INTENSITY OF EEFLECTED AND REFRACTED LIGHT. 415 



and since 



from the natui'e of the case, it follows that 



M , A/I dR\ M, ^,</T 



— cos © I -^j- + -J— ) + COS rf) -7— 



e "^ \ax dxj e, ^ dx 



■^/^-^71¥=^ ^'> 



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

 in the five equations 



(I-R)sin^ + I, = Tsin^' + T,....(l) 



(I + R)cos0 = Tcos^' (2) 



I.+ T, = (3) 



M . ^(d\ dR\ M, . ^ </T _ 

 7^"^*^U-^j+i:^'^'^'^ = (4) 



and 



M ^ (d\ </R\ M, ^ </T F d\, F, dl, 



— C0Sa)(;7- + -^-) + C0S9,-j-+-;::5— + -^ -5— = O (5) 



e ' \dx dx) e, ^'dx f dy f dy *■ ' 



Now, we have already shewn that 



M M, ^ 

 -+—'=0 



e e, 



and in precisely the same manner it appears that 



F F, ^ 



7-^7=° 



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

 cos ^ (I' - R') - cos (p'T-0 



where I', 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 d> cos d)' a cos <b' sin cb 



A A' A AsirKp ^ 



s\n(p 



Hence equation (4) becomes 



M^ cos^^ ^ sin ^ cos =^(^' ^) F j s'n (p ^ sin (^ ^ ) _ 

 e I A \ sin 0' j / | / A ' A ' ) ~ 



