558 
THE JUMP OF A ROLLING CYLINDER. 
bracket in the value of Y will vanish. This corresponds to a minimum value of Y 
represented by the formula 
Y=W-p^|3(^»“-l) +1| , . (42.) 
But if 3 (^“ 2 pr be numerically greater than unity, then the minimum of Y 
will be attained when and when 
Y=W-^/ 
^9^ 1 
'k^ + h9y 
9 I 
The jump of an axis. 
(43.) 
If Y be negative in any position of the body, the axis will obviously jump from its 
bearings, unless it be retained by some meehanical expedient not taken account of 
in this calculation. But if Y be negative in any position, it must be negative in that 
in which its value is a minimum. If a jump take place at all, therefore, it will take 
place when Y is a minimum ; and whether it will take place or not, is determined 
by finding whether the minimum value of Y is negative. If therefore the expression 
(42.) or (43.) be negative, the axis will jump in the corresponding case. An axis of 
infinitely small diameter, such as we have here supposed, becomes a fixed axis ; and 
the pressure upon a fixed axis, supposed to turn in cylindrical bearings without 
friction, is the same whatever may be its diameter; equations (40.) and (41.) deter- 
mine therefore that pressure, and equation (42.) or (43.) determines the vertical strain 
upon the collar when the tendency of the axis to jump from its bearings is the 
greatest. 
The jump of a rolling cylinder. 
Whether a jump will or will not take place, has been shown to be determined by 
finding whether the minimum value of Y be negative or not. 
\ rk^ h a\ 
Substituting a for redueing, equation (35.) becomes 
Y=w(l-^cos^) 
or 
Y=w(i--costf) 4T (44.) 
W {k^ + P) {g + aaP) 
fcos^S— 2acosfl-t-l 
4ga'^ 
( («— cos5)^ 
dt ” 
2ga^{a— cos S)^ 
(k^ + l^)(g + aco^)(a^-l} 
cos 6)^ 
sin ^ 
' 2ffa^{<x— cosdp ’ 
(45.) 
dY h [k^ + P){g + aca^){ct^ — \) i n i 
-^=0, 1st, when- ^ a\a- col^f 2ndly, when ^=7r, 3rdly,when 
)= 0 . 
