THE JUMP OF A FALSELY-BALANCED CARRIAGE-WHEEL. 
56l 
the centre of gravity, ^2 = and cos In this case, therefore, (equation 51.) 
Y=W(|-^*); (52.) 
and there will be a jump if | (53.) 
Pendulum oscillating on knife-edges. 
In this case a is evanescent, and &»=0. Equations (31.) and (33.) become, there- 
fore, 
2^/i(cosa-cosfli) gh sin a 
— 2^2 + 7*2 ■ 
Substituting these values of M and N in equation (30.), 
WA2 r I r 
—2 (cos ^ — cos 61 ) sin ^ — cos ^ sin Y=W+pq:^2| (cos 0 - cos 6 ,) cos 0 — sin"^ 
Wh^ 
.. X =^2;;;pp(2 cos ^1 — 3 cos sin ^ (^^v 
Wh'^ ( li\ 
Y=-pq7^(^3 cos^ ^ — 2 cos ^ cos^i+^J (55.) 
Y is a minimum when cos ^=1 cos 6 ^, in which case 

There will therefore be a jump of the pendulum upon its bearings at each oscilla- 
tion, if the amplitude of the oscillation be such, that 
1 . 2 . 3^2 
^ cos or cos^ d, > -Ji . 
The jump of the falsely-halanced Carriage-wheel. 
The theory of the falsely-balanced carriage-wheel differs from that of the rolling 
cylinder, — 1st, in that the inertia of the carriage applied at its axle influences the 
acceleration produced by the weight of the wheel, as its centre of gravity descends 
or ascends in rolling ; and 2ndly, in that the wheel is retained in contact with the 
plane by the weight of the carriage. The first cause may be neglected, because the 
displacement of the centre of gravity is always in the carriage-wheel very small, and 
because the angular velocity is, compared with it, very great. 
If Wi represent that portion of the weight of the carriage whieh must be over- 
come in order that the wheel may jump (which weight is supposed to be borne by 
the plane), and if Yi be taken to represent the pressure upon the plane, then (equa- 
tion 52.) 
y.=w.+Y=w.+w(i-^') 
(57.) 
