RAILWAY CONSTANTS. 219 



If we suppose the train to move down an inclined plane whose 

 gradient is h, the effective moving force will then be the excess 

 of the gravitation of the train down the plane above this resist- 

 ing force : this excess will be 



and the moving force which that will impart in the time d T 

 will be 



(MA-R)^(/T. 

 This, by the principle of D'Alembert, must be in equilibrium 

 with the moving force which in the same time shall be received 

 by the train ; and since this moving force, including as before 

 that which is absorbed by the revolution of the wheels, will be 



(M + M')f^V, 

 we shall have 



(MA-R)^cZT = (M + M')(/V; ■ 



and substituting for R its value already found, this will become 



{M (A -/) -aY'}gdi:=: (M + M)dY. 



To integrate this, let 



M(/i-/) ' ''^ V a •'^•*- 



Hence we have 



V M «(A-/)(1 - A-2) ^ rfT = (M + M') dx 

 .. ^^Ma{h-f) dx 



M + M' i' " ^ - 1 - ;j;2' 



which being integrated gives 



VM a{h-f) _ 1 1 +x 

 M + M' ^ ^ - 2 ^ T^r^ + ^' 

 the logarithm being hyperbolic. 



If x' be the value of x, which corresponds to T = 0, the above 

 integral will become 



VMa{h-f) _l {i+x){l-x') 



M + M' ^ ^ - 2 ' (l-a;)(l +0;')' * ' ^^^'' 

 The relation between V and S may be found by eliminating T 

 by V rf T = rf S, by which we obtain 



{M (A -/) - a V^} ^ c/S = (M + M') V(/V 

 • ytZS _ NdY 



* * M + M' ~ M (A -/) - a V^' 



