Page:Encyclopædia Britannica, Ninth Edition, v. 11.djvu/623

and this tried in (9) gives

(35).

${\frac {1}{f(t)}}{\frac {df(t)}{dt}}=\kappa {\frac {{\frac {d^{F}}{dx^{2}}}+{\frac {d^{2}F}{dy^{2}}}+{\frac {d^{2}F}{dz^{2}}}}{F}}$

The first member being independent of x, y, 2, and the second being independent of t, the common value of the two must be independent of x, y, z, t, that is to say, must be an absolute constant. Let it be denoted by - f ; we have

(36),

$f(t)=C_{\epsilon }-\rho t$

(37),

${\text{and}}\ {\frac {d^{2}F}{dx^{2}}}+{\frac {d^{2}F}{dy^{2}}}+{\frac {d^{2}F}{dz^{2}}}+{\frac {\rho }{\kappa }}F=0$

or in terms of polar coordinates, by (6),

$\left.{\begin{aligned}&{\frac {d^{2}U}{dr^{2}}}+{\frac {1}{r^{2}\sin \theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {dU}{d\theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {d^{2}U}{d\phi ^{2}}}+{\frac {\rho }{\kappa }}U=0\\{\text{where}}&\qquad U={\frac {F}{r}}\end{aligned}}\right\}(38).$

Of this we have a spherical harmonic solution,

(39),

$U=[\mathrm {A} \phi _{i}(r)+\mathrm {B} \Psi _{i}(r)]S_{i}$

where S; denotes a spherical surface-harmonic of order i, and <pi(r), 4>;(r), two particular solutions of the equation

(40),

${\frac {d^{2}u}{dr^{2}}}+\left[{\frac {\rho }{\kappa }}-{\frac {i(i+1)}{r^{2}}}\right]u=0$

Then (36) and (35) give finally

(41),

$\nu =\epsilon ^{-\rho t}{\frac {\mathrm {A} \phi _{i}(r)+\mathrm {B} \psi _{i}(r)}{I}}s_{i}$

This solution is in its generality applicable to an infinite solid occupying all space except a hollow round the origin. The solid may of course be bounded externally also by a finite closed surface. If there be no hollow, A/B must fulfil the condition that [A</>;(r) + B<J/,-(r)]/r is finite when r=0. If there are two bound aries, concentric spherical surfaces, with their common centre at the origin of coordinates, the boundary condition obviously requires uniform emissivity over each, but not necessarily equal for the two. If the two emissivities be denoted by h and h , and the radii of the surfaces by a (outer) and a (inner), the boundary conditions are

$\left.{\begin{aligned}&{\frac {dv}{dr}}=-hv,\ {\text{when}}\ r=\alpha \\and\quad &{\frac {dv}{dr}}=h'v,\ {\text{when}}\ r=\alpha '\end{aligned}}\right\}(42).$

From these we may find h and h , so as to let the harmonic character of the solution be fulfilled in the subsidence. Or if h and h be given, we have in (41) two equations which determine the two unknown quantities A/B and p. Eliminating A/B, we thus find a single transcendental equation for p, which is proved to have no imaginary or negative roots, and an infinite number of real positive roots, each s-/rt(t + l)/r 2 . In the case of i = 0, or temperature independent of </> and 6, (40) gives

$u=\mathrm {A} \cos r\surd {\frac {\rho }{\kappa }}+\mathrm {B} \sin r\surd {\frac {\rho }{\kappa }}$.

For this case the transcendental equation for determining values for p is very simple, and its roots are calculated numerically with great ease. With the further restriction of no central hollow, we must have A = 0, so that u/r may be finite when r = 0. This case was fully investigated by Fourier, and very beautifully worked out in his fifth chapter. The more general problem of a solid sphere, with any given initial distribution of temperature, without the restriction of temperature independent of and </>, was solved first we believe by Poisson in the 11th chapter of his Theorie Mathematique de la Chaleur, in terms of the formula? (36), (38), (40) above. XVII. The equation of the transference of heat in terms of columnar coordinates, (7) above, affords naturally another beautiful case of harmonic solution. Assume

(43);

$\upsilon =\epsilon ^{-\rho t}u_{i}\ {\overset {\sin }{\underset {\cos }{}}}\ i\phi \ {\overset {\sin }{\underset {\cos }{}}}\ mz$

we find by (7)

(44);

${\frac {d^{2}u_{i}}{dr^{2}}}+{\frac {1}{r}}+\left({\frac {\rho }{\kappa }}-m^{2}-{\frac {i^{2}}{r^{2}}}\right)u_{i}=0$

The treatment of this equation and its integral (obviously derivable by i differentiations from u w which is a Bessel s function) for the full solution of the thermal problem is most interesting, and very instructive and suggestive in respect to pure analysis. It was splendidly worked out for the case of ??i = and i = G by Fourier in his 6th chapter, "The Motion of Heat in a Solid Cylinder," truly a masterpiece of art. When it was printed in 1821, and published after having with the rest of Fourier s work been buried alive for fourteen years in the archives of the French Academy, and when Bessel found in it so thorough an investigation and so strikingly beautiful an application of the " Besselsche Function," we can imagine the ordinary feeling towards those " qui ante nos nostra dixeruut " reversed into the pleasure of genuine admiration.

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HEATH, the English form of a name given in most Teutonic dialects to the common ling or heather, but now applied to all species of Erica, an extensive genus of mono- petal ous plants, belonging to the order Ericaceae. The heaths are evergreen shrubs, with small narrow leaves, in whorls usually set rather thickly on the shoots ; the per sistent flowers have 4 sepals, and a 4-cleft campanulate or tabular corolla, in many species more or less ventricose or inflated ; the dry capsule is 4-celled, and opens, in the true Ericce, in 4 segments, to the middle of which the partitions adhere, though in the ling the valves separate at the dissepiments. The plants are mostly of low growth, but several African kinds reach the size of large bushes, and a Spanish variety, E. arborea, occasionally attains almost the aspect and dimensions of a tree.

FIG. 1. Erica cinerea. FIG. 2. Calluna vulgaris.

One of the best known and most interesting of the family

is the common heath, heather, or ling, Calluna vulgaris, placed by most botanists in a separate genus on account of the peculiar dehiscence of the fruit, and from the coloured calyx, which extends beyond the corolla, having a whorl of sepal-like bracts beneath. This shrub derives some eco nomic importance from its forming the chief vegetation on many of those extensive wastes that occupy so large a portion of the more sterile lands of northern and western Europe, the usually desolate appearance of which is enlivened in the latter part of summer by its abundant pink blossoms. When growing erect to the height of a yard or more, as it often does in sheltered places, its purple stems, close-leaved green shoots, and feathery spikes of bell-shaped flowers render it one of the handsomest of the heaths ; but on the bleaker elevations and more arid slopes it frequently rises only a few inches above the ground. In all moorland countries the ling is applied to many rural purposes ; the larger stems are made into brooms, the shorter tied up into bundles that serve as brushes, while the long trailing shoots are woven into baskets. Pared up with the peat about its roots it forms a good fuel, often the only one obtainable on the drier moors. The shielings of the Scotch Highlanders were formerly constructed of heath

stems, cemented together with peat mud, worked into a