394 H. S. TJhler — Deviatian Produced by Prisms. 



Consequently 



1 d&„ 



sin 17/ d^ 



is always positive, so that the minimum deviation increases 

 with the obliquity of the ray. This includes the theorem that 

 for symmetric passage in a principal plane the deviation is an 

 absolute minimum. Since D increases with 77/, or rj^ the 

 equation 



sin \E = sin %D sec Vl 



shows that E Q increases with r\ x . [The same conclusion follows 

 at once from the usual relation sin \{E Q + /3) = v sin -J/3, since 

 v increases with r] 1 and \{E^ 4- j3) is acute.] 



The first valid proof of the existence of the absolute min- 

 imum seems to have been given by Sir Joseph Larmor,* 

 although the truth involved was generally recognized for a 

 number of years be fore, f Nevertheless, Larmor's article has 

 been overlooked by later writers because, apparently without 

 exception, they have continued to give a fallacious demonstra- 

 tion based on the incorrect formula 



cos \D = cos -Ji^cos 77 x . 



[See foot-note page 389.] In the year 1909 the present writer 

 published a rigorous proof of the theorem of the absolute 

 minimum.:): He desires to state that, at the time, he too was 

 not aware of the existence of Larmor's article. However, the 

 force of the later paper is not entirely lost because (a) the 

 demonstration is altogether different from Larmor's, (5) its 

 main object was to direct attention to the fallacy of the cosine 

 relation, and (c) it gives a table of minimum deviations for the 

 special case of /3 = -J-7T and n = 1*65. 



Keeping 77/ constant, as X increases from zero, 

 cos (X + j-/3) cos 97/ — cos c decreases until it passes through 

 zero and eventually becomes negative. In equation (8), the 

 second and third radicals become imaginary when 



cos (A 4- ifi) cos 77/<;cos c, 



hence the equation of the small-circle boundary C'AC, of the 

 region of real Z>, may be written 



cos (A + 1/3) cos 77/ = cos c (9) 



In like manner, the equation of the small-circle C'BC is found 

 to be 



cos (A — ^/3) cos 77/ = cos c. (10) 



*J. Larmor: Proc. Camb. Phil. Soc, vol. ix, p. 109, 1896. 



f E. S. Heath : A Treatise on Geometrical Optics, Cambridge, 1887, p. 32. 



\ H. S. Uhler : This Journal, vol. xxvii, p. 223, 1909. 



