Practical conclusions ances of sca may be remarked that, as the disturbances are supposed to l as to disturb- small, we may superimpose such as we have now described, « level, and any other small disturbances, as, for instance, on the general direction of oblateness of the earth's figure, with which we shall be occupied amount and gravity. Determinateness of potential through its value over every point of a surface. presently. Practically, then, as the density of the upper crust is somewhere about the earth's mean density, we may say that th effect on the level surface, due to a set of parallel mountainchains and valleys, is, of the general character explained in § 791, but of half the amounts there stated. Thus, for instance. a set of several broad mountain-chains and valleys twenty nautical miles from crest to crest, or hollow to hollow, and of several times twenty miles extent along the crests and hollows, and 7,200 feet vertical height from hollow to crest, wou'd raise and lower the level by 2 feet above and below what it would be were the surface levelled by removing the elevated matter and filling the valleys with it. 793. Green's theorem [App. A. (e)]1 and Gauss's theorem (§ 497) show that if the potential of any distribution of matter. space from attracting according to the Newtonian law, be given for every point of a surface completely enclosing this matter, the poten tial, and therefore also the force, is determined throughout all space external to the bounding surface of the matter, whether this surface consist of any number of isolated closed surfaces, each simply continuous, or of a single one. It need scarcely be said that no general solution of the problem has been obtained. But further, even in cases in which the potential has been fully determined for the space outside the surface over which it is given, mathematical analysis has hitherto failed to determine it through the whole space between this surface and the attracting mass within it. We hope to return, in later volumes, to the grand problem suggested by Gauss's theore of § 497. Meantime, we restrict ourselves to questions practically useful for physical geography. Example (1.)-Let the enclosing surface be spherical, of radi as a; and let F(0, 4) be the given potential at any point of it, 1 First apply Green's theorem to the surface over which the potential is given Then Gauss's theorem shows that there cannot be two distributions of peterta agreeing through all space external to this surface, but differing for any part of the space between it and the bounding surface of the matter. specified in the usual manner by the polar co-ordinates 0, 4. Green's solution [§ 499 (3) and App. B. (46)] of his problem for the spherical surface is immediately applicable to part of our present problem, and gives 1 (r2—a2) F(0', ')r sin 'de'd' 4zalo lo 0 Απα 0 0 {r2-2ar[cos 0 cos 0' + sin ✪ sin 'cos (p—')]+a2} { for the potential at any point (r, 0, 4) external to the spherical surface. But inasmuch as Laplace's equation V2u=0 is satisfied through the whole internal space as well as the whole external space by the expression (46) of App. B., and in our present problem V'V=0 is only satisfied [§ 491 (c)] for that part of the internal space which is not occupied by matter, the expression (3) gives the solution for the exterior space only. When F(0, 4) is such that an expression can be found for the definite integral in finite terms, this expression is necessarily the solution of our problem through all space exterior to the actual attracting body. Or when F(0, 4) is such that the definite integral, (3), can be transformed into some definite integral which varies continuously across the whole or across some part of the spherical surface, this other integral will carry the solution through some part of the interior space: that is, through as much of it as can be reached without discontinuity (infinite elements) of the integral, and without meeting any part of the actual attracting mass. To this subject we hope to return later in connexion with Gauss's theorem (§ 497); but for our present purpose it is convenient to expand (3) in ascending powers of as before in App. B. (s). The result [App. B. (51)] is α a r -)*F1(0, V=—F.(0, 4) + (—)*F,(0, $) + r where F(0, 4), F1(0, 4), etc., are the successive terms of the expansion [App. B. (52)] of F(0, 4) in spherical surface harmonics; the general term being given by the formula where Q is the function of (0, 4) (0', ') expressed by App. B. (61). In any case in which the actual attracting matter lies all within an interior concentric spherical surface of radius a', the harmonic expansion of F(0, 4) must be at least as convergent as the geometrical series (3) Determination of potential from its value over a spherical surface en closing the mass. Determina tion of potential from the form of an approximately spherical equipotential surface round the mass. and therefore (3 bis) will be convergent for every value of r Example (2.)-Let the attracting mass be approximately centrobaric (§ 526), and let one equipotential surface complety enclosing it be given. It is required to find the distribution force and potential through all space external to the smallest spherical surface that can be drawn round it from its centred! gravity as centre. Let a be an approximate or mean rad.us. and, taking the origin of co-ordinates exactly coincident with the centre of inertia (§ 230), let be the polar equation of the surface; F being for all values å Φ). tials (§ 486) between them is MF(0, 4). And if a be the proper a mean radius, the constant value of the potential at the given surface (5) will be precisely Hence, to a degree of approxi M mation consistent with neglecting squares of F(0, 4), the potential at the point (a, 0, 4) will be Hence the problem is reduced to that of the previous example and remarking that the part of its solution depending on the term M α M of (6) is of course simply we have, by (3 bis), for the potential now required, 1 a r U=M{~+~F,(0,0)+—F,(0, = M{ — + F,(0, $) + —~ ~ F, (0, 4)+etc. }; r where F is given by (4). F, is zero in virtue of a being the If further be chosen in a proper mean position, that is to say, Determinasuch that ffQ,F(0, 4) sin 0d0dp=0 tion of from the (9) potential F, vanishes and [§ 539 (12)] O is the centre of gravity of the form of an attracting mass; and the harmonic expansion of F(0, 4) becomes approxiF(0, 4)=F,(0, 4)+F,(0, 4)+F,(0, 4)+etc. mately equipoten round the (10). spherical If a' be the radius of the smallest spherical surface having O for tial surface centre and enclosing the whole of the actual mass, the series (7) necessarily converges for all values of 0 and 4, at least as rapidly as the geometrical series. for every value of r exceeding a'. Hence (7) expresses the a a a mass. force. The direction and magnitude of the resultant force are of Resultant course [SS 486, 491] deducible immediately from (7) throughout the space through which this expression is applicable, that is all space through which it converges that can be reached from the given surface without passing through any part of the actual attracting mass. It is important to remark that as the resultant force deviates from the radial direction by angles of the same order of small quantities as F(0, 4), its magnitude will differ from the radial component by small quantities of the same order as the square of this: and therefore, consistently with our degree of approximation, if R denote the magnitude of the resultant force dU M ~{{ 1 + 3 ( ~—- ) ̊F. (0, 4) + 4 (—~—-) ̊F.(0,4)+etc.} (12). R= dr = a For the resultant force at any point of the spherical surface agreeing most nearly with the given surface we put in this formula r=a, and find M {1+3F,(0,0)+4F,(0,0)+etc.} (13). And at the point (r, 9, 4) of the given surface we have r=a M {1—2F(0, 4)} = = = {(1—2[F,(0,0)+etc.]} (14), a2 Resultant force at any point of approximately spherical level surface, for gravity alone. and we find for the normal resultant force at the point ( M ,{1+F,(0,6)+2F,(0, 4)+3F,(0, 4)+.....} (15) Taking for simplicity one term, Fi, alone, in the expansion of F, and considering, by aid of App. B. (38), (40), (p), and §§ 779...784, the character of spherical surface harmonies, we see that the maximum deviation of the normal to the surface r=a{1+F;(0, 0)} from the radial direction is, in circular measure (§ 404), just i times the half range from minimum to maximum in the values of F(0, 4) for all harmonics of the second order (case i=2), and for all sectorial harmonics (§ 781) of every order; and that it is approximately so for the equatorial regions of all zotal harmonics of very high degree. Also, for harmonies of high degree contiguous maxima and minima are approximately equal We conclude that 794. If a level surface (§ 487), enclosing a mass attracting according to the Newtonian law, deviate from an approximately spherical figure by a pure harmonic undulation (§ 779) of order i; the amount of the force of gravity at any point of it will exceed the mean amount by i-1 times the very small fraction by which the distance of that point of it from the centre exceris the mean radius. The maximum inclination of the resultant force to the true radial direction, reckoned in fraction of the unit angle 570-3 (§ 404) is, for harmonic deviations of the second order, equal to the ratio which the whole range from minimum to maximum bears to the mean magnitude. For the class described above under the designation of sector al harmonics, of whatever degree, i, the maximum deviation in direction bears to the proportionate deviation in magnitude from the mean magnitude, exactly the ratio i÷(i-1); and approximately the ratio of equality for zonal harmonics of hig degrees. ω Example (3.)—The attracting mass being still approximately centrobaric, let it rotate with angular velocity & round (Z, and let one of the level surfaces (§ 487) completely enclosing it be expressed by (5), § 793. The potential of centrifugal force (§§ 800, 813), will be w(x+y), or, in solid spherical har monies, {w2r2+fw2(x2+y3 — 2z3). |