THE JUMP OF A PENDULUM OSCILLATING ON A CYLINDRICAL AXIS. 559 
d'^Y 
The first condition evidently yields a positive value of since it causes the first 
term of the preceding equation to vanish ; and the second term is essentially posi- 
tive, a being always greater than unity. 
If, therefore, the first condition be possible, or if there be any value of 6 which 
satisfies it, that value corresponds to a position of minimum pressure. Solving in 
respect to cos 6, we obtain 
a—\/- 
id + [a? — 1 ) 
COS L 
2gah — V.V.O (46.) 
The first condition will therefore yield a position of minimum pressure, if 
3/]F+g^T^g2Ei)>-l or if + + <(«+!) 
V 2gah < + 1, ^ 
or if 
2gah 
>(«—!), 
+ + <(cc-\-iy {k^ + P){g + au>^){cc-l) 
“^gah 2gah[ct+lf 
and 
(F + p) [g + (a 4- 1 ) 
or if 
and 
„ 2gah{<x.-\-\Y , 
{k^-\-P-) (« — 1)’ 
2gah{a.—\'f 
2gah{oL — Vf 
2gh{a.-\- 1)^ 
(F-l-/2)(«_l) 
>1 
(4?.) 
2gh[x—\Y 
or 0 )^ > jri- 
whence, substituting for a and reducing, we obtain finally, the conditions 
(9\ {F+(« + A)2}2 
(9\ {k^+{a-hyY 
\aj {k^ + l'^)S^k^-\-[a—hy] 
\aj 
\aJ{B + P)i^k^+{a + hy} 
\a) 
(48.) 
- I I 'V / 
Of these inequalities the second always obtains, because 
{]ej^(^a-hyY<{k^-^P){1f+{a-{-hy}, 
whatever be the values of h, a and h. And the first is always possible, since 
{k^-^{a->rhyY>{k^-\-P){Je+{a-hy). 
If the Jirst obtain, there are two corresponding positions of CA on either of the 
vertical, determined by equation (46.), in which the pressure Y of the cylinder upon 
the plane is a minimum. 
dY 
Substituting the other two values (-r and 0) of d which cause -j- to vanish, in the 
value of 
d^Y 
dS^ 
we obtain the values 
h {k^ + P){g-\-a<d^)[a. — \) 
a 
or 
.-J 1. 
a\ 
2ga^{ct+ 1)2 
{k^ -t- l'^){g + aw^) (a — 1) 
2gha{(x-\-iy 
1^ (F + Z2)(^ + fla>2)(a + l) 
2ga%cc-iy 
}andj{l 
(F +P){g + aia^) (« + 1 ) 
2gha[a, — \y 
(49.) 
4 c 2 
