82 Mr. W. Baily on the Vibrations of a Film 



and either 



or 



cos 



cos 



x- 1 ^ 



*-i 



?-* 



Z 



Z 



l + i 



X-\£ 



2m + f 



2m-i 



2m 



2m + 1 



2m— § 



"W3 



2n 



— 2n 



2n 



-2n 



2n 



|27r\-Y^+ |)}=1j and cos(ttX-V3) = 1 7 



| 2ttX- 1 (^+ |) }.= -1, and cos (ttX" 1 \/3) = -1. 



These conditions are satisfied by the sets of values given in 

 the following table ; I, m, n being any integers : — 



2m + % 



-2n 



Let Z be the amplitude of the vibration at any point, then 

 Z is the maximum value of z with respect to t. Obtaining 

 this we get 



A" 2 Z 2 = sin 2 (wX-*3*) + {cos (ttX"^) + 2 cos (wX-ty/*)}*. 



At the nodes Z = 0, and therefore 



sin(7rX- 1 3^) = 0, 

 and 



cos(ttX- , 3a') + 2cos (ttX- 1 ?/v / 3) = 0. 



These conditions are satisfied when 



\- 1 3x = 2m and X~ 1 y>/3 = 2n± J ? 

 or 



X" 1 3.£ = 2m+1 andX- 1 t z/ v /3 = 2n±f. 



Putting y = in the above equations, we get 



h~ l z = cos \2-jr\- l (vt — a)\ +2cos| 2ir\- J (vt + ^\ \, 



A-2Z 2 = 5 + 4cos(ttX- 1 3^). 



The former of these equations gives a section of the film at 

 time t, through a ventral segment in the direction of any one 

 of the waves ; and the latter gives a similar section of the sur- 

 face which encloses the space within which this film vibrates. 

 By putting # = 0, we get 



h- l z =cos (2wX- 1 «0 {1 + 2 cos (TrX-yv/3)} , 

 A- 2 Z 2 ={cos(7rX- ] 3 < 20 + 2} 2 ; 



and these equations give similar sections to the former ones, 

 but in directions parallel to the fronts of the waves. 



The next case we will examine is that of six waves of the 

 same amplitude meeting each other in pairs, the directions of 



