396 H. S. Tlhler — Deviation Produced by Prisms. 



adapted to showing the general contour of this surface. The 

 next problem is, therefore, to derive a formula for A in terms 

 of ft, v, and K Division of (6) by (5) gives 



CSC y/ sin y 2 = esc y l sin y 2 ', 

 hence (sin y/ ± sinyjcscy, = (sin y 2 ± sin y/) esc y/, 



whence 



sin |(y 2 ' + 7l ) COS i (y/ - yj = v sin |(y 2 + y/) COS J(y 2 - y/) 

 and cos^(y 2 ' + y 1 ) sin|(y 2 ' -yj = vcosi(y 2 + y/) sin J(y 2 - y/). 



Introducing relations (1) and (2) it results that 



. w , x v cos A/3 sin A 



and . , , N v sin -J/3 cos A 



cos4(y, +y,) = sSni(jg+ft 



v 2 cos 2 -J/3 . _ v 2 sin 2 -J/3 2x , , oM 



so that ^H^-J) SmX + sin^+ffl °° S A = * ^ 



Consequently 



_ v* [sin 2 QE + 0) + sin 2 |g| - sin 2 (E +/3) 

 C0S 2A _ 2 v 2 sin |^sin (£# + j8) 



which is rational, or 



. /sin 2 J (J£ + fi) - v 2 sin 2 £0 / 91 \ 

 tanA = ± cot^ + « V^^L-4 ( 21 ) 



For A = the last equation reduces to the well-known form 

 sin^ +£)=!/ sin |/3 (22) 



Formula (21) may be adapted to logarithmic computation by 

 the introduction of certain auxiliary angles. Assuming a 

 numerical value for 77/, rj x can be found at once from (4), since 

 c or n is supposed given. Then define an angle ty by the rela- 

 tion 



sin \p = n cos ?;/ sec rj l sin -J/3, 

 i.e., sin if/ = v sin -J/3. (23) 



Also let sin $ be determined by means of 



tan^tanccos^ <\/sin {$E+%fi + \p) sin (-Ji£-h-J/3— i/>) , (24) 



which is possible because /3 and 7> are given constants and 

 hence \E is obtainable from (3). Lastly, since 



v 2 — 1 = cot 2 c sec 2 rj 



formula (21) reduces to 



tan A = ± sin <f> cot (\E + £0). (25) 



