725 
MECHANICS. 
force of the fluid, and of confequence the compreffl.ve 
1 
force C, is as —x — or as I"* 2 . But the denfity of the 
I" l 2 
fluid contained in the given cubical fpace is inverfely as 
the cube of the diftance between the centres of the par¬ 
ticles; that is, Da-, and I OC-^7 5 whence, by fubfii- 
t 3 Ds 
h of the mercury in 
jj — hcl — } 3 6o ° x 
tuting D~‘s for I, in the expreflion C CX I" F 2 , we have 
«+a_ 
c a d 3 . 
Cor. 1. Converfely, if D 3 vary as C, the repulfive 
-For, the 
»F 2 
quantity of matter being given, D a --^Ot — and D 3 
force of each particle, or F, mult vary as 
1“ 
1 1 
will vary as I"+ 2 ; but F varies as C divided by the num- 
ber of particles in L 2 , or as Cx I 2 , or CX ^ 3 X I 2 , or 
a I 2 
+2 I 
I or a - • 
Cor. 2. Flence again we fee, flnce 72 + 2 muff be always 
pofitive to make C pofitive, that n mint be either lb me 
whole pofitive number, ora negative number lefs than 2, in 
orderto conllitutea fluid of particles which repel each other. 
Cor. 3. If water be fuppofed comprefiible in a very fmall 
degree, the particles mull be kept at a diltance by forrje 
repulfive force, while D remains nearly conltant. Now, 
n-1-2 a 
fince C CX D 3 , we (hall have D OC C “ F 2 ; in which, 
3 
that C ’• F 2 may be nearly invariable, n mud be a very 
great number; hence, according to this hypot hefts, the re¬ 
pulfive force of the particles of water varies inverfely in a 
very high power of their diftances. 
Cor. 4. When the denfity of the fluid varies as the force 
2 
■which comprefles it, orDaC, the expreflion CctD 3 
1+2 
becomes CaD 3 and n— 13 whence FOC— becomes 
Far. or the force of each particle is inverfely as the 
interval between two contiguous particles. 
Cor. 5. Hence, becaufe the denfity of the air is nearly 
proportional to the force which comprefles it, its confti- 
tuent particles mu ft repel one another with forces varying 
inverfely as their diftances. 
Cor. 6. The denfity of the air varying as the comprefling 
force, and that perpetually decreafing in afcending the at- 
Hioipjiere, the denfity and elafticity of the air alio perpe¬ 
tually decreafe. 
A homogeneous almofphere is an atmofphere fuppofed to be 
of the fame weight as that which actually furrounds the 
earth ; its denfity being uniform, and every-where equal 
to the denfity of the air at the earth’s furface. 
Prop. XXXIX. To find the altitude of a homogeneous at¬ 
mofphere. —Let H be the height of the homogeneous atmof¬ 
phere, its uniform denfity being D, the fame as the den¬ 
fity of the air prefling upon the mercury in the bafin of 
t.he barometer 5 h the height of the mercury in the ba¬ 
rometer tube, and d the denfity of that fluid ; then 
hd 
(Prop. XXXV.) M =m, or HD=.hd-, whence, H=—• 
Now it appears from experiment, that, when the denlities 
of air and mercury D andd are as if and 13600, the height 
Vol. XIV. No. 1008, 
the barometer is feet. Hence, 
— —27818 feet, =5’268 miles. So that 
D 1 
the height of the homogeneous atmofphere is rather more 
than 5J miles. 
Cor. If it were not for the changes of temperature, the 
height of H of the homogeneous atmofphere would be in¬ 
variable, whatever might be the height of the mercury in 
the barometer. For, if d be conftant, becaufe the fpecinc 
gravity of air varies as D its denfity, and this again as k, 
the height of mercury in the tube, it follows that — isin- 
h d 
variable, and confequently H=— is conftant likewife. 
Prop. XL. Suppofng the force of gravity to vary as the nth 
power of the difiance from the centre of the earth, and the com- 
prefive force to vary as the denfity ; to find the relation between, 
the denfity of the air and the altitude above the furface of the 
earth.- —Let x reprefent the variable diltance from the fur- 
face of the earth, the radius of the earth being unity, d the 
denfity of theairat the diltance x, and H the height of the 
homogeneous atmofphere. Now, fince by hypothefis the 
comprefling force varies as the denfity, the fluxion of the 
former will vary, as the fluxion of the latter; while, at any 
diltance, x the fluxion of the comprefiing force mult vary 
as the force of gravity, the denfity, and the fluxion of the 
altitude, conjointly ; fo that the fluxion of the comprefling 
force will he to that of the denfity in the conftant ratio 
of x”dx to— d, the latter fluxion having the negative fign, 
becaufe the denfity decreafes while the altitude increafes. 
Confequently, lince by the definition of a homogeneous 
atmofphere H will reprefent the comprefling force at the 
furface of the earth, we have H : 1 :: x"dx : — d, whence 
d 1 
x n x— —Hx—; > an <3 —7—=—H .hyp. log. r/ 4 -C. 
d ' n -1 
Now, to correct the fluent, we mult confider that, when 
x—i, d~ 1 ; whence we find C=:—— for the value of the 
72+1 
conltant quantity; and the correft fluent is 
hyp. log. d. 
ttF 1 
, — ,-H. 
n-\-i n-\-i 
liyp. log. d■, which is the 
l - A TJ 
Hence —- u 
n-\-i 
general equation exprelling the relation between the alti¬ 
tude and the denfity. 
Cor. 1. When the force of gravity varies inverfely as the 
. 1—jv'T 1 
fquare of thediftance, ?z=—2,and =H . hyp, log. d, 
becomes - — i=H . hyp. log. d. So that, if x increafe in 
harmonical progreflion, i will decreafe in arithmetical pro- 
greflion ; and confequently hyp. log. d will decreafe in 
arithmetical progreflion. 
Cor. 2. If the force of gravity be fuppofed conftant, 71—0, 
and 1—x=H . hyp, log. d. Confequently, if * increafe 
in arithmetical progreflion, fince 1— x will then decreafe 
in arithmetical progreflion, the hyp. log. of d will decreafe 
in arithmetical progreflion. 
Cor. 3. Since the hyperbolic logarithms are to the common 
logarithms in aconllant ratio, viz. that of 1 to ’43429448, 
&c. it follows that, when x increafes in arithmetical pro¬ 
greflion, the common logarithms of the denlities will de¬ 
creafe in arithmetical progreflion, and the denflties them- 
felves in geometrical progreflion, oil the fuppofition of 
equal gravity. 
Cor. 4. Hence, retaining the fame hypothefis, different 
altitudes above the earth’s furface will vary as the negative 
logarithms of the denflties or weights of air at thofe al¬ 
titudes. So that, if D and d denote the denflties at the 
heights H and /;, fince Hot—log. D, and hQ, —log. d, the 
8 Y differencs 
