Reflexion and Refraction. 123 



for determining the polarizing angle s^, when the axis of the 

 crystal lies in the plane of incidence. It is manifest, from the 

 nature of the formula, that this angle is the same, whether the 

 azimuth is or 180; that is, whether the light is incident at 

 the right or left side of the perpendicular to the surface of the 

 crystal. 



This formula might be deduced more briefly by recollecting 

 what we have already proved, that the corresponding masses m^ 

 and m\ are proportional to the ordinates y of the points where 

 the incident ray and the extraordinary refracted ray meet their 

 respective wave surfaces ; whence it follows that these ordinates 

 must be equal at the polarizing angle ; and thus the question is 

 reduced at once to a geometrical problem. For as both rays are 

 in the plane of incidence, the axis of x will be intersected in one 

 and the same point by right lines touching the wave surfaces, 

 or their sections, at the extremities of the ordiuates. Now the 

 sections in the plane of xy are a circle and ellipse with their 

 common centre at the origin, the radius of the circle being unity, 

 and the semiaxes of the ellipse being a and 6, of which b is in- 

 clined at the angle A to the axis of #; and therefore it is re- 

 quired to draw, parallel to the axis of a?, a right line intersecting 

 the circle and ellipse, so that if tangents be applied to them at 

 two points of intersection which lie on the same side of the axis 

 of y, these tangents, when produced, may cut each other on the 

 axis of x. The angle which the tangent to the circle makes with 

 the axis of x is then the polarizing angle ^1 ; and the solution 

 of the problem just stated leads directly and easily to the for- 

 mula (68). From this way of viewing the matter we see the 

 reason why the polarizing angle is the same in the azimuths 

 and 180 ; for if tangents be applied at the two remaining 

 points where the parallel that we have spoken of intersects the 

 circle and ellipse, it is evident that these tangents also will cut 

 each other on the axis of x\ since tangents drawn at the ex- 

 tremities of any chord, either of a circle or an ellipse, intersect 

 the parallel diameter at equal distances from the centre. 



Let the reflecting surface of the crystal be in contact with 



