in reference to the Phoneidoscope. 87 



wave-length equals £ of the height of the triangle when no 

 allowance is made for change of phase at the reflections. The 

 form of vibration of the film is shown in fig. 4, which has been 

 already explained. 



We may obtain a similar set of waves in a rhombus having 

 one of its angles = 60°, if we suppose sets of waves to start 

 simultaneously from the four sides, and in each direction from 

 the shorter diagonal. 



If in such a rhombus a set of waves starts from the longer 

 diagonal, we get the three sets ; and if two sets of waves in 

 opposite directions start from the longer diagonal, we get the 

 six sets. 



With a right-angled isosceles triangle we may start a set of 

 waves from the hypotenuse, and so get two opposite sets of 

 waves || , and two opposite sets J_ to the hypotenuse; and 

 we may get four similar sets of waves in another position by 

 starting waves simultaneously from the two sides of the tri- 

 angle. 



With a square we can get four sets of waves meeting two and 

 two, the directions being J_ to each other, by starting waves 

 simultaneously from all the sides. In fig. 7 the continuous 

 lines show the coincidence of the maximum displacement in 

 one direction of two waves, and the clotted lines show a similar 

 coincidence in the other direction. The black spots show the 

 ventral segments which move together, and the small circles 

 those which move in the opposite directions. The wave-length 

 = the shortest distance between the ventral segments x n/2; 

 and with the number of ventral segments shown in the figure, 

 the wave-length, not allowing for change of phase at the reflec- 

 tions, is J of the side of the square. 



A similar arrangement of waves in another position may 

 be obtained from a square by starting two sets of waves in 

 opposite directions from one of the diagonals. 



To obtain four sets of waves meeting each other two and two, 

 the angles between their directions being 2a, take a rectangle 

 having its diagonals inclined at an angle 2a, and start two 

 sets of waves from one of the diagonals ; these will by reflec- 

 tion give two sets of waves with fronts parallel to the other 

 diagonal. 



With any rectangle two sets of waves meeting each other 

 can be obtained by starting a set of waves from one side of 

 the rectangle. 



The case of two sets of waves meeting each other not in the 

 same direction is impossible in a limited film ; and I have not 

 been able to discover any form of film which could maintain 

 three sets of waves not making equal angles with one another, 



