20i< EIGHTH REPORT— 1838. 



\ = 'L^'.'YdY = g sin 9 d x, 



which being integrated, supposing that when a- = o, V = o, gives 



V^ = 2 ^' X sin 9. 



But if the load be subject, as it always is in practice, to fric- 

 tion, then let the retarding force of friction be/, and the above 

 equation will become 



Y2 = 2 (^ sin 6 -/) x ; 



and if the load descend a succession of planes of different gra- 

 dients, passing from one to the other without any shock by 

 which it M'ill lose velocity, let a,', x", &c. represent the spaces 

 over which it moves on each plane. Its motion will be then 

 represented by the equation, 



V2 = 2 (^ sin 6 -/) X + 2 {g sin 5' -/) x' + 2{g sin fl" -/") x" + &c., 

 or, V^ = 2S{(5-sin9-/).r} (3.) 



Siich is the equation obtained by M. de Pambour for the mo- 

 tion of a train down one or more inclined planes. 



But this is manifestly erroneous and does not really express 

 that which it professes to express : 



1st. Because the condition (2.), from which all the others are 

 deduced, would be only true on the supposition that all the par- 

 ticles of the load moved in lines parallel to the inclined plane 

 ■wdth a common velocity V, M'hich in fact is not the case, since 

 the wheels and axles of the wagons or carriages have a motion 

 compounded of a progressive and rotatory motion ; and the mass 

 of these bears a considerable proportion to the whole weight of 

 the load. 



2nd. Admitting that the error just mentioned were corrected, 

 it is assumed that the excess of the gravity down the plane over 

 the resistance opposed to the motion is independent of the velo- 

 city. Now, if any resistance be produced by the aii", that re- 

 sistance will increase, according to some law, with the velocity. 

 It is therefore implicitly assumed in the reasoning of M. de 

 Pambour, either that the resistance of the air in his experiments, 

 or any other resistance depending on the velocity, is so incon- 

 siderable that it may be disregarded, or that even at the greatest 

 velocity it bears so small a ratio to the friction, that it may be 

 confounded with the friction, and that the result will exhibit the 

 mean resistance with sufficient accuracy for practical purposes. 



Let usj in tl)e first place, see to what extent the error arising 

 from the omission of the consideration of the wheels operated 

 on the result of M. de Pambour's experiments. To accomplish 



