1831.] 
Barometer is liable . 
53 
doubt, that if the experiment be made on the barometer, by which the Calcutta Re- 
gister is kept in the Surveyor General’s Office, a similar result will be obtained. 
The following is an investigation of the depression produced on the barometric 
column, by a quantity of contained air. 
Let h be the height of the mercury in a perfect barometer, or the true atmospheric 
pressure; / the length of the barometer tube, from the surface of the basin to the 
top. It is required to find at what height the mercury would stand on introducing 
a quantity of air which, at the given pressure, would fill the tube to the depth = n. 
Let x be the height required. 
The air at the pressure A occupies a portion of the tube = n, but when introduced 
at the top of the mercury, it occupies the space / — x, and is then under the pressure 
k — x, hence we have hn =. (A — x) (/ — x ); (A):=:# 2 — (l-f-h) x -f- /A; or,# 2 — 
l «L 
(/ + h ) x hn — Ih ; (B) : hence, x = 
± ( ■ 2~ ' ~) — ^ + hn. 
but since by the nature of the case x /_ h /_ l, the substractive root is the only one 
„ n 7 h v a 
which can obtain here ; and hence x 
/+ h 
■vW)’ 
l — k 
hn A*/i 2 
+ 
l -f- h 
-f- hn; (C).If x be required by a series, we have #= 
h % n 2 
2 t — h ' {l—hf (/—A) 5 
S A 3 n 3 
2 A 3 « 3 , hn , 
; or, x = A ; — "f- 
{l — A) B 
rect, that is, x = A — 
l—h ' {l — A) 3 
+ &c. ; (D) . In most cases the first term would be sufficiently cor- 
hn 
l — h 
very nearly ; (E) . This result may be obtained in 
another way. Resuming the original equation (A) hn z= (A — x) {l — x) whence 
fill 
h — x =: ; (F). But when the depression is small A = x very nearly, and 
/ — x 
fljl 
l — x / — A very nearly, hence A — x zr very nearly : also h — x — 
l — A 
hn 
hnx 
T T* 
hnx * 
hnx 3 
hnx 4 
-f- &c. ; (G): by ac- 
tually dividing (F) . 
The depression is, therefore, very nearly inversely as the vacuum in the tube 2 , 
and hence the comparative advantage of a long tube over a short one imperfectly 
filled : — as an example, 
Suppose / = 36 r 27,9965 
/ = 32 ) \ 27,993 
/ = 31 f , i u ) 27,991 
l = 30 y - 28 *■ 1- ’ rehaTe ' 27,986 
/ = 29 
l— 28 
27,973 
27,947 
2 This is true without sensible error, when the vacuum at the top of the tube is not 
less than an inch in length. 
