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



The small dots represent points at Avhich Z = —Sh. Putting 

 this value for Z, the equation becomes 



0=sin 2 (7t\-Vn/3)+ {cos(7r\- 1 y>v/3) + 2cos(7r\- 1 ^)} 2 . 



This is satisfied when 



2 cos (ttX"" 1 ^) = — cos (jr\~ l y s/ 3) = ± 1 ; 



that is, when 



ys/Z=2rik, a?=(2m±§)\, 



and when 



yV3 = (2w + l)\, a=(2m±£)\. 



The dotted lines give the locus of points at which Z = — 2A. 

 Putting this value into the equation, we get 



0= cos (ttX, -1 ^) J cos (7rX _1 «^) + cos (nrX~ l y s/ 3)^ . 



This equation is satisfied when 



and when 



a;±y^/3 = (2n + l)\; 



so that the locus consists of three sets of parallel straight lines. 

 The nodal lines are obtained by putting Z = 0. The equa- 

 tion to them is 



0= cos (27rX'~ 1 ^) + 2 cos (7rX _1 ^) cos (ttX -1 ?/ V '3). 



It is obvious from the loci already obtained, that these lines 

 must be closed curves surrounding the points for which Z = Qh; 

 and that they must approximate to an hexagonal form, the 

 greatest radii being towards the corners, and the least per- 

 pendicular to the sides of the hexagons formed by the locus 

 of Z=-2A. 



Putting y=0, we have 



«=-.ooB-».^=i«-381.X 



7T 2 



Putting #=0 ; we have 



y= __ = . 385 .x. 



These are the values of the greatest and least radii; and 

 therefore the nodal lines are very nearly circles, with radii 

 = -383.X, and centres at the points for which Z=6/i. The 

 nodal lines are represented by the circles in fig. 4. 



We will next consider the case of four waves meeting two 

 and two, the angle between these directions being 2a, the 

 amplitude of all the waves being the same, with the condition 



