Illustrations of Certain Optical Phenomena. 229 



Equation (1) applied to the mth particle gives 



. f_ . . /ma t\ . , fm — 1 t\ . Q /m + 1 t \*) 



= -A^(2.m27r^ T -- T j-sm2^- r - a- T j~sin 2^^-a-^j 



. -2A/.J an 1tt( x -^ -sm fcr^ -^ cos — j 



. A . . /ma ^ \ . 9 7ra , . q ira 



= — 4A/a sm '2ttI — - 7j j snr -— = -4/xy sin 2 — . 



But by direct differentiation of (2), y = — ( 7^ ) y- 



Hence 



/ 2<7r V ^ • a 7766 1 /* • 2 7m /qn 



(t) =4//si11 T ; P = 7? snr X- ■ • (3) 



Let V denote the velocity of propagation ; then 



X* gA? tto. / sin TO /X \» 



v - T2 - ^ sm ^ _/*a ^ ^ /x ^ . . W 



This is equal to fia 2 when \= & , and diminishes continuously 

 to zero as 7ra/X increases from to 77- . Hence a*/' /uu is the 

 velocity of infinitely long waves, zero is the velocity of waves 

 of length a, and there is continuous diminution of velocity 

 between these limits. 



The expression— 2 sin 2 — obtained in (3) for the square of 



the frequency shows that the frequency vanishes for X = a 

 and \=yo , and attains its maximum value ^//jl/tt when 

 \ = 2a. 



The maximum value of the acceleration-factor — y/y or 

 (27T/T) 2 is 4yLt, and is also attained when \ = 2a. Every 

 frequency less than the maximum corresponds to two different 

 values of A between a and go . Calling them X 1 and \ 2 > we 

 have 



£+^ = 1 (5) 



It thus appears at first sight that two different modes of 

 undulation correspond to each frequency. The following 

 investigation shows that they are merely two different speci- 

 fications of one and the same motion of the particles. 



§ 4. The two undulations 



y l = Asm27rf^-- -^j, y 2 =-Asin27r^+ -^, (6) 

 which have equal amplitudes and opposite directions of pro- 



