in reference to the Phoneidoscope. 83 



the pairs being inclined to one another at angles of 120°, with 

 the condition that at some one point all the vibrations shall be 

 in the same phase. These may be divided into two sets of 

 three waves each ; and the position of the ventral segments of 

 the first set may be represented as in fig. 2. The position of 

 the ventral segments of the second set may be represented by 

 a similar figure, except that we should have to put 2 instead 

 of 3, and 3 instead of 2, in numbering the ventral segments. 

 In superposing the one figure on the other, we must make a 

 ventral segment of the one figure coincide with a ventral seg- 

 ment of the same numeral of the other. Let a one of each 

 figure coincide, then all the ones will coincide, and will indi- 

 cate the points of maximum vibration of the film. On the 

 other numerals the film will not have its maximum vibration, 

 as one set of vibrations will partly destroy the other. 



We can get a simple algebraic expression for the form of the 

 film. 



Divide the waves into two sets as before, and let Z\ be the 

 displacement due to one set, z 2 that due to the other. Then 

 we have 



hr ] z ± = cos 1 27rX~ 1 (vt — x) j- 



+ 2cos|2ttX- 1 ^ + |^ \coa(Trk- l y</$). 



By changing the sign of x and y we turn the whole figure 

 through 180°, and so reverse the motions of the waves; hence 

 we get 



hr %= cos \2ir\- l (yt + x)\ 



+ 2 cos -TbwX- 1 ^*- |) j cos (TrX-tyv/3) ; 



,\ h~ l z = h~ 1 z 1 + h~ 1 z 2 



= cos (2ir\- l vt) |2 cos (2tt\- 1 x) 



+ 4 cos Qtt\- l x) cos (tt\- ! y V 3) j- , 

 h-'Z = 2 cos (2ttV- 1 #) + 4 cos (ttX-'x) cos (ttX-^S). 



The results of this equation are represented in fig. 4. The 

 large dots represent the points at which Z is at its maximum, 

 viz. 6 A. They occur when 



— cos7rA, _1 #= coS7rX _1 y= ±1; 

 that is, when 



x = 2mX, y */ 3 = 2n\, 



and when 



x = (2m + l)\, y v / 3 = (2w + l)X. 



