RELATING TO THE BREAKING OF RAILWAY BRIDGES. 709 



the value of y' not being wanted beyond the limits and 2c of as . It will appear in the course 

 of the solution that the first of the conditions (2) is satisfied by the complete integral of (l), while 

 the second serves of itself to determine the two arbitrary constants which appear in that integral. 



The equation (1) relates to the problem which has been explained in the introduction. It was 

 obtained by Professor Willis in the following manner. In order to simplify to the very utmost the 

 mathematical calculation of the motion, regard the carriage as a heavy particle, neglect the inertia 

 of the bridge, and suppose the deflection very small. Let m , y be the co-ordinates of the moving 

 body, x' being measured horizontally from the beginning of the bridge, and y vertically downwards. 

 Let M be the mass of the body, V its velocity on entering the bridge, 2c the length of the bridge, 

 g the force of gravity, S the deflection produced by the body placed at rest on the centre of the 

 bridge, R the reaction between the moving body and the bridge. Since the deflection is very 

 small, this reaction may be supposed to act vertically, so that the horizontal velocity of the body 

 will remain constant, and therefore equal to V. The bridge being regarded as an elastic bar or 

 plate, propped at the extremities, and supported by its own stiffness, the depth to which a weight 

 will sink when placed in succession at different points of the bridge will vary as the weight 

 multiplied by {9.cx' - x'^Y, as may be proved by integration, on assuming that the curvature is 

 proportional to the moment of the bending force. Now, since the inertia of the bridge is neglected, 

 the relation between the depth y to which the moving body has sunk at any instant and the 

 reaction R will be the same as if R were a weight resting at a distance x from the extremity of 

 the bridge ; and we shall therefore have 



y' = CRiZcx -*'-)', 

 C being a constant, which may be determined by observing that we must have y = S when R = Mg 

 and x' = c ; whence 



^^Jjg?' 

 We get therefore for the equation of motion of the body 



rfV_ gc'y' 



df ^ S(ficx'-w'y' 



da)' 

 which becomes on observing that — - = V 



dt 



dx' V V-S (2ca/ - x'y ' 

 which is the same as equation (l), a and b being defined by the equations 



a = — , 6= — (3) 



2. To simplify equation (l) put 



y = 2cx, y' = IficVfc-'y, b = -tc'/S, 



which gives 



'^=/3--^^^ (4.) 



dx' ^ {.v-ai'y ^ ' 



It is to be observed that w denotes the ratio of the distance of the body from the luginning of 

 the bridge to the lengtli of tiie bridge ; y denotes a quantity from which the depth of the boily 

 below the horizontal plane in which it was at first moving may be obtained by multiplying by 

 I')C*a/>"' or \C) S \ and (i, on the value of which depends the form of the body's path, is a constant 

 defined by the equation 



Vol.. VIII. Paut V. 4Y 



