REFLECTED ON THE SURFACE OF A CRYSTAL. 63 



(sin + 0, cos 0, sin . , . , \ /j 

 2- 2^-: i r.'_/_- - . sin 2 sin- . 1 cos 

 cos sin X sin (p ; / 



sin 



cos<sm(p' sm 



^^_ 



' ^ 



/sin d> <p, cos <p + (p, sin (p - . \ /, 



/= ( -- -t V . r 1~= Jr sin" sin 2 0,1 cos V 

 V cos (f> sin 0, sin ^), r 7 



/sin &-<$>' cos 04-0' sin < -. =-,- - T-^-T \ a, 

 '= ( -L ^s -. XT r t -. -j-jsuro) sur q>' I sin (7 

 V cos <p sin <|) sin <p' r / 



therefore the factor of (11) gives us 



cos 6 cos & sin < + </>' /sin 0-0, cos + 0, sin \ 



- "~ - JT/ jl * IL ~~" * " - H" alii lit bill \u . I 



sin <p \ cos 9 sm (p, sin 0, r '/ 



sin 0sin & sin + 0, /sin rf> d)' cos <b + d>' sin . . - . \ 

 + - - - ' L -'' sm2 ^- sm = 



sin (p, \ cos sin 



tan & / \ 



' "fh fh' ( ^' n ^' ^"^ ^ "^ *?' ^ ^^ *P ^ sin 2 sin 2 0' I 



cot 



_ \ 

 es i n( t ) cos0sin y sin- 0,1 =0 



, /tan ^' (sin 2 A -sin 2 A') cot^rsin^-sin 2 *,^ 

 The part which multiplies f IB sm0 C os0 (- -W^^^ ^ ( f + ^ ~) 



J 



___ 



sin cos sin (0 + 



_ A sin 2 X cos tan 6 

 ~~2 ' sin cos ' 



,. , ir> . . A (sin 2 sin 2 A.) sin 2 rf>. tanc 

 .-. the subtractive part of equation (13) is - . ^- y sin ( ^ /^ sin g' 



Our equation is therefore 



sin (0-0') cos (0 + 0')_ L *fl sin -' cos ^S A (sin 2 - sin 8 0,) sin 2 0, tan f _ n 



~ ~2 sin + sin a" 



O jo 



If we adopt the nota'lon of Mr M'CULLAGH, and put tan e= g sin u cos u, 



where JS 2 == ' 



sm j 



((r b 2 } sin w cos u sin 2 rf) 

 tanf = ^ . 2 , r 



sm 2 0, 



the above equation becomes 



,, sin d>' cos a> + d)' , /, sin <b </> . cos </> + d>, 

 ^^^ sin^ + 00 sin (0 + 0) 



A (a* I-} (sin 2 sin 2 4 J sin 2 * sin CD c"s u 

 2 ' sin (^J + 0,) sin 



This coincides very nearly with M. NEUMANN'S formula. 



