18 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT 



(18.) It appears from (7) and (8) that K and V have the same periodic time as V; it 

 follows, therefore, that the colour of the reflected and refracted ray is the same as that of the 



incident. 



(19.) We shall now in the second place consider the case of cylindrical or plane-waves 

 perpendicular to the plane of xz, the vibrations taking place parallel to that plane. 



In this case 3 = 0, and a and y are independent of y : therefore the six equations of con- 

 nection, (Z») and (E) Section 11, reduce to four, viz. 



n = a, y = y, 

 da da dy dy 



d% dss dss dz 



If we attempt to satisfy these equations by three sets of waves, as in the preceding case, 

 we shall immediately arrive at the conclusion, ja^= 1; which shews that these equations cannot 

 be satisfied in this manner. The reason of this is obvious ; for, in the case of vibrations per- 

 pendicular to the plane of incidence, it is clear that no normal waves will be produced by the 

 refraction and reflexion : but in the present case, supposing, as of course we do, that the in- 

 cident vibrations are transversal, we have every reason to suppose that normal vibrations will 

 be generated by the reflexion and refraction. Therefore, since normal waves are in general 

 propao-ated with a different velocity from that of transversal, we shall have to take into account 

 a set of normal waves in the lower medium, and one in the upper also, not coinciding with 

 the transversal waves. 



Let a + Qi+oo, and 7 + 71 + 72; be the whole displacements at any point of the lower 

 medium, and a + a", 7'+ 7"; at any point of the upper; the parts oj, 72, a", y", arising 

 from the normal waves brought into existence by the reflexion and refraction. Then, the four 

 equations of connection become, 



a + Oi + 02 = a' + a" (l), 7 + 71 + 72 = 7'+ 7" (2), 



£(a+a, + a2) = ^(a'+a") (3), _ (^ + ^, + ^,) = _ (^ + y') (4). 



From (1), or (2), by the equations (A), and by Article (3), we have, as in the preceding 

 case, (Article 14), 



P _P> _P2 _P' _ P" f,. 



Also (3) - -- — , and (4) + -— , give us, by the equations (B) and (C), Article (2), 



V f^i v ,^, v., V" 



- + - = — (6), — = — (7). 



rf(l) d(2) 



Also —r— and —7—) give us, by the equations (S) and (C), 



U £ dltr 



Vs + V^s^- V,p, = V's' - V"p" (8), 



Vp + r,p, + v,s, = v'p'+ v"s" (9). 



These equations, namely (5), (6), (7), (8), (9), completely solve the problem as in the pre- 

 ceding case. 



From (5) we get p^ = p, and therefore «, = ± «. As in the previous case we must take 

 the lower sign; for otherwise V and F, would enter into each of the equations, (6), (7), (8), 

 (9), in the form V + Vi, and therefore we might eliminate altogether the quantities V, F,, V, 

 Kj, F", from these equations, and so obtain an equation which would not be generally true, 



