Reflexion and Refraction of Light. 421 



the phases of all the three waves must agree at the interface. 

 Thus if, to express the disturbance in the lower medium, we 

 take 



tu= sin (oat -+l/c+m$) } (x negative) . . . (21), 

 where 



l,= cosi,a)/pi, m,= sin t y oo/'/Sj . . (22), 



i t denoting the angle of refraction ; we must have, for the 

 disturbance in the upper medium, 



w —f sin (cot + lx + my) + g sin (cot — Ix + my), (x pos.) (23) , 



where 



1= cosi a>//3, m = sin i co /j3 .... (24). 



The agreement of phases all along the interface, that is for all 

 values of y, requires 



m = ??? v ; 



and therefore, by (22) and (24), 



sin i //3 = sin i / 1$, (25), 



which proves the " law of refraction." It, with (22) and (24) , 

 gives 



l = m cot i; l J = mcoti l .... (%6). 



The other interfacial conditions are simply w continuous; and 

 [§ 9, (6)] T, continuous ; which give 



f+g=l; and B;(f-g) = B,l, . . . (27); 

 whence 



1 B[+B/ /# 1 Bl-BJ, , R . 



J-* Bl > 9 ~ 2 Bl ' ' ' K }i 

 g _ Bl-B,l, 



f~Bl+B,l, ^ 



In the case of equal rigidities, or B = B n this becomes 



g _ I— I, _ cot t'—cotty _ _ sin (t-.—ty) .^ 



f l + l, cot i + cot ij ~~ sin (2 + i y ) ^ ' 



which is Fresnel's " sine-law." 



13. In the more difficult case of vibrations in the plane of 

 incidence, we have two displacement-components, u, v, to 

 consider, instead of only the one, w : and two surface-pull 

 components, P and U, instead of the one, T : and our inter- 

 facial conditions now are u, v, P, U all continuous. 



We have now condensational-rarefactional waves, besides 

 distortional waves, to deal with: and it is therefore convenient 

 to divide the solution according to (19) ; and, as a two- 



