390 J. H. ROHRS, ESQ. ON THE 



s' 4 



— which would but slightly modify the result). Of course ll is supposed small compared 



with 1. We should then have an additional term /'(«) on the right hand side of the equation 

 (1), which would have one of these three sets of values. 



(1) till s = a — pt and thence = - ng till s = a. 



(2) till s = a - (pt + b) - thence = - pg till s — a - pt and thence = till s = a. 



(3) — fig till s = b - pt and thence = till s = a. 



And on the right hand side of equation (4) we shall have a term ff'(s) ds, or f(s), which 

 will have one of these three sets of values. 



(1) till s = a — pt, and thence - ng \s — (a — pt)\ till s = a. 



(2) till s= \a - (pt + b)}, and thence - /xg {s - a+ (pt + b)\ till s = a - pt, and 

 thence = — ngb till s = a. 



(3) — ngs till s = b — pt, and thence = - fxg (b — pt) till s = a. 

 In accordance with these conditions f(s) will be represented by 



„ 8a 1 f . 2n+ 1 . 2w + 1 la - pt\) . 2ra + l its ) 



(1) -ngz.-~ .-- rs^sin — ~ — 7r-sin tt >sin — .} 



w ^ 5 7T 2 (2w + l) 2 \ 2 2 \ a )) 2 a ) 



„ 8a 1 [ . Zn + 1 (a - pt\ 



(2) -,lgZ. — - Sin 7T *-] 



w 6 «* (2w + l) 2 I 2 V a y 



. 2» + l (a -pt -b\ . 2n + 1 7r«] 



- sin — - — 7T sin } (8), 



2 \ a J 2aj w ' 



„ 8a 1 f. 2« + l /6 -pA) . 2w + 1 7T3 ) 



- m#2. — r — ^sm— — — 7T > sin . I 



^ e *■* (2n + l) 8 \ 2 V. a /J 2 a J 



(3) 



As will immediately appear if we perform the operation - fs sin — ds re- 



a J a 2 a 



membering the values of/* and/'s. 



The first set of values being employed till the load has entirely cleared the piers, the 

 second till the formost part of the load has reached the middle of the platform, and the last 

 from the time it has reached the middle till it has cleared the first half of the platform. 



If we would calculate the exact amount of disturbance due to the passage of a single load 

 at a given velocity, we must not of course suppose the chain to be always symmetrical about 

 the axis, but must include the functions L and AT,, and compute for the two sides of the 



fiV' 



chain at the same time. Denoting by u v — — , &c. the corresponding values on the one 



(t s 



dV 

 side of the chain to u v , &c. on the other we should have to determine 



an 



L a , Mi, L' and Af/, 



' 3 o ds\ ' 



