The integral of product of two Surface Spherical Harmonics Ym of different de grees taken over the surface of the unit sphere is equal to zero.
The integral over the surface of the unit sphere, of the product of a Surface Spherical Harmonic by a Laplacian of the same degree, 47r 2m by the value the Spherical Harmonic assumes at the Pole of the Laplacian. These theorems enable us to solve many problems in the theory of Gravitation and the theory of Electrostatics by direct integration. Bessel's Functions.—A Bessel's Function or Surface Cylindrical Harmonic of the nth order Jn(x) may be defined as the coefficient of sn in f.
the development of zi into an ing Power Series in .7. It can be shown that V= cosh (liz) (A cos nO + B sirrn0).1X(iir) and V= sin!' (ps) (A cos nii$ + B sin ali)./.(zr), whereµ is any constant, are solutions of La place's Equation in Cylindrical Co-ordinates a I av - — art r ar ass' O. (IV)The Bessel's Functions most used are J,(x) and .11(x), which are appropriate when the problem has axial symmetry about the Axis of Z.
and is convergent for all values of x.
— • Important properties are given by the for mulas fx x./o(x)dx x. i(x)
0 and f x x ..10(x) I'dx = len J o(x) + and the following formulas for development in Cylindrical Harmonic Series, the development holding good for values of r between o and a. f (r) = where Pk is a root of the transcendental equation is p, o(Pa) = 0, or of Ji(lia) = 0, or of isa-TI(na) — o(pa) = 0, and A 12+ r f(r).10(ite)dr.
For the important case where f (r) 1.
2 A 2=-- ..11(p04).
Pra[f 19 As an example in the use of Bessel's Func tions let us find the stationary temperature of any point (r, z) in a homogeneous cylinder of radius a and altitude b if the convex surface and one base are kept at the temperature zero and the other base at the temperature 1.
Here we seek a solution V of equation (1V) which reduces to zero when z=0, and when r = a, and to 1 when z = b. By the aid of the formulas above this is easily formed and is V— 2 sinh (Ms) ..G.Ostr) sinh (0) 2 sinh (pa) Isla .1 i(1120) sink Jo (p•r) 2 sinh (ths) + -roOnr) ...
lila .1 i(Pla) sink 04) If numerical results are desired, tables for J,(x) and Mx) are needed. Such tables have been computed and are accessible. We give here a small three-place one.