whence E, and all following quantities are = 0, and བ This, it will be remembered, is on the supposition that we confine ourselves to the divergent wave, and consider G as a function of v or at- r only; if, to complete the solution, we desire also to introduce the convergent wave, we must introduce H a function of u or at +r only, and the sign of its differential coefficient must be changed. y 2 zR dR + x velocity in y+x velocity in 2= +2 dr The displacement in the direction of r From this it follows, exactly as in Article 47, that the particles originally in a spherical surface whose center is the center of divergence will always be in a spherical surface of the same diameter with oscillating center. And the displacement of the center of the dR B+r in the spherical surface at the time t is R+r direction of z. The elastic pressure of the air at any point (Arti 52. Application of this theory to an oscillating pendulum with spherical bob; first, symbolical integration for the pressure on the whole surface. A spherical solid body moving in the direction of z will have for its coating of air one of the spherical sur faces of which we have spoken, and from it the waves will diverge according to the law found in the last Article; the necessary condition being, that the displacement of center of that air-coating must be the same as the displacement of the center of the solid sphere, or that the function R must be so determined that, when applied to the sphere of the same radius as the solid sphere, B+rd must be equal to the displacement of the center of the solid sphere. Now let us examine the value of the elastic pressure upon the surface of the solid sphere. First, we will examine the pressure upon any spherical surface whose radius is r, and whose center was originally at the origin of co-ordinates. Conceive the surface of the sphere divided into annuli by planes parallel to xy; let two of these planes be at distances from the center of the solid sphere z and z+dz. (Thisz and 8z are not exactly the same as the original z and Sz, because the particles of air have a motion in each spherical surface; but the difference depends on the first order of displacements; and its effect on the variable part of the elastic force, which itself is of the first order, will be of the second order, and may be neglected.) If we put for √(x2 + y3), so that ¿3 + z′ = r3, which for this investigation is constant, and +8 will be the radii of the circular intersections by the two planes. The resolved part of the elastic force which retards the motion of the sphere in the direction of z is = elastic force x area included between circles of diameters & and +8; but as, with positive and increasing z, diminishes, we must say that the resolved part of the elastic force which accelerates the motion of the sphere elastic force x 2π8=- elastic force x 2πżdż = We arrive at the same expression if we confine our attention to that side of the sphere where is negative. Therefore, to find the total pressure acting to accelerate the sphere in the direction of z, we must integrate the quantity with respect to z, from z=-r to z=+r. The first term produces 0. The second term produces Now the ordinate Q of the center of the sphere is or, as R= dG 1 G du, where G is a function of v or dv 53. Pendulum investigation continued; determination of the form of the function, and evaluation of the entire pressure. In the case of a vibrating pendulum, whose bob is a sphere of radius p, the ordinate Q, of the center of the bob or of the atmospheric sphere whose radius is p, will move according to this law, When p is put for r, Q must = b. sin ct. Therefore the function G must be so taken that dQ dt' as expressed above, will, when P is put for r, assume the shape bc. cos ct. There is no hope of doing |