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An approximate value of a must be first obtained from the equation

x= {60345+121. (s+T)} {log1h — log10 k − (0,00008) (s −t)}, and this substituted for a in the small terms will give a nearer value of x.

g

The values of, and E are adapted to the mixture of air and watery vapour, constituting the atmosphere in its ordinary state. The vapour of water is lighter than air, under the same pressure, and the quantity of it contained in a given quantity

μ

of air increases with the temperature. Hence and E are g larger than if the atmosphere consisted of perfectly dry air.

37. The pressure of vapour not in contact with the fluid from which it was produced, is found to be inversely proportional to the space it occupies, and its expansion, on being heated, is the same as that of air. If however the temperature, or the volume of a given quantity of vapour, be diminished beyond a certain point, a portion of it will return to the state of a liquid; and then, if the temperature of the vapour be invariable, its volume may be diminished till the whole becomes liquid, without increasing its pressure.

It appears probable from the experiments of Mr. Faraday, that every gas may be made to assume the form of a liquid by diminishing its volume. When the condensation of a gas is carried on nearly to the point at which it begins to liquefy, the ratio of its pressure to its density, at a given temperature is no longer constant. The value of this ratio for dry atmospheric air does not perceptibly change under the pressure of a column of mercury nearly ninety feet high.

SECTION IV.

ON THE EQUILIBRIUM OF FLUIDS ACTED ON BY ANY

FORCES.

ART. 38. To find the pressure at any point in a mass of fluid at rest acted on by any forces.

Let PQ (fig. 21) be the edge of a very small prism of fluid in the interior of a mass of fluid at rest, R the accelerating force at P, S the resolved part of R in the direction PQ. Let the prism become solid; then (Art. 4) it will remain at rest; and since S. (mass prism), and the pressures on its ends are the only forces that act upon it in a direction parallel to PQ, they must be in equilibrium,

.. press. on the end Q-press. on the end P=S. (mass prism).

Let x, y, z; ∞ +dx, y+dy, ≈+dx be the co-ordinates of P, Q referred to rectangular axes Ox, Oy, Ox. Construct a parallelopiped LMN, of which PQ is the diagonal, having its edges PL, PM, PN parallel to Ox, Oy, Ox respectively. Let X, Y, Z be the components of R resolved parallel to Ox, Oy, Ox; the area of the base of the prism; p the density of the fluid; p the pressure at P, and therefore p+d.p. dx +d ̧p.dy+d2p.dx, ultimately the pressure at Q. Then if the sides of the base of the prism be very small compared with its length, pressure on the end Q-pressure on the end P

=к(d ̧p.dx+dyp.dy+d2p.dx).

S=X.cos QPL+ Y. cos QPM + Z. cos QPN,

and the mass of the prism=pk. PQ, .. S. (mass prism)
=pK. PQ. (X. cos QPL + Y. cos QPM + Z .cos QPN)
= pK (X. Sx + Y. Sy + Z. Sz),

··· d ̧p. Sx + d ̧p. Sy + d2p. dx = p(X. Sx + Y. Sy + Z. S≈), and da, dy, dx are independent of each other;

..d ̧p=pX, dyp=pY, d2p=pZ;

if, then, we can find a quantity p, such that

d.p=pX, dyp=pY, d2p=pZ,

p, taken between the proper limits, is the pressure at P.

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ud, log.p=X, ud1logep=Y, μd, log.p=Z,

and if u be a quantity such that du=X, du=Y, du=Z, p=Cer.

40. d,d.p=d, (pX), d2d ̧p=d ̧ (p¥); and d ̧d ̧p=dd ̧p; .. dypX)=d, (pr),

similarly d2 (px)=d, pZ), and d2 (p¥) =dy (pZ). If we perform the differentiations and eliminate p, we obtain

X (d2 Y— d1Z) +Y (d„Z− d2X) + Z (d1X − d2 Y) = 0, this equation expresses the relation that must exist between the forces X, Y, Z, in order that the equilibrium may be possible.

When the density is constant,

d1X=d2Y, d2X=d„Z, d2Y=d1Z.

41. If c be the pressure at any point P in the fluid, p=c, in which is an implicit function of a and y, is the equation to the surface of equal pressure passing through P,

The derived equations of p=c are

d2p + d2p. dr≈=0, dyp+dzp.d1 =0,

x, y, z being considered independent of each other in forming the differential coefficients dp, dyp, d2p;

... X+Z.d2z=0, Y+Zd,x=0;

therefore if a, ß, y be the angles between Ox, Oy, Ox, and the normal to the surface of equal pressure passing through P, at P,

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But

X Y Z
are the cosines of the angles between Ox,
R'R' R

Oy, Ox, and the direction in which the force at P acts. Therefore the force at any point acts in the direction of a normal to the surface of equal pressure at that point.

The equation to the surface of a fluid, is p=0. And if the fluid be contiguous to another fluid with which it does not mix, and which exerts a pressure II at the common surface of the fluids, p=Пl will be the equation to the common surface of the fluids.

42. When p is variable, and a quantity u can be found, such that X=du, Y=d ̧u, Z=du, p must be a function of u.

For

d.p=pdu, dyp=pd1u, d2p=pd2u,

and these equations cannot be satisfied unless p is a function of u.

Let p=fu, then dpfudu, .. pfufu. Hence u and p are functions of p; and when p is constant, u and P must be constant; therefore u=c is the equation to a surface of equal pressure; also p is the same at all points in a surface of equal pressure.

Hence when an elastic fluid of variable temperature is at rest, the temperature is the same at all points in a surface of equal pressure.

=

43. The conditions dyXd,Y, d2X=d2Z, d1Z=d2 Y, are satisfied whenever the forces tend to fixed centres, and the intensity of each force at any point P, is a function of the distance of P from the centre to which the force tends.

For if a, b, c be the co-ordinates of the centre to which one of the forces tends, r its distance from P, or the intensity of the force at P,

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we should have obtained the same expression for d, Y, therefore d1X=d, Y, in like manner d2X=d, Z,

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and d1 Z=d2 Y.

(pr),

d ̧u=2(pr."=")=x, d,u=2 (pr. b) = Y,

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dy

d_u = (pr) = Z.

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44. Each particle of a fluid attracts with a force which vanishes when the distance of the particle from the attracted point is finite; to find the pressure at any point in the interior of the fluid.

Let the plane a Oy (fig. 22) be a tangent to the surface of the fluid touching it in O, Ox perpendicular to Oy, xOz, yOz the planes of greatest and least curvature through any point M in the surface of the fluid; draw NMQ parallel to Ox, meeting x Oy in N.

Let R, S be the radii of curvature of the surface of the fluid in the planes Ox, y0z; NOx =0, NO=p, NM=%,

OP=r, MP=u, NQ=≈, PQ=u1;

pu the attraction of a portion of the fluid whose volume is unity on a point at the distance u, V the attraction of the fluid on P, which manifestly acts in the direction Ox.

The equation to the surface of the fluid is

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and u12 = p2 + (≈1− r)2, u2 = p2 + (x − 1)2 ;

=

u2 = p2+r2 −2rx nearly,

D

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