INTENSITY OF EELECTED AND REFRACTED LIGHT. 4X3 



Now the sum of the two quantities which constitute the first hne is 



- c^ (I - R sin cp + T sin (p') - c^ (I, + T,) 



And, from the nature of the functions, the last two hues of the above equation 

 cannot, when x^O, give any part of the quantity -c^(a + a), for the one involves 

 sines of the same quantities whose cosines constitute the other ; hence we must 

 have separately equal to the two following expressions, viz. 



(c^-2J))I, + c'-2D,T (1) 



M . ^ /dl dR\ M^ . ^ dT 

 ^^d y^^^^U-^j+i;^^^^'^ • • . • (2) 



The former equation gives l^ + T,=0, for D, and D are the same thing. Hence 



a + «^= I — R sin ^ + T sin (^' 



and — j^ = -c^ia + a) 



as it ought to be. 



On the second of the above equations we shall make some remarks after we 

 have deduced the equations for the motion parallel to the surface. 



15. It would be rather difficult to write down the equation for /3 from the 

 equation for a, I shall therefore briefly deduce it. 



Now 2 (<pr + ^8i/A 8l3~ 



X i — 2Isitf — ^r— 4--T-smKa:' — 2Rsin^— -— +- -3— smK.r" 

 I Z e aw ■ 2 e ax ) 



= — ^ cos « 



i0(I + R)(cos2^-2sin20) + — (^^ + j-^ cos (cos 2^ - 2 sin 2<^) 

 Again, if we denote 2— d^d^e—'^^^smfS^ by F, we obtain 



~ — X Oy a= SH 



r ^ 2 



3M . , , ^ /dl dR\ F dJ, 



2 ^-— d X 8 1/ d a= — —- sin'^cp cos <p (I + 'R) 



+ sm 20 cos O) ( T~ + ^— ) + "> • 'J~ 



e '. ' \ax dx) j dy 



by adding this term to the former we obtain 



«' ^.T 15N M ^ (dl dR\ Ydl, 



VOL. XIV. PART II. 3 N 



