346 On Rail Roads. 
Arr. XV.—On the elevation required for rails on Rail Roads of a 
given curvature; by J. THomson, Engineer, and late Professor 
of Mathematics in the University of Nashville, Tenn. 
TO THE EDITOR. 
Sir—I ebserve, in a valuable little work on rail roads, by Col. 
Long, formerly Engineer in the service of the Baltimore and Ohio 
Rail Road Company, an article on the comparative elevation of rails, 
on the same track, when the rails are curved. ‘The exterior rail re- 
quiring some elevation above the interior, when the road is curved, 
the question is to find what that difference of elevation ought to be, 
when the road has a given curvature and the carriage moves with a 
given velocity. é 
This question is investigated by Lieut. Dillahunty, who introduces 
into the investigation the centres of gravity and percussion ; for what 
reason I cannot clearly perceive. When a loaded car moves along 
a rail road in aright line, the direction of the pressure of the load is 
perpendicular to the horizon; but when the car moves in a curved 
line, the direction of pressure is no longer vertical, but inclined to- 
wards the centre of motion. The object, therefore, of the investi- 
gation is to determine what elevation should be given to the exterior 
rail, so that the plane of the rails will be perpendicular to the direc- 
tion of the pressure of the load. If this be (and it certainly ought 
to be) the object of the investigation, the results, as given by Lieut. 
Dillahunty, appear to be erroneous. ‘These results are expressed 
by the two following formulas, 
(Ran)? Ron) V* 
2h(R+7r) AW(R-+7) 
In these formulas, E represents the elevation of the exterior rail 
above the interior; R andr, the radii of the curves made by the 
rails; & the height of the centre of gravity of the load and carriage 
above the track; W the weight of the load and carriage, and V the 
velocity of the carriage. From the first formula it is evident (as 
observed by Lieut. D. himself,) that the elevation of the exterior 
rail will vary inversely as the height of the centre of gravity above 
the track, supposing R, r and V to be constant; and from the sec- 
ond formula, the elevation will vary inversely as the weight of the 
load and the height of the centre of gravity. 
