724 
MECHANICS. 
tinned, and increafed high enough. But, if the bladder be 
removed from the fire, as it cools it will contract again, 
as before. Indeed it was upon this principle that the firli 
air-balloons were made by Montgolfier; for, by beating 
the air within them by a fire underneath, the hot air dif- 
tends them to a fize which occupies a fpace in the atmof- 
phere, whofe weight of common air exceeds that of the 
balloon. 2. Alfo, if a cup or glafs, with a little air in it, 
be inverted into a veflel of water; and the whole be heated 
over the fire, or otherwise; the air in the top will expand 
til! it fill the glafs, and expel the water out of it ; and part 
of the air itfelf will follow, by continuing or increafing 
the heat. Many other experiments to the fame effect 
might be adduced. The expanfion of air, though expofed 
to the fame degree of heat, is not the fame in experiments 
made at different times ; owing to the difference of denfity, 
coldnefs, humidity, &c. The expanlive force of hot (team 
may exceed the force of gunpowder more than 30 times, 
and indeed is irrefiftrble when the force is intenfe. Hence 
it follows, that, when air is much impregnated with water, 
it will poffefs an expanfive power, by heat, much greater 
than that of pure air. Whether the degree of expanfion 
in pure air be proportional to that of the heat by which it 
is produced, is not known ; but it is manifeft that the va¬ 
riation of fpace occupied by a portion of air expofed to 
different degrees of heat may be fufficient to convey a 
tolerable idea of the afhial quantity, of heat. Upon this 
principle, therefore, have been conflrucled air-thermome¬ 
ters, to exhibit finall variations of heat. 
There is no general rule for the degree of expanfion to 
which different bodies are fubjecl by being expofed to the 
fame degree of heat. Taking an average, however, it has 
been found, by experiment, that for r?ch degree of heat 
meafured by Fahrenheit’s thermometer, mercury, water, 
and air, expand by the following parts of their own 
bulk, viz. 
Mercury the 9600th "J 
Water 6666th >part of its own bulk. 
Air 43 5th J 
Tn mercury the correfponding expanfions for i° gra¬ 
dually diminifh, being expreffed by ’0001177 at 2 ° of the 
thermometer, and by ’0000783 at 212°; but at 12° the 
expanfion correfponding to a degree of variation in heat 
is ’0001160, and at 102 0 it is -0001003, fo that between 
thefe limits the variation in the meafure of expanfion is 
very trifling. Taking into the effimate the changes in 
the expanfion, See. the fpecific gravities of thefe fluids at 
different temperatures have been hated as below. 
Spec. giav. of ail 1 1 when the barom. is. at 29^27 
water 836 h nd the thermom. at 53°. 
mercury 113 65 J 
Oi thus, aii 1 I w h en t he barom. is 29-27 and 
| the thermom. at 55 0 . 
mercury 11315J - > - > 
Or thus, air 1 when the barom. is 29^5 and the 
water 826 > thermom. is 5 5 0 which are their 
mercury 11227 J mean heights in this country. 
Orthus,air 1201 or i-ti 
water 1000 >in the laft circumftances. 
mercury 13592 J 
Or thus,air i’222 or i-IA , . . 
’ ■ 9 I nearly, when the barom. is 30. 
water 1000 , /, ’ , . J 
, f and the thermometer 
mercury 1 3600 J 33 
On this fubjecl: the ftudent may advantageoufly confult 
General Roy’s paper in the Phil. Tranf. vol. Ixvii. Alfo 
Sbuckburgh’sand M. de Luc’s papers in the lame volume. 
Prop. XXXV. To invefligate equations of equilibrium for 
elaflic fluids. —This will be very eafy, if we confider that 
fuch fluids mull, from the nature of perfedl elafticity, oc¬ 
cupy a fmaller fpace in proportion as the forces which 
comprefs them are greater, and re It ore themfelves to their 
primitive volumes when the aflion of the comprelTing 
forces ceafe. Let, then, P be a prefl'ure exerted upon a 
quantity M of the fluid, whofe denfity is D; p another 
prefl'ure, m the mafs or volume the fluid takes in confe- 
quenee of this prefl'ure, and d the denfity of this mafs; 
fo (hall we have thefe equations: 
PM —pm, MD —md, and Vd—pD - (I.) 
When P —p, then M— m, as is obvious. 
Thefe values only give the p refill res exerted upon a unit 
of iui face j but, if we drop the confideration of gravity, or 
any other force which may caufe a variation of denfity in 
the different parts of the fluid, we may then reafon from 
the principles of hydroftatics. This granted, the prefl'ure p 
exerted upon any furface denoted by a will be 
a d a M „ 
p——?,orp=s -P - - - (II.) 
Preflures being commonly valued, as we have often feen, 
by weights, we may reprefent that which is exerted upon, 
a unit of furface by the weight of a prifm of the fame 
fluid whofe height will be given. Let H be the height 
correfponding to P, and h to 'p ■, then we have HD %r 
the mafs of this prifm, and HD# its weight, (g denoting 
the force of gravity ;) hence P=HD#; and in like manner 
p=/idg. Subflituting thefe for P and p in the equations 
marked (I.) above, we have kD — Wd, and /usaiHM, 
Whence we learn that the property of non-elaliic fluids 
obtains likewife with regard to elaflic fluids. 
Inftead of taking the fame fluid, we may employ another 
whofe denfity is ^; and, by proceeding in a fimilar manner, 
get the following general equation for the prefl'ure : 
M _ d 
p~~a g m-~ag^n - - (III.) 
Many other equations might be deduced with equal faci¬ 
lity ; but tbofe here given are among the molt ufeful. 
Prop. XXXVI. If an elaflic fluid be quiefeent, and com - 
pofed of particles equally repulflve at equal di/lances, and at un¬ 
equal defiances, repelling each, other according to any law of the 
difiance, its denfity will be uniform .—For, it the diflances of 
any two particles from an intermediate particle be un¬ 
equal, their repulfive forces mull be unequal, and, of con- 
fequence, motion mull enfue ; which is contrary to the 
hypothefls; therefore the fluid muff have its particles at 
equal diftances, or be uniformly denfe. 
Cor. If any portion of an elaflic fluid be uniformly denfe, 
and equally comprefled on all tides, it mud be quiefeent. 
Prop. XXXVII. If the compcment particles of an uniform cu¬ 
bical mafs of fluid repel cadi other, with forces varying according 
to any inverfe or diretl ratio of their diflances (lefs than the diredl 
duplicate) the fluid will be elaflic .—For the whole repulfive 
force of any l'urface of the fluid is as the number of par¬ 
ticles in that furface, and the force of each, or as the num¬ 
ber in L the length of that furface, into the number in B 
the breadth, into the force of each particle; or, fubftitut- 
ing R for the whole repulfive force of the furface, I for the 
interval or diftance between two contiguous particles, and 
F for the fpree with which they repel each other, R will 
F 
vary as —< Hence, if F vary in any inverfe ratio, or any 
J 1 2 
direft ratio lefs than the duplicate ratio of I, R will vary 
in fome inverfe ratio of I; which is a neceflary condition 
of elaflic fluids. 
Prop. XXXVIII. If the particles of an elaflic fluid repel 
each other with forces varying inverfely, as the n lh power of 
their diflances, that is, as —, and the compr effing force C upon 
any furface be equal to its whole repulfive force R, then will C 
n ~Y z 
vary as that power of the denfity D whofe exponent is —-—.— 
For, let a portion of the fluid be contained in a given cubic 
fpace, one of whofe faces is the reftangle of LxB, the 
comprefling force being applied to that furface. Now, 
the number of particles in the given fquare furface is 
as ; and, by hypothefls, the force F, with which two 
particles repel each other, is as -i; 5 therefore the elaflic 
force 
