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a given velocity and a given load, and investigates the extent to which 

 these resistances may be modified by laying the outer rail of the 

 curve higher than the inner. He assigns a formula for the de- 

 termination of the height which must be given to the outer rail, in 

 order to remove as far as possible all retardation from these causes ; 

 which formula is a function of the speed of the train, the radius of the 

 curve, and the distance between the rails. 



In the latter part of the paper, the author investigates the method 

 of estimating the actual amount of mechanical power necessary to 

 work a railway, the longitudinal section and ground-plan of which 

 are given. In the course of this investigation he arrives at several 

 conclusions, which, though unexpected, are such as necessarily arise 

 out of the mechanical conditions of the inquiry. The first of these is, 

 that all straight inclined planes of a less acclivity than the angle of re- 

 pose, may be mechanically considered equivalent to a level, provided 

 the tractive power is one which is capable of increasing and diminish- 

 ing its energy, within given limits, without loss of effect. It appears, 

 however, that this condition does not extend to planes of greater ac- 

 clivities than the angle of repose ; because the excess of power re- 

 quired in their ascent is greater than all the power that could be saved 

 in their descent ; unless the effect of accelerated motion in giving 

 momentum to the train could properly be taken into account. In 

 practice, however, this acceleration cannot be permitted ; and the 

 uniformity of the motion of the trains in descending such acclivities 

 must be preserved by the operation of the break. Such planes are 

 therefore, in practice, always attended with a direct loss of power. 



In the investigation of the formulae expressive of the actual amount 

 of mechanical power absorbed in passing round a curve, it is found 

 that this amount of power is altogether independent of the radius of 

 the curve, and depends only on the value of the angle by which the 

 direction of the line on the ground-plan is changed. This result, 

 which was likewise unexpected, is nevertheless a sufficiently obvious 

 consequence of the mechanical conditions of the question, if a given 

 change of direction in the road be made by a curve of large radius, 

 the length of the curve will be proportionably great ; and although 

 the intensity of the resistance to the tractive power, at any point of 

 the curve, will be small in the same proportion as the radius is great, 

 yet the space through which that resistance acts will be great in pro- 

 portion to the radius: these two effects counteract each other; and 

 the result is, that the total absorption of power is the same. On the 

 other hand, it the turn be made by a curve of short radius, the curve 

 itself will be proportionately short ; but the intensity of the resistance 

 will be proportionately great. In this case, a great resistance acts 

 through a short space, and produces an absorption of power to the 

 same extent as before. 



In conclusion, the author arrives at one general and comprehensive 

 formula for the actual amount of mechanical power necessary to work 

 the line in both directions ; involving terms expressive, first, of the 

 ordinary friction of the road ; secondly, of the effect of inclined planes, 

 or gradients, as they have been latterly called 3 and, thirdly, of the 



