In the particular case of n = 0, these two are not distinct. Either II. -4-1 = dz1 The other harmonic of the same degree and type is To obtain the harmonics of the same types, but of degree i, III. multiply each of the preceding groups I. and II. by p1, in virtue of (g) (13) below. I. Degree - 1. Generally, in virtue of (g) (13) below, any of the preceding functions of degree zero divided by r; or, in virtue of (g) (15), the differential coefficient of any of them with reference to x, or y, or z. For instance, d dx (the former being of III. 2 degree 1, and the latter being - fdz of VI. degree 0 with n = 1). The Rational Integral Harmonics of Degree 2. I. Five distinct functions, for instance, 2x2x2-y2; x2 - y2; yz; xz; xy. With same notation and same references for proof as above for Examples of spherical harmonics. (v denoting √1, and e and ƒ any real quantities.) Partial 1 or {q€+vƒ [ev{e+vf)$ + €−v{e+vf) $]; 2v qe+vƒ [ev{e+uf)$ − e−v[c+vf)] : qe{ef [cos (flog q − ep) + v sin (ƒ log q − ep)] - + €-[cos (ƒ log q + ep) + v sin (ƒ log q + ep)]}; 3п 2 the same with +ep instead of ep. or III. or ; {r−2e−1q€ [€S$ €v(♬ log q−2ƒ log r−ep) + € ̄S$ €v (ƒlog q−2f log r+ep)]; 4r-te-1qc {cs [cos (ƒlog 2-ex) + v sin (log-e) (b) A spherical surface harmonic is the function of two angular co-ordinates, or spherical surface co-ordinates, which a spherical harmonic becomes at any spherical surface described from 0, the origin of co-ordinates, as centre. Sometimes a function which, according to the definition (a), is simply a spherical harmonic, will be called a spherical solid harmonic, when it is desired to call attention to its not being confined to a spherical surface. (c) A complete spherical harmonic is one which is finite and of single value for all finite values of the co-ordinates. A partial harmonic is a spherical harmonic which either does. not continuously satisfy the fundamental equation (4) for space completely surrounding the centre, or does not return to the same value in going once round every closed curve. The "partial" harmonic is as it were a harmonic for a part of the spherical surface: but it may be for a part which is greater than the whole, or a part of which portions jointly and independently occupy the same space. (d) It will be shown, later, § (h), that a complete spherical Algebraic harmonic is necessarily either a rational integral function of the complete co-ordinates, or reducible to one by a factor of the form quality of harmonics. (e) The general problem of finding harmonic functions is Differential most concisely stated thus :— To find the most general integral of the equation equations of a harmonic of degree n. the second of these equations being merely the analytical expression of the condition that u is a homogeneous function of x, y, z of the degree n, which may be any whole number positive or negative, any fraction, or any imaginary quantity. Let P+vQ be a harmonic of degree e + vf, P, Q, e, f being Differential real. We have equation for real constituents of a homogeneous function of imagi nary degree. (ƒ) general expressions. (f) Analytical expressions in various forms for an absolutely Value of general integration of these equations, may be found without symbolical much difficulty; but with us the only value or interest which any such investigation can have, depends on the availability of 12 VOL. I. Use of complete spherical harmonics in physical problems. Uses of incomplete spherical harmonics in physical problems. Working formula. its results for solutions fulfilling the conditions at bounding surfaces presented by physical problems. In a very large and most important class of physical problems regarding space bounded by a complete spherical surface, or by two complete concentric. spherical surfaces, or by closed surfaces differing very little from spherical surfaces, the case of n any positive or negative integer, integrated particularly under the restriction stated in (d), is of paramount importance. It will be worked out thoroughly below. Again, in similar problems regarding sections cut out of spherical spaces by two diametral planes making any angle with one another not a sub-multiple of two right angles, or regarding spaces bounded by two circular cones having a common vertex and axis, and by the included portion of two spherical surfaces described from their vertex as centre, solutions for cases of fractional and imaginary values of n are useful. Lastly, when the subject is a solid or fluid, shaped as a section cut from the last-mentioned spaces by two planes through the axis of the cones, inclined to one another at any angle, whether a submultiple of or not, we meet with the case of n either integral or not, but to be integrated under a restriction differing from that specified in (d). We shall accordingly, after investigating general expressions for complete spherical harmonics, give some indications as to the determination of the incomplete harmonics, whether of fractional, of imaginary, or of integral degrees, which are required for the solution of problems regarding such portions of spherical spaces as we have just described. A few formule, which will be of constant use in what follows, are brought together in the first place. (g) Calling O the origin of co-ordinates, and P the point x, y, z, let OP=r, so that x2 + y2 + z2 = r2. Let 8, prefixed to any function, denote its rate of variation per unit of space in the direction OP; so that If H denote any homogeneous function of x, y, z of order n, we |