RAILWAY CONSTANTS. 205 



this we must obtain the correct solution of the problem of a 

 train of wheeled carriages moving down an inclined plane, sub- 

 ject only to a resistance which is independent of the velocity, 

 that being the condition on which M. de Pambour's investiga- 

 tion proceeds. 



Let M = the gross load in tons. 



g = the velocity produced by gravity in a fall- 

 ing body in one second. 

 / = the ratio of friction to gravity. 

 '.' f g = the velocity destroyed by friction in one 

 second. 

 Let h = the gradient or the ratio of the height of 



the plane to its length. 

 •.• g h = the velocity which would be imparted to 

 a body in one second moving down the 

 plane without the friction. 

 '.' g (h — f) = the velocity which would be imparted to 

 a body descending the plane by the ex- 

 cess of the gravity over friction. 

 Let T = the time in seconds. 



••• Mg{h—f)dT = the moving force which would be impart- 

 ed to the descending load in the time 

 dT. 

 Let V = the velocity of the train when started down 



the plane in feet per second. 

 V = its velocity after T seconds. 

 '•• rf V = the velocity it acquires in dT. 

 Let m = the weight of a pair of wheels and their 



axles. 

 dm = a, particle of this mass. 



z = the distance of that particle from the 



centre of the wheel. 

 r = the semi-diameter of the wheel, 

 ft) = its angular velocity round its centre. 

 •.' Z(o = the linear velocity of dm. 

 z dm = the increment of its velocity in fZT. 

 zdm d m = the increment of its moving force in d T. 



= this increment i-educed to the point of 



^ contact of the wheel with the rail. 



/z^ dw dm , , . . c • r • J 1 

 = the increment or moving force received by 

 *' the entire mass of the wheels and axle 

 in the time T; and this being applied 

 to each pair of wheels in the train. 



