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where x, y, z are rectangular co-ordinates of the element 8S referred to the centre as origin, and a, b, c, f, g, h are constants. Find the value of the potential at any point, whether internal or external.
14. If the radius of a sphere be r, and its law of density be p= ax + by + cz, where the origin is at the centre, prove that its
47705 potential at an external point ($, n, $) is (aš + bn + cf) where
15 R R is the distance of ($, n, S) from the origin.
15. Let a spherical portion of an infinite quiescent liquid be separated from the liquid round it by an infinitely thin flexible membrane, and let this membrane be suddenly set in motion, every part of it in the direction of the radius and with velocity equal to S, a harmonic function of position on the surface. Find the velocity produced at any external or internal point of the liquid. State the corresponding proposition in the theory of Attraction.
16. Two circular rings of fine wire, whose masses are M and M, and radii a and a', are placed with their centres at distances b, b', from the origin. The lines joining the origin with the centres are perpendicular to the planes of the rings, and are inclined to one another at an angle 0. Shew that the potential of the one ring on the other is
Cimti n (n − 1) where Bu=6"
4.4 and B', and Q, are the same functions of b' and a' and of cos 0 and
sin @ respectively, and c is the greater of the two quantities Na + b* and Va + b^?.
17. A uniform circular wire, of radius a, charged with electricity of line-density e, surrounds an uninsulated concentric spherical conductor of radius c; prove that the electrical density at any point of the surface of the conductor is
2.4 a the pole of the plane of the wire being the pole of the harmonics.
18. Of two spherical conductors, one entirely surrounds the other. The inner has a given potential, the outer is at the potential zero. The distance between their centres being so small that its square may be neglected, shew how to find the potential at any point between the spheres,
19. If the equation of the bounding surface of a homogeneous spheroid of ellipticity e be of the form
prove that the potential at any external point will be
where C and A are the equatoreal and polar moments of inertia of the body,
Hence prove that V will have the same value if the spheroid be heterogeneous, the surfaces of equal density differing from spheres by a harmonic of the second order.
20. The equation R=a (1 + ay) is that of the bounding surface of a homogeneous body, density unity, differing slightly in form and magnitude from a sphere of radius a; a is small quantity the powers of which above the second may be neglected; and y is a function of two co-ordinate angles, such that = y=Y,+Y, + ... + Y.+ ..., yo =2, +2, + ... +2....
= where Yo, Y, ... 2, Z,... are Laplace's 'functions. Prove that the potential of the body's attraction on an external particle, the distance of which from the origin of co-ordinates is r, is given by the equation
21. If M be the mass of a uniform hemispherical shell of radius c, prove that its potential, at any point distant r from the centre, will be
according as r is less or greater than c; the vertex of the hemisphere being at the point at which u = 1,
22. A solid is bounded by the plane of my, and extends to infinity in all directions on the positive side of that plane. Every point within the circle ac* + y = a*, = 0 is maintained at the uniform temperature unity, and every point of the plane xy without this circle at the uniform temperature 0. Prove that, when the temperature of the solid has become permanent, its value at a point distant r from the origin, and the line joining which to the origin is inclined at an angle o to the axis of z will be
Po-P, + Port +...
-(-1)1.3...(21 – 1)
23. Prove that the potential of a circular ring of radius c, whose density at any point is cos my, cy being the distance of the point measured along the ring from some fixed point, is
1 d" P. C" 21 cos mo (sin 6)" 2.4.6... 2m. dum ponti
d" P. 2.4.6... (2m + 2) de 1.3.5...(2k-1) d" P.. 2.4.6...2(m + k du"
**+2+1 where r is greater than c. If y be less than c, n and c must be interchanged.
m + 2
24. A solid is bounded by two confocal ellipsoidal surfaces, and its density at any point P varies as the square on the perpendicular from the centre on the tangent plane to the confocal ellipsoid passing through P. Prove that the resultant attraction of such a solid on any point external to it or forming a part of its mass is in the direction of the normal to the confocal ellipsoid passing through that point, and that the solid exercises no attraction on a point within its inner surface.
CAMBRIDGE : PRINTED BY C. J. CLAY, 2.A. AT THE UNIVERSITY PRESS.