134 Trans. 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 



sin i = n sin r 



sin r' — n sin i' 



= n sin (vl — r) 

 = n sin A cos r — n cos A sin r 

 = n sin A V 1 — sin 2 r — cos A sin i 

 or finally 



sin r'=sin^4 Vn 2 — sin 2 i — cos A sin i (2) 



By considering the physical conditions, it is easily seen 

 that Hie quantities r' and i 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 i, which gives 



sin£:=sin vl V ' n' 1 — sin 2 r' — cos A sin r' (3) 



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

 simultaneous values of i and v' . When r' = 90, we have 



sin i 



A l/n 2 — 1 — cos A. 



The values of i that will be physically possible must lie 

 between the value determined in the last equation, and 90°. 

 These values of ?•' and i determine a curve on the plane of ?•', 

 i of Fio-. 1. This curve is convex toward the axes r' and i, 

 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 i. The conditions of symmetry in- 

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

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



