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



plane symmetrically located with respect to the axes ?•' 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 

 2 -|- 2 cos A 



Sin ^ I = n '■. 



Since sin i = n sin r we have 



sin 2 A 



sin^ r 



2 -f 2 cos A 



The angles ?• = «" within the prism then become inde- 

 pendent of ?i, 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'^ -\- X- -{-2 y X cos A = n'^ sin '^ 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'^-j- x"^ = n"^. 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 zh sin A ]/ ?? ^ — ^'^ 



The line y = A: 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 



