230 Prof. J. D. Everett on Dynamical 



pagation, will specify the same motion of the particles if 

 //i— y 2 vanishes for all values of t when as is a multiple of a, 

 say 7)ia. This gives 



. (ma t\ . /m t\ r . 



" 17r xT" 1 " xj=°- 



sin 



£+■$—• •■•-•(') 



n denoting any integer either positive or negative. This 

 includes equation (5) as a particular case. 

 In like manner the two undulations 



!h 



=Asin2*(£-^), ^ 3 =Asin2^-4), 



which have equal amplitudes and the same direction of propa- 

 gation will be equivalent if 



. (ma t\ . a (ma t\ 



m H\ --*)-***"(*; --TF ' 



(ma ' ma\ ~ 



r-|=» ( 8 ) 



One and the same undulating motion of the particles can 

 accordingly be represented in a unlimited number of different 

 ways by the uniform motion of a harmonic curve in the 

 direction of +x. The amplitude, being the amplitude of the 

 motion of a particle, must be the same for all, but the wave- 

 length may have any value consistent either with equation (7) 

 or equation (8). 



The simplest specification is obtained by employing the 

 greatest admissible wave-length. This wave-length, which 

 we will denote by X 1? will be unique, and will lie between 2a 

 and co , except when it is equal to 2a. In general, the order 

 of magnitude will be \ l9 A 2 > ^3? & c o as defined by 



a 1 a a __ a a _ g a a _ a . 



and so on. " 



\ x is the value obtained by regarding ira/\ as an angle in 

 the first quadrant. 



