226 H. S. U liter — Deviation of Rays oy Prisms. 



find that D\ is a true minimum, as would be expected from 

 our knowledge of the existence of a minimum when i 1 =0, 

 that is, when the projection of the ray coincides with the ray 

 itself and the latter lies in a principal section. Hence, we 

 can conclude from equation (1), since sines affect both D and 

 D', not only that D has a stationary value simultaneously with 

 D / but further that D attains a minimum value D at the 

 same time that D' acquires its minimum D' . 



Again, since by hypothesis n exceeds unity, 



cos i x 



> n. Hence equation (2) shows that when the deviation of 

 the projection of a ray not in a principal section assumes a 

 minimum value, this value is greater than the minimum of 

 deviation for rays in a principal section. Now equation (1) 

 gives D < D' , [i x ^ 0), and equation (2) implies D' > A , where 

 A represents the minimum deviation for principal sections. 

 Consequently, on the face of it, nothing can be concluded 

 about the relative sizes of D and A - For example, how do 

 we know that the deficit of D with respect to D f may not be 

 numerically greater than the excess of D' over A so as to 

 make the minima of deviation for rays not in principal sec- 

 tions less than the minimum of deviation for rays in principal 

 planes ? In other words, how can we logically deduce the 

 usual and correct theorem that : " The deviation of a ray by a 

 prism is least when the ray passes through the prism in a 

 principal plane and when the angles of incidence and emerg- 

 ence are equal " ?* 



The usual argument is to say that D r exceeds A in value, 

 that the equation " cos JD= cos JD'cos i" shows D to be 

 greater than D', that D passes through a minimum value 

 simultaneously with D r and that therefore a fortiori D > A . 

 This assumes that D has a minimum, whereas both the D and 

 the I)' involved in the cosine formula pass through maxima 

 simultaneously and the general theorem does not follow. 



Consequently we shall now outline the proof of a formula 

 for D as a function of a\ i x and n in order to see explicitly 

 what the properties of D are. This formula may be obtained 

 as follows: First, expand sin %(T)\ + a') of equation (2) in 

 terms of sines and cosines of the angles %D\ and \a' . Next, 

 reduce all cosines to sines, except cos -J-a 7 , and, after proper 

 transposal of terms, square the members of the resulting 

 equation. A quadratic in sin JD ; results and this equation 

 is then solved for sin -JD^, care being taken to retain only the 



*R. S. Heath, Geometrical Optics, p. 32, 1887; or Ozapski, Kayser, 

 Wink elm arm, etc. 



: 



