Illustrations of Certain Optical Phenomena. 2>>1 



In the exceptional case\ 1 = 2a = \ 2 , successive particles are 

 in opposite phases, the frequency has the maximum value 

 v/yu 7r, and the acceleration-factor has the maximum value 4/x. 

 The waves in this case can be more simply regarded as 

 stationary. 



In the exceptional case Xi = oc , we have \ 2 = a = \ 3 . The 

 particles are in one straight line, there is no acceleration, and 

 the frequency is zero. 



The above investigation establishes the following 



Geometrical Proposition. 



Through any number of points lying on a harmonic curve 

 and having equidistant ordinates, it is always possible to draw 

 an unlimited number of harmonic curves, having the same 

 amplitude but different wave-lengths. Calling the common 

 distance a, and the wave-length of any curve X, the curves 

 can be divided into two sets such that for any two of the same 

 set the difference of the values of a/\ is an integer, and for 

 any two of opposite sets the sum is an integer. At the points 

 in question, curves of the same set slope the same way, curves 

 of opposite sets slope opposite ways, and the tangents of the 

 slopes are inversely as the wave-lengths. If all the curves of 

 one set are displaced in one direction, and those of the other 

 set in the opposite direction, along the axis of abscissas, 

 through distances proportional to their wave-lengths, all the 

 curves will still intersect on the original ordinates. 



Figs. 1 and 2 illustrate the different possible specifications 

 of the same motion of the particles. Small circles are drawn 

 round the particles to render them more conspicuous. All 

 the curves pass through the centres of these circles. In fig. 1 

 the values of a/X for the two curves are \ and f, their sum 

 being 1. In fig. 2 the values of afk for the four curves are 

 f, f , ?-, J ; the difference being 1 for the first and third, and 

 for the second and fourth ; and the sum being 1 for the first 

 and second, 2 for the first and fourth, and 2 for the second 

 and third. In fig. 1 the two curves are supposed to be 

 travelling in opposite directions, the flat curve six times as 

 fast as the steep one. In fig. 2, curves 1 and 3 travel one 

 way, and curves 2 and 4 the opposite way. 



§ 5. In our chain of particles, if one particle be constrained 

 to simple harmonic vibration with frequency less than the 

 maximum or critical frequency VW 7 *"? the permanent state 

 for the chain will be an undulation whose wave-length is 

 given by 



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