MECHANICS. 
-teomprefled from wltfiouf. And thus it will happen that 
each column of water, being higher at its outer than at its 
inner end, compreffes the air on the water-column beyond, 
or within it, which tranfmits this preffure to the air be¬ 
yond it, adding to it the preffure arifing from its own 
want of level at the ends. Therefore the greateft com- 
preflion, viz. that of the air next the main, is produced 
by the fum of all the tranjmitted preffures ; and theie are the 
fum of all the differences between the elevations of the 
inner ends of the water-columns above their outer ends; 
and the height to which the water will rile in the main 
will be juft equal to this fum. 
Draw the horizontal lines K'K i, K'K 2, K'K 3, See. 
and mn, mn, mn, Sec. Suppofe the left-hand l’paces to 
be filled with water, and the right-hand fpaces to be filled 
with air. There is a certain gradation of compreflion 
which will keep things in this pofition. The fpaces evi¬ 
dently decreafe in arithmetical progreflion ; fo do the hy- 
droftatic heights and preffures of the water-columns. If 
therefore the air be denfe in the fame progreflion, all will 
■ be in hydroftatical equilibrium. Now this is evidently 
producible by the mere motion of the machine ; for, flnee 
the denfity and compreflion in each air-column is fup- 
pofed inverfely as the bulk of the column, the abfolute 
quantity of air is the fame in all ; therefore the column 
firft taken in will pafs gradually inwards,and the increafing 
compreflion will caufe it to occupy preCifely the whole 
right-hand fide of every fpire. The gradual diminution 
of the water-columns will be produced during the motion 
by the water running over backwards at the top, from 
fpire to fpire, and at laft coming out by the fcoop. 
It is evident that this difpolition of the air and water 
will raife the water to the greateft height, becaufe the hy- 
droftaiic height of each water-column is the greateft polii- 
ble, viz. the diameter of the fpire. This diflpofition may 
be obtained in the following manner: Take CL to C B 
as the denfity of the external air to its denfity in the laft 
column next the rifing-pipe or main ; that is, make C L 
to C B as 33 feet (the height of the column of water 
which balances the atmofphere) to the fum of 33 feet and 
the height of the rifing-pipe. Then divide BL into fuch 
a number of turns, that the fum of their diameters fhall be 
equal to the height of the main ; then bring a pipe ftraight 
from L to the centre C. The reafon of all this is very 
evident. 
But, when the main is very high, this conftruffion will 
require a very great diameter of the drum, or many turns 
of a very narrow pipe. In fuch cafes it will be much bet¬ 
ter to make the fpiral in the form of a cork-fcrew, as in 
fig. 89, inftead of this flat form like a watch-fpring. The 
pipe which forms the Jpiral, may be lapped round the 
fruftum of a cone, whole greateft diameter is to the lead 
(which is next to the rifing-pipe) in the fame proportion 
that we afligned to C B and CL. By this conltruflion 
the water will ftand in every round fo as to have its upper 
and lower furfaces tangents to the top and bottom of the 
fpiral ; and the water-columns will occupy the whole af- 
cending fide of the machine, while the air occupies the 
defeending fide. This form is vaftly preferable to the 
flat : it will allow us to employ many turns of a large 
pipe, and therefore produce a great elevation of a large 
quantity of water. The fame thing will be ftill better 
done by lapping the pipe on a cylinder, and making it 
taper to the end, in fuch a proportion that the contents 
of each round may be the fame as when it is lapped round 
the cone. It will raife the water to a greater height (but 
with an increafe of the impelling power) by the fame num¬ 
ber of turns, becaufe the vertical or prelfing height of each 
column is greater. Nay, the fame thing may be done in 
a more Ample manner, by lapping a pipe of uniform bore 
round a cylinder. But this will require more turns, be¬ 
caufe the water-columns will have lefs differences between 
the heights of their two ends. It requires a very minute 
inveftigation to fhovv the progrefs of the columns of air 
and water in this conftru&ion, and the various changes 
759 
of their arrangement, before one 5 s attained which will 
continue during the working of the machine. 
We have chofen for the description of the machine that 
conftruction which made its principle and manner of 
working molt evident, namely, which contained the fame 
material quantity of air in each turn of the fpiral, more 
and more compreffed as it approaches to the rifing-pipe. 
We fhould otherwife have been obliged to investigate in 
great detail the gradual progrefs of the water, and the fre¬ 
quent changes of its arrangement, before we could fee 
that one arrangement would be produced which would 
remain conftant during the working of the machine. But 
this is not the beft conftruffion. We fee that, in order 
to raife water to the height of a column of 34. feet, which 
balances the atmofphere, the air in the laft lpire is com- 
prefled into half its hulk; and the quantity of water de¬ 
livered into the main at each turn is but half of what was 
received into the firft fpire, the reft flowing back from 
fpire to fpire, and being difeharged at the fpout. 
But it may be conltruffed fo as that the quantity of 
water in each fpire may be the fame that was received 
into the firft; by which means a greater quantity (double 
in the inftance now given) will be delivered into the main, 
and raifed to the fame height by very nearly the fame force. 
This may be done by another proportion of the capacity 
of the fpires, whether by a change of their caliber or of 
their diameters. Suppofe the bore to be the fame, the 
diameter muft be made fuch that the conftant column of 
water, and the column of air, compreffed to the proper 
degree, may occupy the whole circumference. Let A be 
the column of water which balances the atmofphere, and k. 
the height to which the water is to be raifed. Let A be 
to A h as 1 to m. It is plain that m will reprefent the 
denfity of the air in the laft fpire, if its natural denfity 
be 1, becaufe it is prefied by the column A + h, while the 
common air is prefied by A. Let 1 reprefent the conftant 
water-column, and therefore nearly equal to the air-co¬ 
lumn in the firtt lpire. The whole circumference of the 
laft fpire muft be i-j-, in order to hold the water 1, and 
m 
1 A 
the air compreffed into the fpace — or ---. The cir- 
m Aft-/? 
cumference of the firft fpire is 1 -f-1, or 2. Let D and d 
be the diameters of the firft and laft Ipires ; we have, 
2 : j -j-=D : d, or 2 m : ot4-iz^D : d. Therefore, 
m 
if a pipe of uniform bore be lapped round a cone, of 
which D and d are the end-diameters, the fpirals will be 
very nearly fuch as will anfwer the purpofe. It will not 
be quite exaft, for the intermediate fpirals will be fome- 
what too large. The conoidal fruftum fhould be formed 
by the revolution of a curve 6f the logarithmic kind. 
But the error is very trifling. With fuch a fpiral, the full 
quantity of water which was confined in the firft fpiral will 
find room in the lait, and will he lent into the mam at 
every turn. This is a very great advantage, efpechlly 
when the water is to be much raifed. The laving of 
power by this change of conumclion is always in pro¬ 
portion to the greateft compreflion of the air. 
Tiie great difficulty in the conitruetion or any of thefa 
forms is in determining the form and pofition of the horn 
and the fcoop; and on this greatly depends the perform¬ 
ance of the machine. The following inftrubfions will 
make it pretty eafy. Let ABEO (fig. 90.) reprefent the 
firft or outermoft round of the fpiral, of which the axis is C. 
Suppofe it immerfed up to the axis in the water VV" ; we 
have feen that the machine is moft effective when the fur- 
faces KB and Ok of the water-columns are diftant the 
whole diameter B O of the fpiral. Therefore let the pipe 
be firft fuppofed of equal caliber to the very mouth Ef. 
which we fuppofe to be juft about to dip into the water. 
The lurface On is kept there, in oppofition to the pref- 
fure of the water-column B A O, by the compreffed air 
contained in the quadrant O E, and in the quadrant 
which 
