INTENSITY OF REFLECTED AND REFRACTED LIGHT. 415 



and since 



from the nature of the case, it follows that 



M ^ {dl dR\ M, dT 



— cos 01-7-+ "1— ) + COS q> -r- 



e ' \dx ax/ e^ 'ax 



¥ d\ ¥,dT, ^ 



+ -?T- + -i-T-=0 (3) 



f dy f dy 



It only remains that we find the values of M, M^, F, F^, and substitute them 

 in the five equations 



(I-R)sin0 + I, = Tsm0' + T,....(l) 



(I + R)cos0 = Tcos0' (2) 



I.+ T. = (3) 



— smrf)(-T--:5— ) +— sin0'-^ = (4) 



e ^ \dx ax ) e, ^ dx ^ 



and 



M td\ ^\ M, ^ ^ F ^, F, ^, _ 



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



Now, we have already shewn that 



M M, ^ 

 -+—=0 



e e, 



4 



and in precisely the same manner it appears that 



F F 



17. By substituting in equation (4) of the last article, we deduce 



cos (I' - R') - cos (^' T' = O 



where F, R', T' are the differential coefficients of I, R, and T. 



But if we differentiate (2), we obtain the same result ; hence equation (4) is 

 a result of (2), and cannot be employed in our calculation. 



Now 



cos (i) cos (h' a cos rf)' sin (b 



A ' A A A sm <p' ^ 



simp 

 •'"" A 



Hence equation (4) becomes 



M (cos^ , P sin^^os^ ^ ) F ( s'n , sin <^ ^ | _ ^ 

 Ti~A~^^~^^~ Asin0' ^P7irA"^~~A"^'f-" 



