348 On Rail Roads. 
the line EK will always remain parallel to itself, and perpendicular 
to AD, whatever be the weight of the load, other quantities remain- 
ing the same. Again the height of the center of gravity above the 
track ¢annot alter BD, or the angle DAB. For, if EK represent 
-the direction of pressure of all parts of the load, itis evident that 
the center of gravity will tend in the same direction, in whatever 
part of the line EK it be situated, or whatever be its height above — 
the track. It may be observed that the lines EA and AK, repre- 
senting any given ratio, may beso drawn that the line EK may always 
be perpendicular on the middle of AD, in which case, the center of 
gravity of the load and car will always be situated in the line EK. 
We may obtain a very simple algebraical expression for the eleva- 
tion of the exterior rail. Let g = force of gravity, c= centrifugal 
force, d = distance between the rails, and E = required elevation, 
R and V representing radius and velocity. Then by the similar tri- 
2 
d 
angles EAK and ABD we have E a but by central forces, c=; 
g Mi R 
dv? ae 
hence E = Re In this expression, g is always a constant quantity 
and equal to 32:2 feet. : 
To take an example, suppose a car to move with a velocity of 
twenty miles per hour, on a rail way, curving with a radius of four 
hundred feet, the distance between the rails being four feet nine inch- 
es. The velocity in this case will be twenty nine feet four inches. 
dV2 
We then have E = gs 3.8 inches. The table given by Col- 
Long makes the elevation in this case 5.5 inches, too much by near- 
ly two inches. If we assume a radius of seven hundred and six- 
teen feet, the other quantities remaining the same, we find E = 2.1 
inches. The above mentioned table makes the elevation three 
inches. os 
If the velocity of a car on a rail way were always the same, We 
should have no difficulty in assigning the proper elevation of the 
exterior rail. But as there must be necessarily a great variety 7 
rates of traveling, an elevation which would be required by a rate of 
twenty miles per hour, would be much too great for a rate of eight, 
twelve or fifteen miles per hour. Perhaps the elevation required by 
the mean velocity would be the most eligible. ‘There is one view of 
the subject however, which ought to be taken into consideration in 
the location of the exterior rail. When a car moves with great ve~ 
