tinuous at the surface; viz. distinguishing by and dn values just outside, and just inside, respectively, we have But at an internal point we have (Thomson and Tait, Art. 522), a, b, c being the semiaxes of the ellipsoid. Hence So (a2 + x) * (b* + x) * (c* + x) * dr (7), Since X of course satisfies (1), and has its derivatives zero at infinity, it is plain that all the conditions of the question are satisfied by The value of X at an external point (x, y, z) is (Thomson and Tait, l. c.), X= 2πabcx √x * where the lower limit is the positive root of .(9), where the values of G, H, Y, Z may be written down from (7) and (9) by symmetry. by formulæ of the same type as (6) and (8). Since the ratio of the expression last written to ny - mz is b2 (4π −H+G) — c2 (4π — G + H) b2 - c2 a constant. Hence all the conditions of the problem are satisfied The values of X2, X, may be written down from symmetry. The student may, as an exercise, prove the equivalence of the above formulæ, in the case where one of the axes of the ellipsoid is infinite, to those of Arts. 88 (d) and 89 (c). Example 4. In the particular case of a sphere we have a = b = c. We then find with similar formulæ for 2, 4,. The values of X1, X2, X, are zero, as is obvious a priori. The solution in this case may however be obtained ab initio by a simpler analysis, as follows*. Let OX be the direction of motion of the centre O of the sphere at any instant, and its velocity. Let P be any point of the fluid, and let OP=r, and the angle POX = 0. It is evident that will be a function of r, only; and we know from the theory of Spherical Harmonics that any such function which satisfies (1) and has its derivatives zero at infinity can be expanded in a series of the form Q1L Q2 + &c., n $ = const. + + + r where Q is the 'zonal harmonic' of order n, multiplied by an arbitrary constant. The condition which has to satisfy at the surface of the sphere is аф V cos 0, It is easy to verify the fact that this value of really satisfies all the conditions of the problem. If we impress on the whole system-the moving sphere and the fluid-a velocity - V, in the direction OX, we have the case of a uniform stream of velocity V flowing past a fixed spherical obstacle. The velocity-potential is then got by adding the term - Vr cos to (12), so that we now have * This solution, generally attributed by continental writers to Dirichlet (Monatsberichte der Berl. Akad. 1852), was given by Stokes, Camb. Trans. Vol. VIII. (1843). 103. A method similar to that of Art. 88 has been employed by Rankine to discover forms of solids of revolution (about x say) for which 4, is known. When such a solid moves parallel to its axis the motion generated in the fluid takes place in a series of planes through that axis and is the same in each such plane. In all cases of motion of this kind there exists a stream-function analogous to that of Chapter IV. If we take in any plane through Ox two points A and P, A fixed and P variable, and consider the annular surface generated by the revolution about x of any line AP, it is plain that the quantity of fluid which in unit time crosses this surface is a function of the position of P, i.e. it is a function of x and w, where a denotes the distance of P from Ox. Let this function be denoted by 2πy. The curves const. are evidently stream-lines, so that may be called the 'stream-function.' If P' be a point infinitely close to P in the above-mentioned plane, we have from the definition of = and thence, taking PP' parallel, first to, then to x, where u, v are the components of fluid velocity parallel to x and ☎ respectively. For the case of the sphere, treated in Art. 102, we readily find, by comparison of (12) and (13) So far we have not assumed the motion to be irrotational. The condition that it should be so, is This may be derived by transformation from (1), by writing and remembering that is now independent of e, or by repeating the investigation of Art. 12, taking, instead of the elementary volume dady dz there considered, the annular space generated by the revolution about x of the rectangle dada. It appears that and are not, as they were in Chapter IV., interchangeable. х 104. Rankine's procedure is then as follows. Supposing the solid to move parallel to its axis with velocity V, we have at all points of a section of its surface made by a plane through Ox, If in this equation we substitute any value of satisfying (15), we obtain the equation of the meridional section of a series of solids of revolution, any one of which would when moving parallel to its axis produce the system of stream-lines corresponding to the assumed value of y. In this way may be verified the value (14) of for the case of a sphere. Dynamical Investigations. 105. The second part of the problem proposed in Art. 99 is the determination of the effect of the fluid pressure on the |
