560 
THE JUMP OF A PENDULUM OSCILLATING ON KNIFE-EDGES. 
which expressions are both negative if the inequalities (47.) obtain. The same con- 
* 
ditions which yield minimum values of Y in two corresponding oblique positions of 
CA, yield, therefore, maximum values in the two vertical positions ; so that if the 
inequalities (48.) obtain, there are two positions of maximum and two of minimum 
pressure. 
Substituting the value of cos 6 (equation 46) in equation (44.), and reducing, we 
obtain for the minimum value of Y in the case in which the inequalities (48.) obtain. 
If this expression be negative the cylinder will jump. 
In the case in which which is that of a pendulum having a cylindrical axis 
of finite diameter, it becomes 
If the first of the inequalities (48.) do not obtain, no position of minimum pressure 
corresponds to equation (46.) ; and the inequalities (47.) do not obtain, so that the 
d^Y 
values (49.) of given respectively by the substitution of ic and 0 for are no longer 
both negative, but the second only. In this case the value t of ^ is that, therefore, 
which corresponds to a position of minimum pressure, which minimum pressure is 
determined by substituting t for & in equation (35.), and is represented by 
v=w(i+^)-^^S!±M 
o?-\ 
[u+iy 
{k'^ + P) 
2ga{ct+l) 
Y= 
W 
I a-^h 
'1 
|4®-l-(a + 4)®— 4fl4cos®^S, j 
[(^ + «“^)1 
^{4®-f(a + 4)®} J 
W< 
Y=W 1- 
hcP 
4hi 1 ■ """ 
+ 
) cos^ iSil 
2^ + I 
f 
9 F + (« + A)2 
The cylinder will jump if this expression be negative, that is, if 
1 cos® ^5, . „ oPh I Ah cos® 
' Q f 2 or it { , 2 
(51.) 
hop 
A:®-1- (fl + /«)® 
or, substituting and reducing, if 
’ A® + (« + hf- 
>1 + 
44® cos® -6, 
■ 2 ^ 
4® + (a 4- 4)® 
44(ffl + 4) cos® 
4® + Z® 
If the angular velocity u be assumed to be that acquired in the highest position of 
* When the pendulum oscillates on knife-edges a=0, and this expression assumes the form of a vanishing 
fraction, whose value may be determined by the known rules. See the next article. 
