400 Foci of Lines seen through a Crystalline Plate. [June 21, 



accuracy of the Huyghenian law has also been confirmed by the elaborate 

 observations of M. Abria*. 



In the method of prismatic refraction employed by M. Abria and my- 

 self, the difference between Huyghens's construction and the result of the 

 theory just referred to is greatest about 45° from the axis, while extremely 

 close to the axis, or to the equator, it would hardly be sensible. Mr. 

 Sorby's method is remarkable for this, that it brings out into prominence 

 variations of refraction with change of direction, though the absolute 

 refractions which are involved may be nearly the same. Thus Mr. Sorby 

 informs me that his method shows with perfect distinctness the two widely 

 different foci for a plate of Iceland spar cut perpendicular to the axis, even 

 though the inclination of the rays concerned to the axis is so small that, 

 when polarized light is used, which is extinguished by an analyzer, the 

 field remains dark after the interposition of the crystal. 



In the theory referred to above the extraordinary sheet of the wave- 

 surface is generated by the revolution of the curve which is the envelope 

 of straight lines whose distance, v, from the origin is connected with their 

 inclination, 90° — 0, to the axis by the relation 



v - 2 = cr 2 cos 2 + c~ 2 sin 2 d. 



The radius of curvature of this envelope at the axis is c~ 2 (2ac 2 —a 3 ), 

 and accordingly the apparent index, m, is given by 



V-/« 



2' 



(16) 



which exceeds the apparent index, fx 1 li' 2 , given by the spheroid of 

 Huyghens by 



(17) 



The same formula will apply to a point in the equator if we interchange 

 fj, and ^ . 



Putting these excesses into numbers, according to Eudberg's indices for 

 the line D in Iceland spar, we find 0*0536 and 0-1180, giving for apparent 

 indices 1*3858 instead of 1*3322 for a plate perpendicular to the axis, and 

 1*9685 instead of 1*8505 for a plate parallel to the axis when the micro- 

 scope is focused on a line in the equatorial plane. These differences are 

 much too large to escape detection. 



Postscript. 



Being anxious to- complete the theory of the images free from astig- 

 matism by determining in what cases, if in any, they could be formed by 



* 1 Annales de Chimie,' torn, i, (1874) p. 289. 



