RELATING TO THE BREAKING OF RAILWAY BRIDGES. 721 



coefficient in (p (.v) is .0048, whicli is very small, and we get T = 1.14. In these calculations the 

 inertia of the bridge has been neglected, but the effect of the inertia would probably be rather to 

 diminish than to increase the greatest value of T. 



22. The inertia of a bridge such as one of those actually in use must be considerable : the bridge 

 and a carriage moving over it form a dynamical system in which the inertia of all the parts ought to 

 be taken into account. Let it be required to construct the same dynamical system on a different 

 scale. For this purpose it will be necessary to attend to the dimensions of the different constants on 

 which the unknown quantities of the problem depend, with respect to each of the independent units 

 involved in the problem. Now if the thickness of the bridge be regarded as very small compared 

 with its length, and the moving body be regarded as a heavy particle, the only constants which enter 

 into the problem are M, the mass of the body, M', the mass of the bridge, 2c, the length of the 

 bridge, iS*, the central statical deflection, V, the horizontal velocity of the body, and g, the force of 

 gravity. The independent units employed in dynamics are three, the unit of length, the unit of 

 time, and the unit of density, or, which is equivalent, and which will be somewhat more convenient 

 in the present case, the unit of length, the unit of time, and the unit of mass. The dimensions of 

 the several constants M, M', &c., with respect to each of these units are given in the following 

 table. 



g. I -2 



Now any result whatsoever concerning the problem will consist of a relation between certain 

 unknown quantities at', x" ... and the six constants just written, a relation which may be expressed by 



f(,x,a!', ... M, M', c, S, V,g) = (40) 



But by the principle of homogeneity and by the preceding table this equation must be of the 

 form 



[{m) (x ) M c eg] 



where (a?'), (,r") ... , denote any quantities made up of the six constants in such a manner as to have 

 with respect to each of the independent units the same dimensions as a)',ai" ... , respectively. Thus, 

 if (40) be the equation which gives the maximum value T of T in terms of the six constants, we 

 shall have but one unknown quantity ,r', where x = T, and we may take for {x), Meg, or else AfV. 

 If (40) be the equation to the trajectory of the body, we shall have two unknown constants, .r', le", 

 where x' is the same as in Art. 1, and x" = y, and we may take (x) = c, {x") = c. The equation 

 (41) shows that in order to keep to the same dynamical system, only on a different scale, we must 

 alter the quantities M, M', &c. in such a manner that 



M' 0= M, Sa:C, P cc Cg, 



and consequently, since ,§■ is not a quantity which we can alter at pleasure in our experiments, V 

 must vary as .y/c. A small system constructed with attention to the above variations forms an 

 exact dynamical model of a larger system with respect to which it may be desired to oi)taiii certain 

 results. It is not even necessary for the truth of this statement that tiie thickness of tlie large 

 bridge be small in comparison with its length, provided that the same proportionate thickness be 

 preserved in the model. 



