarization in the perpendicular plane. 



Next take the case of a uniaxal crystal cut parallel to the axis. In 

 this case, as regards the extraordinary pencil, the divergence after refrac- 

 tion will be different in the axial and equatorial planes, so that a small 

 pencil diverging from a point at the under surface of the crystal will, after 

 refraction, diverge from two focal lines ; and in order that a line may be 

 seen distinctly, it must lie in one of the planes of symmetry, in which 

 case, at a certain focal adjustment of the microscope, each element of the 

 line would be seen as a short line in the direction of the actual line, and 

 therefore the line as a whole will be seen sharply defined. 



In the equatorial plane the extraordinary ray obeys the ordinary law of 

 refraction ; and as regards divergence, therefore, in this plane, on which 

 depends clear vision of a line parallel to the axis, the apparent index 

 will be the same as the real index, jx . In the axial plane everything will 

 be the same in respect of divergence as in the first case, except that the 

 principal axes of the ellipse which is the section of the extraordinary 

 wave-surface will be interchanged. Accordingly a line in the equatorial 

 plane will be seen distinctly at a focal adjustment which will give an 

 apparent refractive index : fi . 



There will therefore, on the whole, be three focal adjustments of the 

 microscope at which one or other of the systems of cross lines, or both 

 together, will be seen distinctly, namely, one for the extraordinary pencil, 

 which is polarized in the equatorial plane, at which the lines in the axial 

 plane are seen distinctly ; another at which the lines in the equatorial 

 plane are seen distinctly ; and, intermediate between these, a third for the 



[June 21, 



(1) 



