Niplier — Law of Minimum. Deviation of Light by Prism. 135 



plane symmetrically located with respect to the axes r' and i. 

 This plane is determined by the condition r ' = i, which makes 

 the entering and emergent rays symmetrical with respect to 

 the bounding surfaces of the refracting angle A. 

 Putting this condition in (2) it reduces to 



„ . sin 2 A 



" 3. 



2 + 2 cos A 

 Since sin i = n sin r we have 



. „ sin 2 A 



n 



2 + 2 cos A 



The angles r = i' within the prism then become inde- 

 pendent of n, their value being dependent on A only. 



If the sines are regarded as the variables, equations (2) 

 and (3) represent an ellipse. Calling sin r' = y and 

 sin i = x, those equations become, 



y 2 -\- x 2 -\- 2 y x cos A = n 2 sin 2 A. 



When the angle of the prism becomes zero the ellipse be- 

 comes the diagonal of a square whose sides are 2 n, the last 

 equation being y = — x. When A = 90° the ellipse becomes 

 a circle whose equation is y 2 + x 2 = n 2 . For intermediate 

 values of A the ellipse has the square whose side is 2 n as an 

 envelope, the major axis lying in the line whose equation is 

 y= — x. The minor axis always lies in the line whose 

 equation is y = x, which involves the condition i=r'. In 

 its general form (2) becomes 



y = — x cos A ± sin A y n 



2__<v.2 



The line y = ± n sin A laid off on the axis y and the line 

 whose equation is 



y = — x cos A 



are conjugate diameters of the ellipse. Those portions of 

 the ellipse corresponding to values of x or y greater than 



