564 
THE SLIP OF THE DRIVING-WHEEL OF A LOCOMOTIVE ENGINE. 
Still showing the displacement of the centre of gravity necessary to produce a jump 
to diminish with the diameter of the wheel. These conclusions are opposed to the 
use of light engines and small driving-wheels ; and they show the necessity of a care- 
ful attention to the true balancing of the wheels of the carriages as well as the driving- 
wheels of the engine. It does not follow that every jump of the wheel would be high 
enough to lift the edge of the flange off the rail ; the determination of the height of 
the jump involves an independent investigation. Every jump nevertheless creates an 
oscillation of the springs, which oscillation will not of necessity be completed when 
the jump returns; but as the jumps are made alternately on opposite sides of the 
engine, it is probable that they may, and that after a time they will, so synchronize 
with the times of the oscillations, as that the amplitude of each oscillation shall be 
increased by every jump, and a rocking motion be communicated to the engine 
attended with danger. 
Whilst every jump does not necessarily cause the wheel to run off the rail, it 
nevertheless causes it to slip upon it, for before the wheel jumps it is clear that it 
must have ceased to have any hold upon the rail or any friction. 
The Slip of the Wheel. 
If /be taken to represent the coefficient of friction between the surface of the wheel 
and that of the rail, the actual friction in any position of the wheel will be represented 
by Y,/. But the friction which it is necessary the rail should supply, in order that 
the rolling of the wheel may be maintained, is X. It is a condition therefore neces- 
sary to the wheel not slipping that 
Y,/>X,or/>A ( 60 .) 
X 
If, therefore, taking the maximum value ofy- in any revolution, we find that f ex- 
ceeds it, it is certain that the wheel cannot have slipped in that revolution; whilst if, 
on the other hand, / falls short of it, it must have slipped*. The positions between 
which the slipping will take place continually, are determined by solving, in respect 
to cos the equation 
f=Y, («>■) 
The application of these principles to the slip of the carriage-wheel is rendered less 
difficult by the fact, that the value of h is always in that case so small, as compared 
with the values of h and «, that ^ may be neglected in formulae (34.) and (35.), as com- 
pared with unity. Those equations then become 
X= 
W4 sin fir k^{g + aQ:> 
1 - 
(62.) 
a i g{k^ + a^) 
* Of course, the slipping, in the case of the driving-wheels of a locomotive, is diminished by the fact that 
whilst one wheel is not biting upon the rail the other is. 
