80 Mr. W. Bally on the Vibrations of a Film 



ously after crossing distances = an. Hence the difference of 

 phase of the whole vibration at c and that at a 



an a it cos fZ 



Z7T = 



\ sin y sm a 



Draw (fig. 2) a series of straight lines || ab at distances 



jr— : — apart. Along each of these lines (d—e) is constant; 



and its value changes by 2tt in passing from any line to the 



next. Draw another series II be at distances ^—. — apart. 



2 sm a r 



These lines will have similar properties with respect to (e—f). 



Straight lines drawn through the intersections of these two 



series will form a series I] c a, at distances ^—- — n apart, and 



; 2 sm p r ' 



having similar properties with respect to (f—d). At all the 

 intersections (d—e), (e—f)) (f—d) have the same values 

 respectively; and therefore at all the intersections the ampli- 

 tude of the vibration of the film is the same. 



We can find a line such that (d — e) is a multiple of 2ir, and 

 another line such that (e—f) is a multiple of 2tt. At their 

 intersection (f—d) will be also a multiple of 2ir. Such a point 

 is a ventral segment. Suppose (fig. 2) the lines to be moved 

 until one of the intersections lies on a ventral segment ; then 

 all the intersections are ventral segments. 



Now in a A with ventral segments at its //_, letp, q, r be the 

 distances of a point from the sides of the A. Let p f , q f , r' be 

 the differences between the values of (e—f), (f—d), (d — e) 

 at the point and on the respective sides. Then 



p f : 2nr=p : ^—. — ; 



* r 2 sm a 



hence 



r>'\ q r \ r r X 



1 47rsina ? a 47rsin/3 ? 47rsiny 



The displacement of the film at the point at any time may 

 be expressed as 



h cos (d + t) + k cos (e + r) + 1 cos (/+ r), 



where t is proportional to the time after the given moment. 

 The maximum displacement will be 



s/ {h 2 + P + P + 2kl cos/ + 2lh cos q r + 21ik cos /} ; 



that is, 



