454 



Archdeacon Pratt on the Undulation of an 



when the whole train is on the roadway. This, in point of 

 fact, will be the greatest curvature to which, not only the girder, 

 but the roadway itself will be subjected in the several positions 

 of the train. For the roadway will never be able to assume a 

 curvature, as in the diagram, exceeding that of the girder. The 

 instant the girder bore upon the point P, more than half the 

 weight of the girder would be thrown upon the suspending rod 

 QP. But the weight of the train could never communicate 

 through the chain an upward tension in the rod PQ at all 

 capable of sustaining this load. It is clear from this that the 

 weight of the girder will keep down the undulation, and that 

 the undulation of the roadway, in this arrangement of a sus- 

 pended girder, can never exceed the curvature of which the 

 girder itself is capable. The girder and roadway will therefore 

 always remain in contact, and the position of the greatest point 

 of elevation of the undulation shows where the upward pressure 

 is greatest. 



The greatest depression of the roadway and girder will be that 

 corresponding to the deflection of a girder of the length DC. 

 The following Table is constructed as the last was : — 





Distance the train has passed along the roadway. 



40 



feet. 



80 

 feet. 



120 



feet. 



160 200 1 240 



feet.l feet. feet. 



1 1 



280 j 320 360 

 feet.j feet J feet. 



400 



feet. 



Distance of greatest de-1 

 pression from the left- > 

 hand pier, in feet J 



Ratio of this to half-length 1 

 of the roadway J 



Depression of the girder DC. 



33 



0-16 

 0-02 



56 



0-28 

 0-05 



73 



0-36 

 0-09 



86 



0-43 

 0-13 



97 



0-48 

 0-16 



105 



0-52 



0-19 



113 



0-56 

 0-22 



120 127 



0-60 0-63 



0-25 0-28 



1 



0-70 



6. There are two cases which I will now consider when the 

 train is in motion : (1) when the curvature of the roadway is 

 continuous, without any sudden change; (2) when there is a 

 slight break in the curvature. 



(1) When the train is moving with a velocity v, the curved 

 roadway is subject to additional pressure from centrifugal force : 

 the pressure is increased by this effect beyond the simple weight 

 of the train in the ratio of 



9 + 



radius of curvature 



:g (or gravity) 



If h be the depression of the roadway when the train is in motion, 

 k the depression when it is at rest, and 2n the length of DC the 



