^74 GRAY, ON A DOUBLY-KEFLECTING PRISM. 



at which it is reflected to the opposite side. It here under- 

 goes a second reflexion, and falling at right angles on the 

 side opposite that of incidence, it finally emerges, forming 

 with its original direction a given angle. Determine the 

 form of a prism which shall best fulfil the prescribed con- 

 ditions. 



Let ABCD be a section of the prism, and RSTV the 

 axis of the pencil, entering the prism at the side AD at 

 right angles, and emerging, also at right angles, at the side 

 BC, after two reflexions, at the points S and T. Let the 

 angle TVE,, made by the emergent with the incident pencil, 

 Avhich we call the deviation, = d. 



Produce CB, DA, to meet in E. 



ED, EC, being respectively at right angles to RV, TV, 

 the angle contained by the former two lines is equal to that 

 contained by the latter ; that is, z CED= z TVR = f/. 



Denote the angle of incidence at S by i,, and that at T by 

 i.^ ; hence, the angles of incidence and reflexion being equal, 



zRST=2i„and zSTV=2i2. 



Now, z RST= z ST V + z TVS ; 



or, 2ii = 2i2 + d ; whence i2=ii — g^. 



The angle of incidence at S is complementary to the angle 

 ASR, and so also is the angle SAD. Hence the angle of 

 the prism at A=ii ; and, for a like reason, the angle at 

 C = ?'.,=«i — -.V^. 



Also the 'angle at B = 180°- z EBA= 180°— z BAD + 

 zBEA=180°-i, + ^; and the angle at D = 180°- zDCE- 

 zCED = 180°—/2-^=180°-«,-i^. 



The sura of these four angles is 360% as it ought to be. 



Were the sides AB and DC produced^ they would meet at 

 an angle which would obviously be supplementary to the 

 sum of the angles at A and D, and the value of which 

 would therefore be \d. This property may be enunciated as 

 follows : 



The angle contained by the transmitting sides is double of 

 that contained by the reflecting sides. 



We have no immediate concern with this property, but 

 possibly it may aid in the practical construction of the prism. 



The four angles of the prism are thus determined, in terms 

 of i, and d. Of these d is given, and ij remains disposable. 

 Inquire, therefore, whether there is reason for preferring 

 for ij any particular value^ to the exclusion of others. It is 

 desirable that the prism should not be larger than it needs 

 be^ since, the loss of light by absorption being proportional to 



