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, (§ 474, fig. 2, with P infinitely distant,) 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, = ECP, and the angle between the plane of ECP and a fixed plane through CP, we have do a sin 0 do do. = in the two cases respectively, which proves (45). (8) 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, expressed by definite When ƒ is infinitely nearly equal to a, every element of this in- integral. tegral will vanish except those for which D is infinitely small. Green's problem for case of spherical surface, expressed by definite integral. Its expan sion in harmonic series. Hence the integral will have the same value as it would have if F(E) 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 Now, if e denote any positive quantity less than unity, we have, by expansion in a convergent series, T. Q,,Q,, etc., denoting functions of 0, for which expressions will be investigated below. Each of them is equal to + 1, when ✪ = 0, and they are alternately equal to 1 and + 1, when =. It is easily proved that each is >- 1 and <+1, for all values of between 0 and T. Hence the series, which becomes the geometrical series 1e+e2etc., in the extreme cases, converges more rapidly than the geometrical series, except in those extreme cases of 00 and 0. - Hence, for u (46), we have the following expansions: За x=}{ƒƒF(E)}do+ 3a ƒƒQ,F(E)do+5a2ƒƒQ,F(E)do + ... }, when f>a, surface, 1 u = and 3f 5f = {ƒƒ F(E) do + 3 / SSQ, F(E)do + [[Q,F(E)do + ... }, when ƒ <a 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 4ñaF(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)= 1za {ƒƒF(E)do +3ƒƒQ,F(E)do + 5ƒƒQ,F(E)do + etc., }... (52), Απα Green's problem for case of spherical solved explicitly in harmonic series. Laplace's spherical harmonic expansion of function. which is the celebrated development of an arbitrary function in an arbitrary a series of "Laplace's coefficients," or, as we now call them, spherical harmonics. may of series except at abrupt changes in (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 abrupt transition Convergence in the value of F, across any line on S, the series cannot con- never lost verge when P is exactly on, but must still converge, however near it be to, this line. [Compare with last two paragraphs value of the of § 77 above.] The degree of non-convergence is so slight that, as we see from (51), the introduction of factors e, e2, e3, &c. to the successive terms e being <1 by a very small difference, produces decided convergence for every position of P, and the value of the series differs very little from F(P), passing very rapidly through the finite difference when P is moved across the line of abrupt change in the value of F(P). (u) In the development (47) of 1 (1 - 2e cos 0 + e2)1⁄2 the coefficients of e, e3,... e1, are clearly rational integral functions They are given ex But, if x, y, z and function expressed. Expansion of in x, y, denote rectangular co-ordinates of P and of E respectively, we have symmetrical of the two points. noting, as above, by Q, the coefficient of e' 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. Viewed as a function of (x, y, z), Q. is symmetrical round OE; and as a function of (x', y', ') 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 har monic of order i. (v) But it is important to remark, that the coefficient of any term, such as xyz", in it may be obtained alone, by means of Taylor's theorem, applied to a function of three variables, thus:— (1 − 2e cos 0+ e2)* (r2-2rr'cos 0+r22) [(x − x')2 + (y — y')2 + (≈−≈')']} ° F(x+f, y+g, z + h) = Σ Σ Σ figh J-0001.2...j.1.2...k.1.2... dx'dy'dz 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 = ΣΣΣ dj + k + i 1 (-1) + k +113y" 1.2...j.1.2...k.1.2... l dx dy dz1 (x2 + y2 + zo)1 ' a development which, by comparing it with (48), above, we see to be convergent whenever x'2 + y'2 + z12 < x2 + y2 + z3. the summation including all terms which fulfil the indicated condition (j+k+l=i). It is easy to verify that the second member is not only integral and homogeneous of the degree i, in x, y, z, as it is expressly in x', y', '; but that it is symmetrical with reference to these two sets of variables. Arriving thus at the conclusion expressed above by (53), we have now, for the function there indicated, an explicit expression in terms of differential coefficients, which, further, may be immediately expanded into an algebraic form with ease. (v) In the particular case of x'=0 and y'=0, (54) becomes reduced to a single term, a function of x, y, z symmetrical about the axis OZ; and, dividing each member by r', or its equal, z′′, we have Expression for biaxal harmonic deduced. monic with nates trans By actual differentiation it is easy to find the law of successive Axial harderivation of the numerators; and thus we find, with about equal its co-ordiease, either of the expansions (31), (40), or (41), above, for the formed case m = n, or the trigonometrical formulæ, which are of course biaxal. obtained by putting z = r cos and x2 + y2 = r2 sin30. becomes 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). of the biaxal of order i. (x) Some of the most useful expansions of Q, are very readily Expansions obtained by introducing, as before, the imaginary co-ordinates harmonic, (§, ŋ) instead of (x, y), according to equations (26) of (j), and similarly, (',') instead of (x', y'). Thus we have |