H. 8. TJhler — Deviation of Rays by Prisms. 227 



proper sign before the radical. In this manner we find, after 

 substituting in formula (1), that 



[ + V(^ a_ sin2 *i) cos2 i a '~\~ 



[ + ^/{n 2 — sin 2 *J cos 2 ia' — (n* — l)]}sin p.' (3) 



Obviously, this equation could not have been obtained by 

 substituting in " cos -|D — cos iD' cos i'\ and hence, if formula 

 (3) is correct, the cosine equation must be erroneous. 

 Again, formula (3) leads to 



d D o sin 2 i x sin \a c os 2 \a! 



zzz X 



di^ cos %D 



( 1 1 



+ A /(n 2 -sin 2 ^ 1 )cos 2 ^a'-(^ 2 -l) + ^/(n 2 -sin^'Jcos 2 -^' 



(4) 



Since D cannot exceed 7r, and since the first fraction within 

 the braces of equation (4) is larger than the second, it follows 

 that D decreases when i x diminishes in absolute value. There- 

 fore, the minima D of D decrease as i x approaches zero in 

 magnitude until, when ^=0, D =A - Thus we have shown 

 formally that, although D is always less than D' , D is con- 

 stantly greater than A . The general theorem of minima 

 quoted above from Heath is therefore established. 



Equation (3) is interesting because the second radical indi- 

 cates limitations upon a\ i x and n in order that D be real, that 

 is, in order that a minimum of deviation may exist. For 

 illustration, when a' and ware given 



i x <sin- 1 [± v /l-(^ 2 -l)tan 2 ^a'], 



n 2 -2 



which m turn requires cos a' > — . 



n 



Furthermore, combining . with equation (2) the condition 

 that the second radical of formula (3) shall vanish, we find 

 sin J-(D' -f a')~l, so that D\ and a' are then supplementary 

 in value. Of course, the same restricting conditions can be 

 obtained directly from equation (2) by observing that 

 S ini(D'„ + a')<i: y 



Returning from this digression, attention may be called to 

 the fact that when a=0 or when n=l formula (3), which is 

 implicitly formula (1), gives D =0, whereas the relation 

 " cos iD = cos -J-D 7 ,, cos i", combined with the auxiliary equa- 

 tion (2), leads to D =±i 1 for both cases. For a plane-parallel 

 layer of relative index n, or for a prism of finite angle but 

 with the same refractive index as the surrounding medium, 



