Expression of an arbitrary funetion in a series of surface harmonics. Preliminary proposition. will have no difficulty in verifying for himself the present application of the principles developed in that admirable work. Thus it appears that spherical harmonies belong to the general class, to which Sir William R. Hamilton has applied the designation "Fluctuating Functions." This property is essentially in volved in their capacity for expressing arbitrary functions, to the demonstration of which we now proceed, in conclusion. (r) Let C be the centre and a be the radius of a spherical surface, which we shall denote by S. Let P be any external or internal point, and let ƒ denote its distance from C. Let do denote an element of S, at a point E, and let EP=D. Then,// denoting an integration extended over S, it is easily proved that do α Απα when P is external to S D3 ƒ ƒ3 — a2 and = (45). This is merely a particular case of a very general theorem of Green's, included in that of A (a), above, as will be shown when we shall be particularly occupied, later, with the general theory of Attraction: a geometrical proof of a special theorem, of which it is a case, will occur in connexion with elementary investigations regarding the distribution of electricity on spherical conductors and, in the meantime, the following direct evaluation of the integral itself is given, in order that no part of the important investigation with which we are now engaged may be even temporarily incomplete. Choosing polar co-ordinates, 0= ECP, and & the angle between the plane of ECP and a fixed plane through CP, we have is, Let now F(E) denote any arbitrary function of the position Solution of of E on S, and let Green's problem for case of spherical (46). surface, ex pressed by definite When fis infinitely nearly equal to a, every element of this in- integral. tegral will vanish except those for which D is infinitely small. Hence the integral will have the same value as it would have if FE) had everywhere the same value as it has at the part of S nearest to P; and, therefore, denoting this value of the arbitrary function by F(P), we have when differs infinitely little from a; or. by (45, u=4жаF(P) (46'). Now, if e denote any positive quantity less than unity, we have, Its expanby expansion in a convergent series, 1 =1+Q2e+Q,e2 + etc. (47), (1 ze cos 0+e2)} Q1, Q., etc., denoting functions of 0, for which expressions will be investigated below. Each of them is equal to +1, when 0=0, and they are alternately equal to -1 and +1, when 0=. It is easily proved that each is >- 1 and <+1, for all values of between 0 and . Hence the series, which becomes the geometrical series 1±e+e2± etc., in the extreme cases, converges more rapidly than the geometrical series, except in those extreme cases of 0=0 and 0. sion in harmonie series. Green's problem for case of spherical surface, solved explicitly in harmonic series. Laplace's spherical harmonic u = 1 Hence, for u (46), we have the following expansions::{S[F(E)do+7SSQ;F(E)do+S/Q;F(E)do+... }, when ƒ>a, and За 3f 5a2 = = { //F(E)do+3⁄41⁄4ƒƒQ, F(E)do+"/", //Q2F(E)do +...........}, whenƒ<a u = expansion of an arbitrary function, Convergence of series never lost except at abrupt changes in value of the function expressed. Expansion of in symmetrical harmonic functions of the coordinates of the two points. These series being clearly convergent, except in the case of ƒ=a, and, in this limiting case, the unexpanded value of u having been proved (46') to be finite and equal to 4aF(P), it follows that the sum of each series approaches more and more nearly to this value when ƒ approaches to equality with a. Hence, in the limit, 1 F(P) = __1__ {ƒƒF(E)do+3ƒƒQ,F(E)do+5ƒƒQ,F(E)do+etc.,} (52). Απα which is the celebrated development of an arbitrary function in a series of "Laplace's coefficients," or, as we now call them, spherical harmonics. (t) The preceding investigation shows that when there is one determinate value of the arbitrary function F for every point of S, the series (52) converges to the value of this function at P. The same reason shows that when there is an an abrupt transition in the value of F, across any line on S, the series cannot converge when P is exactly on, but must still converge, however near it may be this line. to, Later we shall derive a rule for the degree of convergence of of the series (52) in any case according to the character of F. (1 1 · 2e cos 0+e2 ̧1 (u) In the development (47) of the coefficients of e, e3,...e, are clearly rational integral functions of cos, of degrees 1, 2...i, respectively. They are given explicitly below in (60) and (61), with 0' 0. But, if x, y, z and x', y', z' denote rectangular co-ordinates of P and of E respectively, we have xx'+yу'+zz' cos 0 where r(x+y+2), and r'=(x22+y'2+z'2). Hence, denoting, as above, by Q; the coefficient of ei in the development, we have H¿[(x, y, z), (x', y', z')] denoting a symmetrical function of (x, y, z and (x, y, z), which is homogeneous with reference to either set alone. An explicit expression for this function is of course found from the expression for Q in terms of cos 0. monic. Viewed as a function of (x, y, z), Qiriri' is symmetrical Biaxal harround OE; and as a function of (x', y', z') it is symmetrical round OP. We shall therefore call it the biaxal harmonic of (x, y, z) (x', y', z') of degree i; and Q; the biaxal surface harmonic of order i. 1 (v) But it is important to remark, that the coefficient of any Expansion term, such as x'y'kz', in it may be obtained alone, by means of of by Taylor's theorem, applied to a function of three variables, thus :— theorem. 1 = g Fx+f.y+g,2+h) = Σ Σ Σ fight ; (1—2e cos 0+e2) 4 ̄ ̄ ̄(r2—2rr'cos 0+r'2)} ̄ ̄ ̄ [(x—x' ́)2+(y—y' )2+(z—z' )2]}' Now if F(x, y, z) denote any function of x, y, and z, we have di+k+1F(x, y, z) jo ko to 1.2...j.1.2...k.1.2...l dx dydz where it must be remarked that the interpretation of 1.2...j, when j=0, is unity, and so for k and also. Hence, by taking F(x, y, z) = j=0 k=0 1=0 1 '(x2 + y2+z2) § 1 1 (-1)+k+1x'y' z'l di+k+l Hence 12 (rr')'Q = r2i+1 Σ Σ Σ 2 1 2 D Taylor's (54), (j+k+l=i) ( — 1}j+k+x13ykz'l di+k+l (v) In the particular case of x'=0 and y'=0, (54) becomes Expression for biaxal harmonic deduced. (55). Axial har By actual differentiation it is easy to find the law of successive derivation of the numerators; and thus we find, with about equal monic of order i. Axial har monic with its co-ordin ates transformed becomes biaxal. Expansions of the biaxal harmonie, of order i. ease, either of the expansions (31), (40), or (41), above, for the case m=n, or the trigonometrical formulæ, which are of course obtained by putting z = r cos 0 and x2 + y2 = r2 sin30. (w) If now, we put in these, cos 0 = xx'+yy'+zz' rr' introduc ing again, as in (u) above, the notation (x, y, z), (x', y', z'), we arrive at expansions of Q; in the terms indicated in (53). (x) Some of the most useful expansions of Q are very readily obtained by introducing, as before, the imaginary co-ordinates (§, n) instead of (x, y), according to equations (26) of (j), and similarly, (ŝ', n' ́) instead of (x', y'). Thus we have And, just as above, we see that this expression, obviously a homogeneous function of έ', n', z', of degree i, and also of ŋ, §, z, involves these two systems of variables symmetrically. Hence ri Q, viewed as a function of 2, §, 7, is expressed by these 2+1 terms, each with a coefficient involving z', §', n . And because of the symmetry we see that this coefficient must be the same function of ', n', ', into some factor involving none of either set of variables (z, έ, n), (', n', '). Also, by the symmetry with reference to έ, n' and n, έ', we see that the numerical factor must be the same for the terms similarly involving έ, n' on the one hand, and ŋ, έ' on the other. Hence, |