378 Rm^al Society. 



the precision of the results } the carriages having been treated as 

 sledges moving down an inclined plane. The author has here given 

 the analytical formulse by which the effect of the rotatory motion of 

 the wheels may be brought into computation ; this effect, depending 

 obviously on the amount of inertia of the wheels, and on the propor- 

 tion which their weight bears to the weight of the waggon. 



The properties investigated in this first division of the paper, are 

 strictly those which depend on the longitudinal section of the line, 

 presumed to be straight in every part of its direction. There is, 

 however, another class of important resistances which depend on 

 the ground-plan of the road, and these the author next proceeds to 

 determine. 



The author then gives the analytical formulae which express the 

 resistance arising,— >rs<, from the inequality of the spaces over which 

 the wheels, fixed on the same axle, simultaneously move ; secondly, 

 from the effort of the flanges of the wheels to change the direction of 

 the train ; and thirdly, from the centrifugal force pressing the flange 

 against the side of the rail. He also gives the formulae necessary to 

 determine, in each case, the actual amount of pressure produced by 

 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 j 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. 



