OSCILLATIONS OF A SUSPENSION CHAIN. 393 



Whence when t is considerably great 



kt cos\/cg — t 



2a 

 A = — • 



2\/c£-. 



s %a 

 Neglecting periodic terms, in comparison with the large non-periodic term. 



10. If the ends of the chain be fastened and P have for its type the same value 

 as any one of those which we have found for a chain tied at its ends, and if the disturbing 

 force continue for a considerable time and then cease, the disturbance will have but one 

 type, the same as the type of P. 



For if, as before, we consider equations (5), and assume (sin qt) for one of the periodic 

 functions which satisfy these equations, as a term of M , q being the type of P, after a 

 long time M will be nearly expressed by one term C t cos qt. 



If then we assume for it A u A % , A 3 , &c. values Cjcosqt, C.J cos qt, &c, and equate 



coefficients we shall find that 



Cj C2 C3 



— , — , — &c. 



Co Co C 



have the same values as though we had assumed simply, M = C cos qt. 



Now if at the time t the disturbing force ceases, C 1 t, C 2 t, &c. are the coefficients of 

 A it A 2 , &c. 



And these coefficients are in the same ratio as though we had assumed for M , C cos qt, a 



dA 

 vibration of one type only (q) and -— , &c. are the same therefore the disturbance will be the 



ft t 



same. In this case if we put v = the periodic term divides out, consequently there will be nodal 



points at which there will be no vertical motion. The reason why the type [x/zi.Zcg — ) 



has been so prominently brought forward will now appear ; suppose troops marching at the rate 



of 4 miles an hour ; and let the length of a step be one yard ; then the time of a single step is 



3600" l" 



- = - nearly ; 



4 x 1760 2 

 if therefore, sin l\/24.2cg- — t\ = sin. 5 \/cg — t nearly, 



have a period of l" it is very likely that the impulse of the feet may be communicated to the 

 platform at its highest and lowest positions, and therefore it may be represented by a periodic 

 force P which goes through its periods in l". 



\/ cs 



And P and sin 5ir t will have the same period 



2a r 



if sin I — . \/cs (t + l")} m sin — \/cs t ; 

 \2a 5 J \ 2a s 



\/cg = - a ; 



5 



