Undulation of an unstiffened Roadway in a Suspension Bridge. 445 

 equation, it reduces itself to 



dt dt " dt 



Hence by integration, the function yjr, which, as said before, 

 may include a term which is an arbitrary function of the time, 

 becomes approximately an arbitrary function of coordinates. 

 Hence, while in those instances the motion, so far as it depends 

 on the mutual action of the parts of the fluid, will still be recti- 

 linear where udx + vdy + wdz is strictly an exact differential, it 

 cannot be shown to be such generally, although it may be 

 gathered from that form of the function ^r that the lines of 

 motion will approximately be straight lines. 



The general mathematical theory of vibratory motion in an 

 elastic fluid is reserved for treatment in my next communication. 



Cambridge, April 22, 1862. 



LXI. Calculation of the Undulation of an Unstiffened Roadway 

 in a Suspension Bridge as a heavy Train passes over it ; and 

 Remarks upon the effect of a suspended Iron Girder in deaden* 

 ing the Undulation. By Archdeacon J. H. Pratt, M.A. 



To the Editor's of the Philosophical Magazine and Journal. 



Gentlemen, 



IT is designed to build across the River Hooghly a bridge 

 consisting of five or six spans, of 400 feet each, with sus- 

 pension chains and suspended iron girders, to carry the railway 

 from Calcutta to Delhi across the water. I have been asked by 

 G. Turnbull, Esq., Chief Engineer E. I. Railway, for an opinion 

 regarding the proposed structure upon theoretical grounds. The 

 following calculations and remarks are the result, which I send 

 to your Magazine as they may be useful to some of your readers. 

 The data are as follows : — The span is 400 feet ; the versed 

 sine of the chains at the middle point is 33 feet 4 inches ; the 

 weight of the chains, including suspending rods, 200 tons ; the 

 weight of the girder and roadway 350 tons; the maximum 

 moving load, when occupying the whole span, 400 tons ; the 

 depression of the girder at its middle point, when supported at 

 its two ends, from its own weight alone, 0*7 foot, from the 

 weight of the train 0*9 foot in addition, making 1*6 foot in all. 



Calculation of the Undulation of an Unstiffened Roadway. 



2. The train is supposed to be so long that both ends cannot 

 be on the same span at the same instant. The weight of the 

 train I suppose to be thrown into the chains, and the girder to 



