AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 21 



Assuming then this value of F, we have at the surface of separation, 



j^ = f e''('"-'">^-', 

 2 



2s a 



and therefore r=? !ii ^es("'-MV-i, r' = -, e« "'-'''> ^^. 



' S + ^S 2 « + /US 



Hence if la^e'<''-^'+"'^/-', and ^ a'et'i"''-?'-''*!^^, be the general values of F and F', we have 



s — fis , 2s 

 a, a = o, 



S + ;US S + ftS 



V p 

 -p = - 



V II. 



11 , v p J , / P^ 

 and of course k ii = kv, p = — p = — , and s = ± \/ 1 -. 



Now let us assume 



P / — J f* '"''' / — / — ■ 



I— ^ = (7.\/-l (supposing p > ix), and — = ± — v-l =±\/— i tan w, 



then a=eT2«v:^a, o'= 2 cos to 6^="^-' a, 



and the general values of V^ and V become 



F = -iae!*fr''-i'->^+»-)T2<»)V~ (2), 7*= acostye'*'<'"-.P'-^'T'"'V^**'''* (3). 



Now let us superpose the system (l) (2) (3), taking the lower of the double signs, with 

 another system formed from (l) (2) (3), by putting - A; for k and therefore - k' for k' and 

 taking the upper of the double signs. The result of this superposition will be the following 

 real system, viz. 



V = acosk (yt — pco — sx), 



F,= ocos \k{vt- px + sz) +2a)}, F'= 2acosa)e~*'''* cos {A:'(u'^ - p' '^) + ">}• 



These values of V, V^, and V', since they are real, and satisfy the equations of motion 



and of connection, represent a possible case of motion. The expression for V^ shews that there 



is a reflected ray, of the same intensity as the incident ray, but having its phase altered by the 



quantity 2w. The expression for V' gives 4a^ cos^oje'^*''' for the intensity of the refracted ray, 



. , 27r 



which quantity rapidly diminishes as z increases, since k = —7- , and \' is extremely small. This 



A 



indicates a complete extinction of the refracted ray. If we had taken the upper signs instead of the 



lower, and the lower instead of the upper, in the above process of superposition (as we might have 



done), we should have obtained e^''"', instead of e~^'"", in the expression for the intensity of 



V'. Now this represents an intensity which increases rapidly with z, and therefore a vibratory 



motion which becomes extremely large compared with that which gave rise to it, contrary to the 



principle stated in Article (8). We must therefore take the signs as we have done above. 



The alteration of the phase of the reflected ray is given by the equation 



MO- \/p^- fJ.' 

 tan ft) = — = — . 



This is exactly the first case of total internal reflexion considered in Airy's Tracts, p. 36l. 

 (second edition)*. 



(24.) We shall, in the second place, apply the same method to the case of vibrations parallel 



to the plane of incidence. To do this we have only to put tan w = — = ; , and 



The /i in Airy's Tracta is the same as the - here. 



