Digression on spherical harmonics. Hence [§ 707 (23), writing v for √−1], Cases of solid har monics resolvable Polar har monics. Zonal and sectorial defined. H„=A {(x + y v)" + (x − yv)"} + Bv{(x + yv)" — (x − yv)"}, if x=p cos o, y = p sin 4, H=2p" (A cos no + B sin no), which shows that the n branches cut one another at equal angles round the origin. 781. The harmonic nodal cone may, in a great variety of cases [V, resolvable into factors], be composed of others of lower into factors. degrees. Thus (the only class of cases yet worked out) each of the 2i + 1 elementary polar harmonics [as we may conveniently call those expressed by (36) or (37) of App. B, with any one alone of the 2i+1 coefficients A., B.] has for its nodes circles of the spherical surface. These circles, for each such harmonic harmonics element, are either (1) all in parallel planes (as circles of latitude on a globe), and cut the spherical surface into zones, in which case the harmonic is called zonal; or (2) they are all in planes through one diameter (as meridians on a globe), and cut the surface into equal sectors, in which case the harmonic is called sectorial; or (3) some of them are in parallel planes, and the others in planes through the diameter perpendicular to those planes, so that they divide the surface into rectangular quadrilaterals, and (next the poles) triangular segments, as areas on a globe bounded by parallels of latitude, and meridians at equal successive differences of longitude. With a given diameter as axis of symmetry there are, for complete harmonics [App. B. (c), (d)], just one zonal harmonic of each order and two sectorial. The zonal harmonic is a function of latitude alone (π-0, according to the notation of App. B.); being the given by putting s0 in App. B. (38). The sectorial harmonics of order i, being given by the same with s = i, are sin'e cos ip, and sin'e sin ip............................. .(1). The general polar harmonic element of order i, being the cos sp and sin sp of B. (38), with any value of s from 0 to i, has for its nodes is circles in parallel planes, and s great circles intersecting one another at equal angles round on spherical their poles; and the variation from maximum to minimum Digression along the equator, or any parallel circle, is according to the harmonics. simple harmonic law. It is easily proved (as the mathematical student may find for himself) that the law of variation is approximately simple harmonic along lengths of each meridian. cutting but a small number of the nodal circles of latitude, and not too near either pole, for any polar harmonic element of high order having a large number of such nodes (that is, any one Tesseral for which is is a large number). The law of variation along surface by a meridian in the neighbourhood of either pole, for polar har- polar harmonic elements of high orders, will be carefully examined and illustrated in Vol. II., when we shall be occupied with vibrations and waves of water in a circular vessel, and of a circular stretched membrane. * 782. The following simple and beautiful investigation of the zonal harmonic due to Murphy may be acceptable to the analytical student; but (§ 453) we give it as leading to a useful formula, with expansions deduced from it, differing from any of those investigated above in App. B: "PROP. I. division of nodes of a monic. analytical of the zonal "To find a rational and entire function of given dimensions Murphy's "with respect to any variable, such that when multiplied by invention "any rational and entire function of lower dimensions, the harmonics. "integral of the product taken between the limits 0 and 1 "shall always vanish. "Let f(t) be the required function of n dimensions with respect ....... "Let the indefinite integral of f(t), commencing when t = 0, be "represented by f(t); the indefinite integral of f(t), commencing "also when t=0, by f(t); and so on, until we arrive at the "function f(t), which is evidently of 2n dimensions. Then the "method of integrating by parts will give, generally, Digression on spherical harmonics. Murphy's analytical invention of the above harmonics. "Let us now put t = 1, and substitute for x the values 1, 2, 3, "......(i-1) successively; then in virtue of the equations (a), we get, 66 66 ' (b).......................ƒ,(t) = 0, ƒ,(t) = 0, ƒ.(t) = 0,.........................fi(t) = 0. ....... "Hence, the function f(t) and its (i-1) successive differential "coefficients vanish, both when t = 0, and when t=1; therefore “ť and (1 − t)' are each factors of f(t); and since this function is "of 2i dimensions, it admits of no other factor but a constant c. "Putting 1-t=t', we thus obtain Murphy's analysis. "Corollary.-If we suppose the first term of f(t), when arranged "according to the powers of t, to be unity, we evidently have "The function Q, which has been investigated in the pre"ceding proposition, is the same as the coefficient of e' in the "expansion of the quantity "But if, as before, we write t' for 1-t, we have, by Lagrange's "theorem, applied to the equation (c), "Second Expansion.—If u and v are functions of any variable t, "then the theorem of Leibnitz gives the identity di dt (uv) = v d'u + i dt dv d-lui (i-1) d'v div + + etc. dt dt-1 1.2 dt2 dt-2 "Put u = t' and v = t', and dividing by 1. 2. 3...i, we have Expansions of zonal harmonics. "Third Expansion.-Put 1 - 2t = μ, and therefore tt' Digression on spherical harmonics. Formulæ for zonal, The t, t' and μ of Murphy's notation are related to the we have used, thus :— t = (2 sin 10)3, t' = (2 cos 10)2 μ= cos Also it is convenient to recall from App. B. (v'), (38), (40), and (42), that the value of Q. [or of App. B. (61)], when 0 = 0 is unity, and that it is related to the ", of our notation for polar harmonic elements, thus: and tesseral harmonics. as is proved also by comparing (g) with App. B. (38). We add the following formula, manifest from (38), which shows a derivation of from O, valuable if only as proving that the i-s roots of = 0 are all real and unequal, inasmuch as App. B. (p) proves that the i roots of "=0 are all real and unequal:— (0) i Biaxal harmonic expanded. And lastly, referring to App. B. (w); let Q', and Q. [cos cos' + sin é sin e' cos (-')] i denote respectively what Q, becomes when cos is replaced by cos e', and again by cos e cos 0' + sin 0 sin 0' cos (☀ –☀′): and let μ denote cos 0; and μ', cos e'. By what precedes, we may put (61) of App. B into the following much more convenient form, agreeing with that given by Murphy (Electricity, p. 24): |