442 HYDROMECHANICS [HYDROSTATICS, If we neglect changes of temperature, then, the density at any point being a function of the pressure, it follows that surfaces of equal pressure are also surfaces of equal density, and therefore -j- , JL dP , L ^ are the partial differential coefficients with respect p dy p dz to x,y,z of some function P of x,y,z, such that P=f c -; and there-
- J P
fore also X,Y,Z must be the partial differential coefficients of some function - V, called the potential, such that the force in any direction is rate of diminution of V in that direction , and the con ditions of equilibrium - = X, &c., are equivalent to p ax dP dV dP dV _ dP d~V _ -; I ; = ~~; 1 ; * * i 1 7 ^ J dx dx dy dy dz dz or P + V = constant . If the temperature be variable, then the surfaces of equal pressure and density need not be coincident ; but, since the pressure is a function of the density and temperature only, it follows that the surfaces of equal temperature and of equal density must intersect in curves which lie on surfaces of equal pressure. As an example of the use of the general equations, take the simplest case of a homogeneous liquid at rest under gravity ; then, the axis of z being directed vertically downwards, the equations become A^ = 1-^ = ^ = <7 p dx p dy p dz and therefore p = w + gpz, tt being the pressure at the level of the origin. We are here employing a new unit of force the absolute unit which is defined as the force which causes unit acceleration in the unit of mass. With the same units of length, time, and mass, the gravitation unit of force is g times the absolute unit offeree ; for instance, in the equations p = pz, andp = gpz, ihep in the first equa tion is measured in gravitation units of force per unit of area, and in the second equation in absolute units of force per unit of area. Again, suppose the density to vary as any power of the depth, and put dp -gp-9l* , g n+l then and If the fluid be elastic, and the temperature uniform, p = kp, and 1 dp _ k dp _ ~p ~dz~~p dz~y and integrating, Consequently, as we go up in the air, if the temperature be uniform, as the heights increase in arithmetical progression, the pressures and densities diminish in geometrical progression. If H denote the pressure height, then p-kp-gpR, and therefore H = ; 9 and if p^ p. 2 denote the pressures at levels z^, z 2 , then Pj_ _ C ~H~~ K~ -Hx 2-3026 Iog 10 -. P-2 For dry air at C., taking H = 26,000 feet, ^-2., = 60, 000 logy,^- nearly. P-2 (Maxwell, Heat, chap, xiv.) The Figure of the Earth. Suppose a fluid mass arranged under the gravitation of its parts in concentric spherical strata, the density increasing towards the centre for stability of equilibrium, and p being the density at a distance k from the centre. If we suppose this mass to be rotating without relative motion of its parts about an axis, and to be slightly disturbed in consequence of the rotation from the spherical arrange ment of the strata, we can gain an idea of the figure of the earth on the hypothesis of original fluidity. If <a be the angular velocity, we must suppose a disturbing function, whose potential is ^o> V 2 sin "6, added to the gravitation potential, 6 being the angular distance from the axis of revolution and of figure. Denoting the zonal surface harmonic of the second degree f,u 2 - J, where /z= cos 6, by Q 2 , the disturbing function may be written ^o>V 2 (l - Q 2 ) ; and we shall assume in consequence that the disturbance of each stratum from the spherical form is also a zonal harmonic of the second degree, so that when disturbed the equation of a stratum will be r- &(l-feQ 2 ); and e, which is the ratio of the difference of the equatoi ial and polar axes to the mean axis, is called the ellipticity of the stratum. The gravitation potential of a homogeneous spheroid of density p, and bounded by r-JKl-ifQ,), is the same as that of a homogeneous sphere of radius k and density p, and of a distribution of matter on the sphere of radius k, of surface density - feQ.,p, neglecting e 2 . Therefore, for an internal point the potential is 2irpF - 7rpr 2 - T Vrper 2 Q., ; and for an external point the potential is Therefore, for the shell of density p, enclosed by the stratum r=Jk(l-eQ 2 ) and the consecutive stratum, the potential dJ = kirpkdk - T^irp-fT^Qa^ UK for an internal point, and for an external point ; and, therefore, for any point in the interior of the whole mass on the stratum u = M denoting the mass enclosed by the stratum of mean radius k, and K being the mean radius of the exterior stratum. Neglecting t", /K U = 47T / pkdk - frlTKrl Jk k r d* r* d . The equation of equilibrium is, since the force in any direction is the rate of increase of the gravitation potential, ^ r and, supposing surfaces of equal pressure to be also surfaces of equal density, we must have to our order of approximation U + Jjco 2 /b 2 (l - Q.J = constant, over a surface of equal density, or, equating to zero the coefficient ofQ,, Dividing by k~, and differentiating with respect to Tc, o> 2 disappears, and we obtain and, differentiating again with respect to k, due a differential equation of the second order to determine Me, and therefore e, provided we know what function p and therefore M is ofk. Properly speaking, from the elasticities of the substances of the various strata we should know the relation between the pressure and the density, and then from the conditions of equilibrium of the strata when undisturbed and in this spherical shape we could de
termine what function p is of A", the pressure and density at a
