22 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
a train leaving or entering the yards or stopping at a station. The effect 
is very severe upon the drawbars and couplings. 
Short, choppy grades are conducive to sudden reversals of stress upon 
the couplings; especially when the grades are not properly rounded off 
with suitable vertical curves at the junctions. 
The question of the proper form and length of vertical curve to he em- 
ployed is of first importance; then the best method of setting it out. It 
should he noticed that the absolute value of the gradients has nothing 
to do with the problem — it is only their algebraic difference that concerns 
ns. 
If we consider a train passing from one grade line to another; it is 
evident that the path of the center of gravity of the train as a whole dif- 
fers from that followed by each individual car; this difference becoming 
more marked as the difference in gradients increases. Suppose two grade 
lines; BA and AC; to unite at A; as shown in Fig. 1, and let the train at 
B 
any instant he in the position DAE. For the sake of simplicity; let the 
mass of train per unit length he the same throughout; and call the length 
of train a . Let AE = 2 , then DA=a — 2 . The center of gravity of DA 
will he at F; its mid-point; and of FE at G-, distant \z from A. Let the 
center of gravity of the whole train he at H; then taking BA and AC as 
the axes of y and x respectively; and taking moments at F, 
GFxz = H Fxa 
GF a 
Or M- (1) 
HF % 
From similar triangles, 
G F GA iz a 
= = — = from(l) 
H F AK x z 
From which z = ^ 2 ~ax (2) 
Also from the figure 
GF FA £(c& — z) a 
HF j[KH y z 
