H. S. U/iler — Deviation Produced by Prisms. 405 



tan" 1 (2 tan <?)>/3>0. 



For a given value of r) 1 the deviation on the left boundary is 

 obviously the same as the deviation on the right boundary. 

 On the other hand, the corresponding abscissae are 



y 1 = sin -1 (cot c sin (3 sec rj x — cos (3), 

 and y 1 = |?r. [See formula (36)] 



At the discrete minima 



secrj 1 = 2 tan c cot (3 

 hence y 1 = \it — (3, 



as was obtained before in the case of equal roots of equation 

 (26). Thus, as expected, the equal roots pertain to the discrete 

 minima, four in all. Also, the azimuth of the incident ray is 

 the complement of the prism angle for the two discrete minima 

 of deviation corresponding to grazing emergence. [At the 

 four minima it was shown earlier that E '=ir —2/3.] 



The value of the deviation at the discrete minima is given 

 by formula (27). When tan/3 = 2 tanc, that is, when the 

 discrete minima coalesce with their respective maxima, the 

 deviation too is the supplement of twice the prism angle. 



The case of two discrete minima and one maximum suggests 

 that it may be possible for the deviation at grazing incidence 

 in a principal plane to equal the deviation at simultaneous 

 grazing incidence and emergence. Moreover, since, for a given 

 altitude n ir the greatest deviations correspond to the small- 

 circle boundaries (U and Y of fig. 1), the problem just suggested 

 is very closely related to the question of the location of the 

 greatest deviation which can be produced by a prism of given 

 angle and relative index of refraction. A very incomplete 

 attempt to investigate the greatest deviation has been made by 

 A. Anderson in a paper * entitled : " On the Maximum Devi- 

 ation of a Ray of Light by a Prism." 



Let D x denote the deviation for grazing incidence in a prin- 

 cipal plane and P 2 the deviation for simultaneous grazing inci- 

 dence and emergence. A fruitful formula involving I) 1 — D 2 

 will now be derived. Equation (17) may be written 



cos (D 1 + (3) = cos /3 — 2 cot c sin |/3 cos i(3. 

 The formula 



cos (Z> 2 + £) = cos (3 cos D 2 — sin (3 sin Z> 2 

 becomes 

 cos (Z> 2 + (3) = cos (3- 2 cot 2 c cos £ sin 2 i/3 



— 2 cot c sin j3 sin %/3 V 1 — cot 2 c sin 2 £/?, 

 *Proc. Camb. Phil. Soc, vol. ix, p. 195, 1898. 



