4 
E ds cos. a = P ds cos. (a— /5) 
if a = then P = E ^for cos.^—cos. 
when a is > then P < 11. 
We may now write the equation as follows :— 
cos. /3 + sin. /3 tan. a 
When the swingle-tree hook turns upward the angle a increases also tan. a 
for all angles between 0° and 90°, consequently P diminishes. P disappears 
when a =90° and tan a =gc , that is to say that P varies from E to zero when a 
increases from — to 90°. 
2 
In practice a is at first manifestly about zero, that is to say, the hook is per¬ 
pendicular, or thereabouts, to the traction bars when the brake is applied, the 
hook being vertical, and the bars practically horizontal. In this position the 
formula gives : 
E ds = P ds cos. /3 
whence P > E 
The dynamometer however gave 55 lbs. as the maximum value of P, whereas 
the force of E is about 80 lbs. This discrepancy is accounted for by the fact 
that the chain Y is not taut when the brake is on. There is a certain amount of 
play at both of its points of attachment to the beam and the connecting rods re¬ 
spectively. The horses therefore turn the hooks upwards in a slight degree before 
the springs of the brake act. 
When the springs act a is > ^ and then P<E. 
If a becomes equal to 90° P, as we have said, disappears, for we have, in fact, 
the position given in the annexed figure where the brake is disarmed and has 
really ceased to be automatic. Theory and practice therefore both shew the 
necessity of preventing the swingle-tree hook ever assuming a horizontal position. 
Fio. 2. 
Inasmuch however as the traction bars lie underneath the splinter-bar, the “dead 
angle ” cannot be reached. The slighter the dimensions given to these parts the 
more nearly will the “dead angle 55 be reached, though not attained, and the 
smaller the effort of the team to keep the brake disarmed. When the team comes 
to a halt the springs may be relied on to overcome the inertia of the system and 
apply the brake. 
It has been found that, thanks to the springs, the shocks proceeding from the 
irregularities of the ground acting on the carriage are not transmitted in their 
entirety to the team, and the experiments of M. Marey have proved that this 
elasticity effects an economy of 25 per cent, in horse power. 
The reader will remember that when the brake is applied the swingle-tree hook 
tends to fall to a vertical position. This produces a certain movement of the 
