Manchester Memoirs, Vol xlii. (1898), No. 3. 3 



where the origin oi x has been taken (temporarily) at the 

 undisturbed position of the Jth particle. Similarly, in the 

 interval between the (j— i)th and jrth particles, we have, 

 with the same origin, 



^ ^ , 4cos/^^-£,_i . , . X 



Hence the tensions of the string on the two sides of the 

 s\h particle are 



kE 



i^s+i-^scoska) . (6), 





and 



'dV\ kE 



<4L=^.(^''^°^'''-^-> • ('>' 



respectively. The equation of motion of this particle is 

 therefore 



or, since It °^ ^^^^*^ 



^,+1 - 2^Xcos ka - ifika sin ka) + ^,_i = o . (9), 

 where 



^ = M/pa . . . (10), 



i.e. fjL denotes the ratio of the mass of one of the attached 

 particles to that of the portion of string constituting one 

 of the intervals. 



The solution of the difference-equation (9) will assume 

 distinct forms according as the coefficient of 2^g does or 

 does not lie between the limits +1. In the former case 

 we put 



cos ka - ^fxka sin ka = cos 6 . , (n), 



and we may without loss of generality suppose that d lies 

 between o and tt. This gives 



i. = Fe"'+Qe-"' . ... (la), 

 or, expressing the time-factor, 



£,=^/(^^'+^^> + ^/<*^'-^«' . . (,3), 

 where A, B are independent of .$•. 



