OSCILLATIONS OF A SUSPENSION CHAIN. 



389 



though it obviously produces scarcely any effect ; we should then write for s 2 the following 

 series : 



d' 4« 2 ITS 1 2 7TS 



1- — (- cos — + - cos &c.) 



3 ir* a 4 a 



v/ioi 

 If we take the first term only, we shall find the types increased in the ratio of — -^~ 



J \/l00 



nearly, and the tension at the end will be increased about ^ b tb . 



If a = 300 feet, h' = 1 foot, it hence appears that for a disturbance (A 3 being the leading 



term, and \Z24.2cg — the type) of ±.95 foot at the lowest point of the chain, the increase of 

 tension would be more than 120g, more than ^ th the equilibrium tension at the lowest point. 



7. Let us now suppose M a = 0, or the increase of tension to be at the points where the 

 chain is supported ; in this case 



s d*V 1 dV .„ 1 

 - 3-j- + - — will = - 

 c as c as c 



u — j-j- + - — will = - (Ai — A., + A 3 — &c.) when s = a. 



And if we take one term only, 



d 2 , 25 ir 



It 1 *'- 08 -** 



A 3 = h cos v eg. — t. 

 2a 



™, h Zah ' 



The maximum value of « = - = 



c 5ttc 



= — nearly. 



30 ' 



2 d? 

 The greatest value of w = - — V = 2h'g. 



C 0LV 



We have obtained then this remarkable result, that a " play " of ^j th the maximum value 

 of v in u at the points of suspension is sufficient to reduce the increase of tension to an inap- 

 preciable quantity. This fact would seem to suggest the employment of buffers or saddles, or 

 some kind of yielding elastic mode of attachment for the ends of the chain. For it is clear 

 that if M be always zero, or at all events small, w will never be considerable. 



8. I shall now proceed to consider the effect of the transit of loads, and the marching of 

 troops along the platform, and also of the variable pressure of gusts of wind moving at a high 

 velocity. I shall suppose the load to be travelling at an uniform rate p, and to be equally 

 distributed throughout the portion of the platform it covers ; so that if b were the length of the 

 part of the platform covered by the load, /u the ratio of the weight of the load to that of the 

 chains and platform immediately above and below it ; the effect of the load would be nearly 

 the same as though the force of gravity were increased in the ratio of 1 + /x : 1 , upon the 

 chains and platform just above and below the load ; (that is, neglecting terms multiplied by 



50—2 



