REFLECTED AT THE SURFACE OF A CRYSTAL. 53 



which, since x=Q, have all the same type or the same circular function. 



Our equations of motion are also 

 (I-R) sin 9 + R, = T sin 9,cos 0-T* sin 9' sin & + T, (I) 



(I + R)cos$ = Tcos0 ; cos0-T'cos0'sin0' ' . . ' I ' . . (II) 



r-R'=Tsin0 + T'cos0' . . . . . (III) 



U/dl dR\ , M, dT . M' dT . ,, . l 



M/rfl dR\ , M, dT r, M' dT .,. # F dR, F, dT 



[ - + -3 I cos 9 + -r- cos d), cos -- r -T cos 9' sin 6 + - -T- + -; -j = (V) 

 e \dx^ dx) e, dx e' dx f dy f dy 



M/dl' dR'\ M, dT , a M' dT' fi , , vn 



-(- --- ) + ' -r- sm0 + -j cos0' = . . . . . . . (VI) 



e \dx dx ) e, dx e! dx 



Also equation (IV.) consists of two parts, one depending on cosines, and the 

 other on sines of the same arc. These two must, therefore, separately equal 0, 

 so that equation (IV) is divided into the two following equations : 



M/rfl dR\ M, dT . a M' dT' . ,, . & 



- ( -- ) sin (b + ' - sin d>, cos 6 - r - sin <*' sin tf = .... (IV) 

 e \dx dx) r e, dx e' dx 



and (c-2D)R, + (c-2D / )T,=0 ... ... (VII) 



We can obtain one further reduction of the equations in the following man- 

 ner: 



M, MM' ,M 



Let -2 = - p p'; 



e, ^ e' e' e 



_2 V COS (f) _2 IT COS <f>,_ 2lT COS <p, sin (p -_2 7T COS ^)' _ 2 7T COS <p' sin (f) 

 -- ^r -- , &. -- ^ -- - ^ - ; - -T - 5 -- ^-; --- ^ - : - -7-. - j 



X X, \sm(f>, \' \sin0' 



if we substitute these values in (IV) and divide by sm(fy+ct), retaining the 

 notation of (3'), we obtain 



, ,s , sin (p. cos 6 cos d>. sin d> sin <6' sin & cos d)' sin (b _ 



(a + 6) sin cos (b p c -^ - ; -- fi - L + n'd J- - -. -!- - -*-=0, 



sin 9, sin (f)' 



or (a + 6) cos 9 p c cos 9, cos 6 + p' c' cos 9' sin 0'= 0. 



Also equation (II) gives, by dividing out cos (fy + c t), 

 (a + 6) cos <p = c cos (p, cos d d cos <p' sin 0'. 



Hence, by combining these two, we have 



(1 - p) c cos <p t cos 6 -(!-/)</ cos 9' sin # =0 , . . (VIII) 



F M 



Equation (V) is also reduced by writing ^ = -- (to which it is equal), 



/ e 



F, M 2-7T . ... 



j = + * and /=-^~ sm ^ : it gives 



cos 2 (b, cos 6 sin . cos 2 9' sin 0' sin 9 , . . _, . , n 7 ., 



(rt-6) cos 2 c - r< . , - J-+p'd - y . . , - r Asin9 + sCsm9=0 (V) 



' 



. 

 sm , sin 



VOL. XV. PART I. 



