TRACT 34.

IN GUNNERY.

329

k = — ^ and, by the doctrine of forces,

vv = - 2 hfz= - 2 ho. -

9,hc

VC 1 -* s ) ps ‘VC 1 -* 5 )'

But — is the versed sine or height of the whole arc whosechord is c, and ox (1 — — s s)) = o — o V(1 — >'«) is

the versed sine or height of the part whose sine is os;therefore ^ — o+oV(l — s s') is their difference, or theheight of the remaining part, and which is nearly equalto the height due to the velocity v; therefore h : 4 h h : :;^-0 + oV(l — -ss): V = 4 /i x (y‘ - » + oV{ 1 - s-s))nearly. Then, by substituting this for v* in the value of vv,we have

8 h h f> a . c 4 ■— 2o 2

x (

2 o "

vv = - 2 ho.— ---

J , n VC 1 — **) PSand the fluents give

V = 4 /i 0 -v/(l— ss) -(■ . - z — os ) + a ;

where q is a constant quantity by which the fluent is to

l>e corrected. Now, substituting v 2 for V, and 0 for their

corresponding values at the commencement of motion, the

above fluents become

v 2 ' = 4 h o -j- q ; from which the former subtracted, gives

v 2 — V= 4 ho — 4 ho\/(l—ss) — --(—5-7-2 - os).

And when v = 0, or the pendulum is at the full extentof its ascent, then

v 2 = 4 ho - 4 ho V (1 ~ s-s)-( 2o ,- 2 - ot),

at which point 0 s is the sine of the whole arc whose chord isc, and consequently s = ^V( 4 o 2 — c 2 ).

But the value of s being commonly small in respect ofthese following values will be nearly true, namely,

4c a 0 2 —<"4 C 4 _ , 0 2

8 ? 8 * ~~ ” 27 2>

V(i-Si) = l-is 2 -^ 4 =l-

4 ho — 4 h 0 V (1 -si) = 4 h 0 .

z — os + %os 3 , and

2 o’

os = — — +

2o *

c 2 .? 2o 2 — c 2

12i