208 EIGHTH REPORT — 1838. 



If the train move successively on two planes whose gradients 

 are li and A', we shall have 



If we suppose that on the second plane /> A' the motion 

 will be retarded, and still more if A' be negative, or, which 

 is the same, if the second plane be an ascending gradient. If 

 the train in such case be allowed to move until it come to rest, 

 we should have V^ = 0, which would give 



(A~/)S + (//-/)S' = 0. . . . (17.) 



Now, it is remarkable that this conclusion will follow equally 

 from the correct formulae which include the effect of the wheels, 

 and from the erroneous formulae in which that effect is omitted. 

 This takes place by a compensation of two contrary errors. So 

 long as the motion of the train is accelerated, the error pro- 

 duced on V^ by neglecting the wheels is in excess, and while it 

 is retarded, the error produced on V^ is in defect; and in M. de 

 Pambour's formulas this excess and defect are equal. They 

 therefore neutralize each other, and the final result, so far as 

 respects the effect of the wheels, is thus accidentally correct. 



From the condition (17.) we obtain 



The quantities AS and A'S' are the dift'erences between the 

 levels of the extremities of the spaces S and S', and if A' be 

 taken negatively when the gradient rises, the quantity A S + A' S' 

 will be the actual difference between the level of the point from 

 which the train commences its motion and that of the point 

 where it stops. Thus the resistance would, according to this 

 reasoning, be found by dividing the difference of levels of these 

 two points by the entire space run over by the train. 



The Sutton inclined plane on the Liverpool and Manchester 

 Railway, falling toward Manchester, was staked out in distances 

 of 110 yards, commencing from a point 1100 yards from the 

 foot of the plane. The level of the tenth stake, which marked 

 the foot of the plane, is stated by M. de Pambour to be 34-61 

 feet below the level of tlie first stake. The line extending from 

 the foot of the plane towards Manchester, which continued to 

 fall, but in a very slight degree, was also staked out through a 

 distance of more than a mile from the foot of the plane. 



Five wagons loaded with bricks, and weighing gross .81-31 

 tons, were allowed to descend by gravity from the point 1100 

 yards from the foot of the plane ; and they continued to move 



