744 
MECHANICS. 
if this joint preflure-is greater than that of 33 feet of water, 
the water cannot he retained fo high ; but, if it is lefs than 
the column of 33 feet, the water will continue to rife in 
the pump. 
From thefe confiderations we may readily inveftigate a 
general theorem. Let a = the altitude or vertical diltance 
from the point O to the furface R S of the water in the re¬ 
fervoir, p~ O D the play of the pifton, and x the diltance 
O T ; then we have — p, and ST the height of 
the point T will be a — x. Since the air contained in 
C DTZ has the fame denlity and elafticity as the exterior 
air, its force may be meafured by a column of water of 
the fame bafe ZT and 33 feet high ; and, becaufe when 
this air is fo expanded as to fill the fpace QOTZ the elaftic 
force will be lefs in the ratio of DT to O T, we fhall have 
(rejecting the bafe of the column, as equally affecting 
every part of the procefs) this latter force exprelfed by the 
31 
fourth ternuof this proportion, x : x—-p : 33 : — (x— p). 
But the force which the water comprifed between ZT and 
R S exerts in oppofition to the exterior p refill re of the air, 
is meafured by the height a — x ; confequently, the elaftic 
force of the air in the fpace QOTZ, together with the 
weight of the water between ZT and R S, will be ex- 
prefled by + a— x. Now, in order that the water 
may always rife, this joint preflure mull be lefs than the 
weight of a column of water of 33 feet by fome variable 
quantity, which we will call y ; fo that the following equa¬ 
tion muft always obtain, viz. 
3 fx—p) 
+ a— *=33— y. 
The value of x deduced from this equation is ambiguous, 
being thus expreffed : 
x=h a + hy±V(ia~{-ly') 2 —i3p. 
Now, when the water Hops and does not rife any fur¬ 
ther, y vanifiies, and the equation becomes * = |± 
f%a 2 - —33p; of which the two values are real, fo long 
as \a 2 is greater than 33 p. Hence we conclude that, when 
one-fourth of the fquare of the greatefl height of the pifton above 
the furface of the water in the refervoir is greater than 3 3 times 
the play of the pifton, there art always two points in the Jucking- 
pump where the water may flop in its motion 5 and the pump 
mult be reputed bad when the lowed point to which the 
pifton can be brought is found between thefe two points. 
But if 33/1 be greater than \a 2 , the two values of x, when 
y is fuppofed =0, become imaginary 5 fo that in a pump 
lb conltrufted it is impofiible that y Ihould vanifti; that 
is, the preflure of the exterior air always prevails, and the 
water is not arrefted in its paffage. Hence we conclude, 
fecondly, that, in order that the fucking-pump may infallibly 
produce its effteB, the fquare of half the greatefl elevation of the 
pifton .above the water in the refervoir tnufl always be lefs than 
33 times the play of the pifton. 
This general rule may alfo be eafily deduced geometri¬ 
cally thus: Suppofe the fucker or valve E be placed at 
the furface RS of the water (fig. 49.) the tube to be of 
uniform bore, and Y S to be the height of a column of 
water whofe prelTure is equal to that of the atmofphere j 
that is, YS^33 feet. Conceive the water raifed by work¬ 
ing to N ; then the weight of the column of water S N, 
together with the elafticity of the air above it, exaftly ba¬ 
lances the prelTure of the atmofphere Y S. But the elaf¬ 
ticity of the air in the fpace O M (Q O being the higheft 
and C D the lowed: iituauon of the pifton) is proportional 
DN 
to Y S X —; and, confequently, in the cafe where the 
limit obtains, and the water rifes no further, it will be 
Y N S -J- (Y S . Tranfpofing NS, we have, 
DN 
YS-NS(s=NY)e=YS . ~ j whence .ON ; DNYS: YNj 
or, dividendo, ON—DN(=DO) : ON s: YS—YN(—NS) 
: YS 5 confequently, DOXYS=r,ONXNS. Hence we fee, 
that if SO, the diftance of the pifton in its higheft pofition 
from the water, and O D, the length of the femi-ftroke, 
or the play of the pifton, be given, there is a certain de¬ 
terminate height, as SN, to which the water can be raifed 
by the difference of the preflures of the exterior and inte¬ 
rior air 5 for YS is to be confidered as a conftant quantity, 
and, of courfe, when OD is given, ON . NS is given, 
likewife. To enfure, therefore, the delivery of water by 
the pump, the ftroke mult be fuch that the reftangle 
O D . Y S may be greater than any reftangle that can be 
made of the parrs of SO ; th 3 t is, greater than the fquare 
of | OS, by a well-known theorem. Hence we deduce a 
practical maxim of the fame import as the preceding; 
i. e. No fucking pump can raife water effectually unlefs the play 
of the pifton in feet be greater than the Jquart of the greatefl 
height of the piflon divided by 132. 
• . 
Refuming the equation—--— -j-a—*—33— y, anu 
% 
finding thence the value of y, we have y~- - 
Now let AB (figs. 52, 53.) reprefent the greatefl height 
of the pilton above the furface of the water in the refer¬ 
voir, and AD the play of the pifton ; fuppofe the differ¬ 
ent portions AP of the line A B to reprefent the fuc- 
ceflive values of x, and lay down upon the perpendiculars 
PM the values of y which correfpond to thefe-a flu m erf 
values of x ; fo fhall we have a curve M M C, (fig. 52.) 
which, while \a 2 is greater than 33 p, will cut AB in 
two points I and I', in fuch manner that the ordinates PM 
will lie on different fides of A B ; the ordinates which are 
below AB fhowing the politive values of y, and thofe- 
whichare above AB the negative values. We fee, there¬ 
fore, that, fo long as \a z is greater than 33/, the prelTure 
of the exterior air is ltrongelt, until the water has attained 
the height of B I'. At this point V it will (top (abftraft¬ 
ing from the confideration of the motion acquired), be¬ 
caufe the value of y is = o. But, if the water by the mo¬ 
tion it has acquired continues to rife till it reaches fome 
point between I'and I, it will not Itop there, but will de¬ 
scend, if the fucker does not oppofe its defcending mo¬ 
tion 5 becaufe the value of 7 being there negative indicates 
that the prelTure of the exterior air is weaker than the 
united preflures of the water and the internal air. If the 
water reach the point I, it will flop there, for the fame 
reafon as it would at the point I'; but, if once it gets 
above I, there is then no reafon to'fear that it will defcend 3 
for all the ordinates PM between I and A being pofitive, 
fliow that in that portion of the pump the preffure of the 
external air exceeds the combined efforts of the internal 
air and water. 
When, on the contrary, the value of £ a 2 is lefs than 
that of 33 p, the curve (fig. 53.) will no-w here interleft 
the axis AB; all the ordinates are pofitive, and confe¬ 
quently the preflure of the external air is always the 
Itrongeft. This confirms and illuftrates what has been 
laid down in the general theorem. 
If the fucking-pump were to be placed fo high above 
the ufual furface of the earth (as at the top of a high 
mountain), or fo low beneath it (as in a deep mine), that 
the prelTure of the atmofphere would be feniibly different 
from the aflumed mean prelTure equivalent to 33 feet of 
water, we muft then in all the preceding inveitigation 
change the co-efficient 33 to that which would exprefs 
the height in feet of the correfponding column of water. 
And thefe equivalent columns may always be afcertained 
by means of the height of the mercurial column in the ba¬ 
rometer ; the analogy being this : As 29! inches, the 
mean altitude of the mercurial column, is to 33 feet, the 
mean height of the column of water ; fb is any other mer¬ 
curial column in inches to its correfponding column of 
water in feet* 
Sm 
