RELATING TO THE BREAKING OF RAILWAY BRIDGES. 727 



This equation applies to any point of the bridge between A and P. To get the equation 

 which applies to any point between P and B, we should merely have to write 2c - ^ for f, 

 2 c — a; for x. 



If we suppose the fulcrum to be very nearly in the same horizontal plane with the point of 

 contact, the angle through wiiich the travelling weight turns will be - very nearly ; and we shall 

 have, to determine the motion of this weight, 



Mk'V'-~^=Mghl - Rl- (49) 



We have also the equations of condition, 



»} = when x = 0, for any value of ^ from to 2 c ; 

 r] =y when ^ = uV, for any value of x from to 2c ; 



»7 = when 5=Oor=2c;y=0 and — = when x = oA 



ax 



(50) 



Now the general equations (-ts), (or the equation answering to it which applies to the portion 

 PB of the bridge,) and (49), combined witli the equations of condition (50), whether we can 

 manage them or not, are sufficient for the complete determination of the motion, it being under- 



stood that ») and — vary continuously in passing from AP to PB, so that there is no occasion for- 



mally to set down the equations of condition which express this circumstance. Now the form of 

 the equations shews that, being once satisfied, they will continue to be satisfied provided r] <x. y, 

 ^ az X az c, and 



V ISR ISM'V'y V 



d' Mghc^ Mffhc' & " 



These variations give, on eliminating the variation of R, 



y-'^U-hVM-l^ (^') 



Although g is of course practically constant, it has been retained in the variations because it 

 may be conceived to vary, and it is by no means essential to the success of the method that it 

 should be constant. 'J'he variations (51) shew that if we have any two systems in which the 

 ratio of Ml^ to M' P is the same, and we conceive the travelling weights to move over tile two 

 bridges respectively, with velocities ranging from to oo , the trajectories described in tlie one 

 case, and the deflections of the bridge, correspond exactly to the trajectories and deflections in the 

 other case, so that to pass from the one to the other, it will be sufficient to alter all horizontal 

 lines on the same scale as the length of the bridge, and all vertical lines on the same scale as the 

 central statical deflection. The velocity in tiie one system which corresponds to a given velocity in 

 the other is determined by the second of tiie variations (51). 



We may pass at once to the case of a free weight by putting h = k = I, wiiicii gives 

 J/cc.S', r'S-^gd\ M az M' (52) 



The second of these variations shews that corresponding velocities in the two systems are those 

 which give the same value to the constant /i. When S '^ <■ we get P<x^'-c, whicii agrees with 

 Art. 22. 



In consequence of some recent experiments of I'rofessor Willis's, from which it appeared that 

 the deflection j)roduced by a given weight travelling over the trial bar with a given velocity wan 



5 a2 



