206 EIGHTH REPORT 1838. 



S ( / ) — the increment of moving foi'ce received 



^*^ *" ' by the mass of all the wheels and axles. 



But since r dco =. clY, if day be eliminated we have 



By the principle of D'Alembert the moving forces which act 

 upon the train must be in equilibrium with the moving forces 

 received by it. Therefore the forces M^ (/'—/) (^T must fulfil 

 the conditions of equilibrium with M <i V, the progressive mo- 



mentum of the whole train, and f/ V S I / - — § — ) the re- 

 volving momentum of all the wheels and axles. 

 Hence we have 



M^ {h-f) rfT- j^M + S (y£!^)J. dY = 0, 

 which being integrated gives 



M^o. (h -/) T = {m + 2 (y ^)} (V - V) . . . (4.) 



The quantity / z'^ d m being the moment of inertia of the 



wheels round their centres is equal to m k"-, where k is the di- 

 stance of the principal centre of gyration from the centre of 

 gravity; and this quantity m k" may be determined by observing 

 the vibration of the wheels on any point of suspension, and 

 thence determining the corresponding centre of oscillation. 

 Let d = the distance of the point of suspension from the centre 

 of gravity. 

 I = the distance of the centre of oscillation from the pohit 

 of suspension. 

 Then by known principles we have 

 d{l-d) = k^, 



and hence m k^ may be found for each pair of wheels. 

 We shall therefore consider the quantity 



as thus determined, and for brevity shall call it M', so that the 

 equation (4.) shall be reduced to the form 



Mg (h-f) T={M + M') {Y -\') . . . (5.) 



If S express the space over which the train moves in the time 



T, then \ dT = dS, and we obtain the relation between V and 



