406 H. S. Uhler — Deviation Produced by Prisms. 



since, by (15), 



sin ^J) 2 = cot c sin \p. 

 Hence 



cos (D l + fi) — cos (Z> 2 + p) = 2 cot c sin lp X 



jcot c cos /? sin \p — cos |/3 + sin p V 1— cot 2 c sin 2 ^ /?! 

 or 



sin ^(P 1 — -Z> 2 ) = cos c sin J/3 {sin (c — J/3) + 2 cos c sin 3 -J-/3 



— sin pys'm* c sin 2 J/3 -f sin 2 c — sin 2 J/?} 4- 



sin 2 c sin[|(Z> 1 + Z> 2 ) + P] 



Obviously, the quantity within the braces may be rationalized 

 by multiplying by 



sin (c — \ p) 4- 2 cos c sin 3 J/3 



+ sin p V sin 2 "c~sin 2 i/3~+~(sin 2 c — sin 2 ^]. 



Since <? cannot be less than J 6 each term of the last expres- 

 sion is real and positive, so that the fraction which equals 

 sin J(Z\ — Z> 2 ) will not be made indeterminate or ambiguous by 

 multiplying both numerator and denominator by the rationaliz- 

 ing factor. Denoting the reciprocal of this factor by k% for 

 brevity, it will be found that 



sin J (Z> 1 — 7> 2 ) = k cot c sin J/3 sin (c — J/3) X 



esc [J (Di + DJ + P] (cos IP — cot c sin J/3) (49) 



JSTow 



7T - P > D x > 



and 



tt - j8 > A > 0, 



hence 



i(A+A) + 0O 



or 



sin[J(Z> 1 + Z> 2 ) + P] >0, 



for all cases except the trivial, extreme case where /3 = 2c. 

 Hence, in general, 



sin (c - J/3) esc [J(A + A) + P] 



is positive and determinate. Therefore, the deviation at 

 grazing incidence in a principal plane {D x ) is greater than, 

 or equal to, or less than, the deviation at simultaneous grazing 

 incidence and emergence (D t ), according as cos |/3 is greater 

 than, or equal to, or less than, cot c sin J/3, respectively. For 

 all practical purposes, this answers the question concerning the 



