INTENSITY OF RELECTED AND REFRACTED LIGHT. 4I3 



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



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

 And, from the natiu-e of the functions, the last two lines of the above equation 

 cannot, when x=0, give any part of the quantity -c^ (« + «,), 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=-2D)I, + c^-2D, T, (1) 



, M . ^ (dl f/RN M, . ^di: 



and — smd) (:t-— -^1 +— smd)'-,— .... ('2) 



e ^ \dx dxje^ ^ dw ^ ' 



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



and 



a + a,= I — R sin + T sin 0' 

 d^ (a + a,) 



dt^ 



-=:-c2(a + o,) 



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 ^rate down the equation for )S from the 

 equation for «, I shaU therefore briefly deduce it. 



J2^?- + 2-I(5.r'2sin-0 + oy2cos2<^)| cos^ 



Now 



I -21 



sin^ -JT- + -;j- smKa' -2Rsin^ _^ +_ gmKa-" 

 Z e dx 2 e dx ) 



= -2 cos^(I + R)(cos''0-2sin='^) + — ^^ + -^) cos (cos =<^ - 2 sin «<^) 

 Again, if we denote 2 ^ Sa' 5y «-"'^'^sin/5y by F, we obtain 



2 — dx di/ Sa= ~ sin 20 cos ^ (I + R) 



3M 



■\ sin ' 



e 



^ ^ \dx dx) f dy 



by adding this term to the former we obtain 



-9 cos(/)(I + R) + — COS0 {— + -—)+---' 

 i e ^ \dx dx) fdy 



VOL. XIV. PART 11. 



3n 



