24 
TRANSACTIONS OR THE TEXAS ACADEMY OF SCIENCE. 
short tori, we have a grade corresponding to a rate of 0.3 per cent; then 
calling the train length a, and rate of change per station r, 
ar 0.3 
from which 
_ 0.3 
a 
in which r is in feet and a in stations. If the train he 1200 feet long 
this gives r— 0.025. For an algebraic difference of 1.5 feet, this would 
give 6000 feet for the length of curve, with a correction at the vertex of 
11.25 feet. 
Kecognizing the impracticability of securing such inordinate length of 
curves, Wellington further states that with the modern close couplers 
the length of curve and consequent correction may be reduced one-half 
without danger of breaking the train. 
In regions yielding long, easy grades, with relatively small changes in 
gradients, it may be possible to adhere to this rule; but where heavy 
grades occur, with frequent changes of considerable amounts, this can 
not be done. Generally it will be found that where heavy grades occur 
the local conditions limit the length of curve to such an extent that no 
theoretical rule can be adhered to; yet it is just such regions as this that 
require some sort of curve at summits and sags. Local conditions, then, 
will fix the minimum length of curve — there is no danger of getting them 
too long, unless it necessitate undue expenditure for construction. The 
engine man can control his train by a proper use of air to such an extent 
that even a short curve can be passed without the existence of an undue 
amount of slack, particularly if he is careful to give the engine steam at 
the proper point. 
Now as to the best curve to use. The parabola offers the advantage of 
being the kind of curve the center of mass of the train tends to follow, as 
well as ease in setting out in the field. Two properties of the parabola are 
utilized: (1) that ordinates from tangent to curve vary as the square of 
the distance from the point of tangency; and (2) that the curve bisects 
the vertical intercepted between the vertex and the long chord joining 
the P. C. with the P. T. 
