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XIII. The Vibrations of a Film in reference to the Phoneido- 

 scope. By Walter Baily, M.A* 



TPlate I.] 



THE object of this paper is to consider the superposition 

 of several systems of waves in a plane film, when the 

 vibrations are perpendicular to the film, and the wave-lengths 

 of all the systems are the same, with the view of discovering 

 (1) what combinations of systems give simple results in an 

 infinite film, (2) which of these combinations can exist in a 

 finite film, and (3) which of the latter can account for appear- 

 ances presented in the phoneidoscope. 



Let us examine the case of a plane uniform film of infinite 

 extent traversed by three systems of waves with straight fronts, 

 the vibrations being perpendicular to the film, and the wave- 

 length of each system being the same. 



Draw (fig. 1) Ac, Be, Cc meeting in c; and let ZBcC = 2a; 

 ZCcA=2/3; ZAcB = 2 7 . Take cB = cC = X. Draw 

 QcRJ_Ac; RBPJ_Bc; PCQj_Cc. Join P c, and draw 

 Q b and R b, bisecting IJ_ P Q R and Q R P. Draw ca\\Qb, 

 cmJ_R6, anJ_B.c. It may be easily shown that Lcab — a y 

 Z.abc=fSj Z.bca = y; 



X X cos B X 



cm—^-. , an= ^—. : — , c a = 



2 sin y' 2 sin y sin a ? 2 sin y sin a 



Let X be the wave-length ; Ac, Be, Cc the directions of the 

 three systems of waves ; h, k, I their amplitudes ; and d, e,f 

 their phases at some point. The values of (c— f), (f—d), 

 (d — e) are independent of time, and depend only on the posi- 

 tion of the point chosen. QR, RP, PQ are wave-fronts; for 

 they are J_ to the directions of the waves. 



Consider the line Pc. Every point on this line is equidis- 

 tant from the wave-fronts RP and PQ, so that (e—f) is con- 

 stant along Pc. Similarly (d — e) is constant along ab, and 

 (f—d) is constant along R6; and as we may shift the point c, 

 it follows that along any lines || P6, Q6, and R6 we have 

 c— /), (d — e), and (f—d) constant respectively. Hence 

 f—d) is constant along ca. 



The value of e at c differs by 2-7T from its value along PQ, 

 since Bc = X. Hence the value of (d — e) at c differs by 27r 

 from its value along ab. Similarly the values of (f—d) at b, 

 and of (e—f) at a differ by 2tt from the corresponding values 

 along ca and be. 



Now the wave -fronts Re, cB, BR will reach a simultane- 



* Communicated by the Physical Society, having been read at the 

 meeting on June 12. 



