Royal Society. 87? 



in two parallel rows, consisting of ten cells in each row, on a long 

 table, with their siphon-tubes arranged opposite to each other, and 

 hanging over a small gutter, placed between the rows, in order to 

 carry off the refuse solution when it is necessary to change the acid. 

 Having observed that the uniformity of action may be completely 

 maintained by the occasional addition of a small quantity of acid, he 

 is able to dispense with the cumbrous addition of the dripping funnel; 

 an arrangement which admits with facility of any combination of the 

 plates which may be desired. 



April 28. — On certain parts of the Theory of Railways ; with an 

 investigation of the formulse necessary for the determination of the 

 resistances to the motion of carriages upon them, and of the power 

 necessary to work them. By the Rev. Dionysius Lardner, LL.D., 

 F.R.S. ' 



The author observes, in his prefatory remarks, that an extensive 

 and interesting field of mathematical investigation has been recently 

 opened in the mechanical circumstances relative to the motion of 

 heavy bodies on railways ; and having collected a body of experiments 

 and observations sufficient to form the basis of a theory, he purpose's, 

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

 formulae, 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 inclined planes 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 j 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 



Third Series. Vol. 9. No 55. Nov. 1886. 2 Y 



