3S6 



in the present paper, to lay before the Society a series of mathematical 

 formula?, embodying the most general expressions for the phenomena 

 of the motion of carriages on these roads. 



The author begins by investigating the analytical formulae for 

 the traction of trains over a level line which is perfectly straight, and 

 finds, first, the distance and time within which, with a given amount 

 of tractive power, the requisite speed may be obtained at starting ; 

 and also the point where the tractive power must be suspended, pre- 

 vious to coming to rest. The excess of tractive power necessary to get 

 up the requisite speed is shown to be equal to the saving of tractive 

 power previous to a stoppage; and formulae are given for the determi- 

 nation of the time lost under any given conditions at each stop. 



The motion of trains in ascending inclinedplanes which are straight, 

 is next considered ; and formulae are given combining the effects of 

 friction and gravity, in opposition to the tractive force. The cir- 

 cumstances which affect every change of speed, and the excess of 

 tractive force necessary, in such cases, to maintain the requisite 

 speed, are determined ; as well as the other circumstances already 

 stated with respect to level planes. 



The friction of trains upon descending planes is next investigated; 

 and an important distinction is shown to exist between two classes of 

 planes ; viz., those whose acclivities are inferior to the angle of re- 

 pose, and those of more steep acclivities. A remarkable relation is 

 shown to exist between the tractive forces in ascending and descend- 

 ing the first class of planes. For descending planes of greater acclivity 

 than the angle of repose, the use of breaks becomes essentially re- 

 quisite. The effect of these contrivances is investigated, as well as 

 the motion of trains on the accidental failure of breaks. 



In any attempts which have been hitherto made to obtain the 

 actual velocities acquired by trains of carriages or waggons under 

 these circumstances, an error has been committed which invalidates 

 the precision of the results -, the carriages having been treated as 

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

 the analytical formulae 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, — -first, 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 



