134 Travis. Acad. Sci. of St. Louis. 



angles A, the ordi nates d would terminate in points upon a 

 common plane whose equation is (1). If any two prisms have 

 unequal angles A, the ordinates d would terminate in differ- 

 ent, but parallel planes. This plane is in all cases symmetric- 

 ally placed with respect to the axes i and r'. Its position is 

 shown in Fig. 1. 



There is, however, another condition depending on the index 

 of refraction. We have 



or finallv 



(2) 



By considering the physical conditions, it is easily seen 

 that the quantities r' and % must be symmetrical in equation 

 (2). If the light be made to reverse its direction, it will re- 

 trace its path through the prism. The angles r' and i will 

 then replace each other. The same result will be obtained 

 by solving (2) for sin «*, which gives 



sin I = sin -4 V n'^ — sin'^ r' — cos A sin r'.. (3) 



Equation (2) or (3) may be used in the computation of 

 simultaneous values of ^ and r'. When r' = 90, vve have 



sin i = sin A l/?i ^ — i — cos A. 



The values of ^ that will be physically possible must lie 

 between the value determined in. the last equation, and 90 '\ 

 These values of r' and i dete^'mine a curve on the plane of r', 

 i of Fig. 1. This curve is convex toward the axes r' and 2, 

 and it is symmetrical with respect to them. This curve is a 

 projection in a direction parallel to the axis d, of points on 

 the plane represented by (1), which must represent the rela- 

 tion between d, r' and /. The conditions of symmetry in- 

 volved in equations (1) and (2), both of which must bo 

 satistied, show that the minimum ordinate d must lie in a 



