On Rail Roads. 347 
That these conclusions are erroneous, and that the elevation of the 
exterior rail does not depend on the weight of the load, nor on its 
height above the track, may be shown from the following consider- 
ations. 
EK 
Let CAB represent a horizontal surface, on which a rail way is situ- 
ated; A and B, the rails placed in a circular curve around C as a 
centre. A car in moving over the rails Avand B, around the centre 
C, will be acted upon by two forces; one horizontal and centrifugal, 
arising from the motion of the car in a curved line, and acting in a 
direction from the centre C; the other, the force of gravity, acting 
in a vertical direction. I omit here, as not necessary in the present 
investigation, the moving force, derived from animal or other power, 
acting in the direction of a tangent to the curve. Let the horizontal 
line AK represent the centrifugal force above mentioned, and the 
line EA the force of gravity. It is evident that the resultant of these 
two forces will be EK, which will represent both the intensity and 
the direction of the pressure of the loaded car upon the rails. The 
line EK, therefore, representing the direction of pressure, the rails 
should be so placed that this line may be perpendicular to the plane 
passing through them. Draw the vertical lme BD, and through A 
draw AD perpendicular to EK. BD will be the elevation of the 
exterior rail above the interior, and the angle DAB will be the incli- 
nation of the plane of the rails to the horizon. The centrifugal force 
AK, compared with the force of gravity AE, is easily found, when 
the radius of curvature of the track and the velocity of the car are 
given. ‘The distance between the centre C and the middle of the 
track may be considered as the radius of curvature. 
Now, by a reference to the above figure, it will be seen that a 
change of weight on the car cannot alter the elevation BD of the 
exterior rail, or the angle DAB. For, if we suppose the absolute 
weight of the load to increase or decrease, it is evident that the cen- 
trifugal force will increase or decrease in the same ratio—in other 
words, the lines AE and AK will vary in the same ratio, and hence 
