of Reflexion and Refraction. 295 



2 ' 



y _ ^^ ^V~-i _^ g-[nt-k{px-sz)} V^ll. 

 ' 2 



And superposing these two sets of values we find the follow- 

 ing real values of V, V, and V, which satisfy the equations of 

 motion and coimectioUf viz. 



V =«cos{w^— ^ (pa;+S2)}, 



Vy= — Pacos {nt—Jc{px—sz)—^]i 



y =ae~^^^ \cos[nt—k{px-^ (Tz)) —Vcos[nt —h{px -\- <y^—^\. 



Thus we have proved the laws of reflexion and refraction 

 above-stated for vibrations perpendicular to the plane of inci- 

 dence. 



13. We may proceed in exactly the same way for vibrations 

 parallel to the plane of incidence. 



Assuming 



^^-^-^ = P Q ^- (^+') ^^, 

 we find 



w c — i ~~ I ; > 



^ S— j«.s'/XS + S' + >J 



which gives, by a simple reduction, 



, /— T P -\-i>'Sn 



l-Qg-i^--i _ /^ ^ ^^ ^ 



Now, by section viii. (of the Memoir above-quoted), we have 



where v- is what jtx.^ becomes when B and B' are changed into 

 A and A', and therefore 



A' V » / A'B ^ 



Hence 



