392 J. H. ROHRS, ESQ. ON THE 



d?V 7T 3ir , 



Now — - = — A l + — A 2 + &c. at the centre, 

 ds- 2a 2a 



Z ( 3 ~ a * _ 

 ~ 2 l^c " 



32a 2 . \ 

 + &c.) 



ix /32a*\ , , ,, . a a 2 



= . 1 , a result remarkably close to the true . — . 



2 \31 c / ' 2 c 



81tt ;< c 

 If we take only the first term (ir 3 = 31 nearly ;) 

 d?V 



d^ 



In practice p 2 always will be small compared with eg, for taking a at only 75 feet, c = 300, 

 and eg exceeds 9000; while p 2 would rarely exeeed 400, and for persons walking would not 

 probably be more than 50 (or about 4 miles an hour). In the same manner we may shew that 

 if the chain be tied at its ends, the mere walking, without stamping, across the platform of a 

 body of troops would produce an inappreciable vibration. 



In the case of a gust of wind, however, sweeping across the platform, and exerting a 



variable pressure upon it, rapidly changing from point to point ; p 2 might = eg, or if the 



4a 2 

 chain were fastened at its ends, p 2 , or Qh 2 , or 25p 2 might = — - x the square of some one of 



the types of A x &c. ; of course then terms not periodic would be introduced, and with this 

 condition, a succession of gusts having the same velocity might do much mischief. 



9. To estimate the effect of marching in time, we observe that if when the platform 

 is either in its highest or lowest position, it receives an impulse from the foot, the extent 

 of vibration will be increased each time. Such an effect may be expressed by a periodic 

 function, which goes through all its values in the time occupied by a single step, or perhaps 

 by two steps ; certainly by two steps if the platform be in its highest or lowest position 

 at the epoch of two consecutive steps. Let .'. P be such a periodic function, then formulae 

 (8) must be multiplied by (l + EP) where E is a given constant. 



To take a simple example, let us suppose the bridge covered with people, and that 

 half of them, half way from either end to the middle are stamping in time, it is required 

 to estimate the effect. 



Then /(«) = till « = - , and = - fP (s - -) till s = a. 



Where fP is the periodic force, P the periodic part of it, and f a coefficient. 



Let us suppose the chain to have play at its ends so that M can be kept = 0, 



... _ 8a 1 2w+l . 2ra + l \ . 2n + lTrs 



f («) = - z, . -7- -; . irfP . sin 7r - sin ir . sin — . 



J w (2n + iy J \ 2 4 / 2 a 



Let now P = sin ( \/cg. — t ) . 

 V 2a ) 



Then writing k for — - [ sin sin — ) ./ 



it \ 2 4/ 



d 2 A. 7T 2 . , . / 7T 



-^ = ~ c Z^i A i~ k sin veg' — *, 

 dt 4a" la 



&c. = &c. 



