SPHERICAL HARMONICS , in See also:

• MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

The degree is may be fractional or imaginary, but we are at present mainly concerned with the See also:

• CASE
• CASE, JOHN (d. 1600)

, Denoting e'n by k, and y-See also:

• FIX, THEODORE (180o-1846)

Pry" (cos e), where m = i, 2, ... n - I, and the sectorial harmonics are 2; n4). P:(cos B). The functions Pe(i ),, P' Cu) denote the expressions writing _ ((2n) ! ' —n(n—I)µ„-!+n(n—I)(n—2)(n—3)µn_... (3) 2"n. R 2.27—I 2 .4 . 2n — I . 2n -3 P'' (A) n){(I —u2)ISm mn—'" 2''n! (n—m) (n—?n) (n—1n—I)tin_,,, t+... 2.2n—I Pn(ti) = (22,,',,),!(I —tit) In. Every ordinary harmonic of degree n is expressible as a linear function of the system of 2n-1-I zonal, tesseral and sectorial harmonics of degree n; thus the general form of the surface harmonic is n aoP,(A) +Z(am cos mcti +bm sin m0)P„0u). (5) ~n=l In the present notation we have (z+ix cos a+Iysin a)"=r" Pn(ti)+2EIm(n+m)!

"(µ)cosm(,p —a) if we put a= o, we thus have n (cos 0+i sin 0 cos iy)"=P,(cos 0)+2Eim(n+m)IP" (cos 0) cos m¢, m from this we obtain expressions for Pn(cos 0), P" (cos 0) as definite integrals Pn(cos 0) =! f o (cos 0+i sin 0 cos ¢)"d¢ i.'"(n+m).P" (cos 0) =A f o (cos 0+o sin 0 cos 0)"cos m~d4,. 4. Derivation of Spherical Harmonics by Differentiation,—The linear See also:

• CHARACTER (Gr. xapareri7p, from xap&crew, to scratch)

. +'7 d(r2)n_,d2a+... fn(x, y, z) (71), where fn(x, y, z) is a rational integral homogeneous function of degree n. The harmonic of positive degree n corresponding to that of degree —n — in the expression (7) is I _ r2 2.211—1+2.4.2n y9II.42n—3—.. , tfn(x, y, z). It can be verified that even when n is unrestricted, this expression satisfies Laplace's equation, the See also:

• SOLE (Solea)

In a similar manner, by placing two singular points of degree, unity and strength, el, —el, at a distance al along an axis h2, and at the origin respectively, when el is indefinitely increased, and al diminished so that is finite and =e2, we obtain a singular point of degree 2, strength e2 at the origin, the axes being hi, h2. Proceeding in this manner we arrive at the conception of a singular point of any degree n, of strength en at the origin, the singular point having any n given axes hi, h2,. .. hn. If en_I y.n_i (x, y, z) is the potential due to a singular point at the origin, of degree n—1, and strength en_i, with axes hi, the potential of a singular point of degree n, the new axis of which is hn, is the limit of en_1 QSn-i (x —lna, y—mna, z—nna) —en_i (x, y, z) ; when La=o, Len_I=o^, Laen_i=en; this limit is --en (1 ,, it+mna ~y I+nna a2 i) , or —enahn~n_I• Since ¢o=I/r, we see that the potential V, due to a singular point at the origin of strength en, and axes hi, h2, . . .h,, is given by Vn= (—t)"e"ahiahan..ahnr (8) 6. Expression for a Harmonic with given Poles.—The result of performing the operations in (8) is that V. is of the form n !e'' i, where Y. is a surface harmonic of degree n, and will appear as a function of the angles which r makes with the n axes, and of the angles these axes make with one another. The poles of the is axes are defined to be the poles of the surface harmonics, and are also frequently spoken of as the poles of the solid harmonics Y"r", Ynr -1. Any spherical harmonic is completely specified by means of its poles. In See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

(umxn-2m)rn~m . thus we obtain the following expression for Yn, the surface harmonic which has given poles h2, ...hn; yn=rn+1(—I)" a" . n! ahiah2. .ahn r = S (—I)m "(2n' 2m)! I~(~n-2mtim) 2"-'''n (n—m) m=0 where S denotes a summation with respect to m from m =o to m=in, or %(n—I), according as n is even or See also:

• ODD (in middle English odde, from old Norwegian oddi, an angle of a triangle; the old Norwegian oddamann is used of the third man who gives a casting vote in a dispute)

292-3 (2n)! n(n-I),, , znn!n! (i 2.271 - I the expression on the right See also:

• SIDE
• SIDE (mod. Eski Adalia)

Determination of the Poles of a given Harmonic.—It has been shown that a spherical harmonic Y,,(x, y, z) can begenerated by means of an operator fn (ax' a' a) acting upon A, the function fn being so chosen that 71(271) r-p 2 Y,(x,y,z)=(^ 2n7l! 3I—2.2n—I+..: Sfn(x,y,z); this relation shows that if an expression of the form (x2+y2+2)fn-2(x, y, z) is added to fn(x, y, z), the harmonic Yn(x, y, z) is, unaltered; thus if Y,, be regarded as given, fn(x, y, z) =0, is not uniquely deter-See also:

• MINED

(1876), 5th series, vol. ii., " A See also:

• NOTE
• NOTE (Lat. nota, mark, sign, from noscere, to know)

,sin"0 cosh-'"0 (n-m)(n-m-' — cos"-'"20 sin20+ (I2) 2.2m+2 )) which is an expression for Pn (cos 0) alternative to (4). to. See also:

• LEGENDRE, ADRIEN MARIE (1752–1833)
• LEGENDRE, LOUIS (1752-1797)

(µ*dµ2-I). We now see that 2 -' is expressible in the form co E yn (~~ r'n+IPn V') 0 when r < r', or r'n rn+I Pn 0 when r' < r; it follows that the two expressions r"Pr()s), r-"-IPn(µ) are solutions of Laplace's equation. The values of the first few Legendre's coefficients are Po(µ)=1, PI(µ)=12, P2(µ)=1(31.42-1), Ps (A) =1(5A' - 3A) Pa(µ) =g(35µ4-3oµ2+3), Pa(µ) L(63µ5-70µa+15,) I 1 Pa(µ)=Iti(231µa-313124+1O5µ2-5), P7(A)=i6(429µ'-693µs +315,' - 3512) P„ (1) =1, Pn(—1) = (-On P,,(0)=0, or (—1)40.3.5...n—I 2.4...n according as n is odd or even; these values may be at once obtained from the expansion (13), by puttingµ =1, o, -1. ii. Additional Expressions for Legendre's Coefficients.—The expression (3) for Pn(s) may be written in the form (2n)! n n I—nn I 1 P. (µ) = 2rt 71, F ( 2 2 , 2 — n, 122 with the usual notation for hypergeometric series. On writing this series in the See also:

• REVERSE

. .`` (( =cos2tt-F (—n, —n, 1, —tan22 1 . (16) By means of the identity ( (I -2hµ+h2)-I = (I +h2( —µ2g ( (1-hµ) it may be shown that P„(cos0) =cos"o -n( 2- I)tan20+n(n-I)Zn.42)(n-3) tan 40-... =cos"pF(-2n, I-2n, 1, —tan2o). (17) Laplace's definite integral expression (6) may be transformed into the expression (µ+-122-1 cos $)(µ— yµ2— cos jf) =I. Two definite integral expressions for Pn(µ) given by Dirichlet have been put by Mehler into the forms Pn(ccs 0 __ 2 e_cos (n+2)0 d __ 2 n sin (n+3)~ d o112 COS 0—2 cos 0 7r,J 0 s 2 cos 0—2 cos ~ When n is large, and 0 is not nearly equal to o or to r, an approximate value of Pn(cos0) is {2/nir sin 01I sin {(n+*)0+17r}. 12. Relations between successive Legendre's Coefficients and their Derivatives.—If (I -2hµ+h2)-I be denoted by is, we find (I -2hµ+h2) ah+(h -µ)u =0; on substituting Zh"Pn for u, and equating to zero the coefficient of h", we obtain the relation nPn-(2n- 1),Pn-1+(n—I)Pn-2=0. From Laplace's definite integral, or otherwise, we find (,2 - 1) d--n = n (12Pn - P,_l) = - (n + 1) (,sP" — Pn+1) We may also show that dPs dPn_I µ dµ — dµ =nPn dPn dPn+1 (n+I)Ps=-µdµ+ d12 (2n+I)PP=d +1+I—d (2n+1), d (n+I)d i+nd (2n+I) ('2— 1) =n(n+1) (P.+1- P.-0 dP,, _ =(2n-1)Pn_1+(2n-5)Pn-a+(2n-9)Pn-a+. the last term being 3P1 or Po according as n is even or odd. 13. Integral ) rop,erties of Legendre's Coefficients.—It may be shown that if P,+a7s) be multiplied by any one of the numbers 1, µ, 122, ..-. and the product be integrated between the limits 1,- I with respect to As, the result is zero, thus f 1 µkPn(µ)dµ=0, a=0, I, 2, ... n-I. (18) To prove this theorem we have /..

1 I do(~~ I,kPnV*)d12=2nn7 -Iµkdun(,2—I)nd12, We find also (15) 1 in d4, 0(12—dµ2—1 COS 4)n+1' by means of the relation on integrating the expression k times by parts, and remembering that. (p2 — On and its first n — I derivatives all vanish when p = =1, the theorem is established. This theorem derives additional importance from the fact that it may be shown that AP,(µ) is the only rational integral function of degree n which has this See also:

• PROPERTY

(2I) The See also:

• PROOF (in M. Eng. preove, proeve, preve, &°c., from O. Fr . prueve, proeve, &c., mod. preuve, Late. Lat. proba, probate, to prove, to test the goodness of anything, probus, good)

Vo=a P.(14), we see that V4, Vo have the characteristic properties of potential functions for the spaces internal to, and external to, the spherical surface respectively; moreover, the condition that Vi is continuous with Vo at the surface r=a, is satisfied. The density of a surface distribution which produces these potentials is in accordance with a known theorem in the potential theory, given by r (Arc 0Vo _ ° 4r or a)r—a, hence =4 a2E(2n+r)A,Pn(p); on comparing this with the series (2r), we have An=2a2f f(A')P„(µ)dµ', 1V4 = 27raFaaPn(µ)fI If (P') Pn (P') dµ lvi = 27raF.an+1+ P,(p) f 121010 Pn(lt )dtl' are the required expressions for the internal and external potentials due to the distribution of surface density f(µ). 15. Integral Properties of Spherical Harmonics.—The fundamental harmonic property of spherical harmonics, of which property (19) is a particular case, is that if Y,,(x, y, z), Zn,(x, y, z) be two (ordinary) spherical harmonics, then, f fYn(x, y, z)Zn,(x, y, x)dS=0, (22) when n and n' are unequal, the integration being taken for every See also:

• ELEMENT (Lat. elementum)

I'n(ax'ay' a.)f(x, y, z) where x, y, z are put equal to zero after the operations have been performed, the integral being taken over the surface of the sphere of radius R (see Hobson, " On the Evaluation of a certain Surface Integral," Proc. Lond. Math. Soc. vol. See also:

• XXV

+V,(si, 4') +... ; change µ, 4, into A', 4,' and multiply by P,,(cos 8 cos 8'+sin 8 sin 8' cos 4,—0), we have then f on f See also:

• IIF

. r'n +r+l+IPn(cos y)+... r'2dr'dµ'do' which is, 1 fOnfIt I 3r(1+3€u)+4-(1+4€u')Pl+.. +n+3 yn(i+n+3eu')Pn(cos y) dµ'do' on substituting for u' the series of harmonics, and using (22), (23), this becomes 4xa2C3 y+s 3r2Y1(ts, 4,)+5r3Y2(JA, 4')-i-...functions of his h2, ha, which depend on the form of these parameters; it is known that Laplace's equation when expressed with h1, h2, h as independent variables, takes the form a H1 aV\ a / H2 aV\l a (H3 AT al\ \l Ha Ohl/ TA; \MI I ah21 + A3h3 \1W.2 ah,J =0. (25) In case the orthogonal surfaces are concentric See also:

• SPHERES, MUSIC OF THE

2n — 1 and 5 I (n-1-I)(n+2) I (n+I)(n+2)(n+3)(n+4) I n+1+ 2 . 2n-{-3 µn+a+a I 2 .4 . 211+3 .2n+5 µn+5 f ... } the first of these series is (n integral) finite, and represents P,,(µ), the second is an See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

. which is the required potential at the external point (r, 8, 4'). 17. The Normal Solutions of Laplace's Equation in Polars.—If h1, h2, ha be the parameters of three orthogonal sets of surfaces, the length of an elementary arc ds may be expressed by an equation of the form ds2 = II2dh; + 1dh= + H dh21, where H1, H2, H3 are HARMONICS If, in Legendre's equation, we differentiate in times, we find 1 (I—t~E)dµ11+z-2(m+1)pd/2—R+(n—m)(n±m±I) ,~,=o; 656 From this form it can be shown that Q, (/A) =2 P,,(/A) See also:

• LOG (a word of uncertain etymological origin,possibly onomatopoeic; the New English Dictionary rejects the derivation from Norwegian lag, a fallen tree)

See also:

• NEUMANN, FRANZ ERNST (1798-1895)
• NEUMANN, KARL FRIEDRICH (1793-1870)

It can be shown that, whenµ is real and between I f— I)m I+µ Ic'n do Pn(u)=211(11—m)!~IdµnJ(A—I)n+m(!A+I)n—'1 _ I I —µ\) im dn' n—m/ n+m 211(11—ni)! \I+pJ dµnl(K—I) ./2+I) ). In the same case, we find Pn+2(cos 6)—2(112+I) cot 0 Pm11(cos 0) +(n—m)(n+m+I)P,, (cos 0) =o, (n—m+2)1'„+2(cos 0) — (2n+3)µI'01 l(cos 0) +(n+m+1)Pn (cos 0) =o. 20. Bessel's Functions.—If we take for three orthogonal systems of surfaces a system of parallel planes, a system of co-axial circular cylinders perpendicular to the planes, and a system of planes through the axis of the cylinders, the parameters are z, p, ¢, the cylindrical co-ordinates; in that case H1= I, Ho= 1, Ho= and the equation (25) becomes ,32V a2V i aV i a2V a2+ap2+p ap+'°2 a02=o. To find the normal functions which satisfy this equation, we put V=ZR'h, when Z is a function of z only, R of p only, and shof 0, the equation then becomes i d2Z I (d2R i dR\ i i d24, Zd+K dp2+ dp)+p2 doe=o. d2Z That this may be satisfied we must have 2 0 constant, say =k2, 2- constant, say = —m2, and R, for which we write u, must satisfy the differential equation d2u ids (, 1112 ap2+p ap+ k P' U=0, it follows that the normal forms arc e 3ks' inm~. u(kp), where u(p) satisfies the equation 1216 I Lit 117 dp2 +pdp+ ~L— p2~71=0. (29) This is known as Bessel's equation of order m; the particular case den. I du dpz+v dp+u=0, corresponding to m=o, is known as Bessel's equation. If we solve the equation (29) in series, we find by the usual See also:

• PROCESS

The equation (29) is unaltered by changing m into-m, it follows that J_,,,(p) is a second solution of (2n). thus in general n = AJm(P) + BJ-m (P) is the complete primitive of (29). However, in the most important case, that in which nz is an integer, the solutions J_,,,(p), J,,,(p) are not distinct, for J_,,,(p) may be written in the form sz, +(-I)m(P ( - I)P p 2p z) II(m-11p)II(p) ~2J P=0 now II(n -nz) is infinite when m is an integer, and n < m; thus the first part of the expression vanishes, and the second part is (-I)'^J,,,(p), hence when m is an integer )"Im(p), and the second solution remains to be found. Bessel's Functions of the Second Kind.—When in is not a real integer, we have seen that any linear function of Jm(p), J_m(P) satisfies the equation of order m. The Bessel's function of the second kind of order m is defined as the particular linear function vrem+n'J-m(P) - cos Mir. Jm(P) sin 2ma - and may be denoted by Ym(p). This definition has the See also:

• ADVANTAGE

If um(p)=J,„(p), we have um+p=Jm+P(P). and by comparing the coefficients of pet,' . we find C=(-2)P, hence Jm+p(P) = (-2)PP,"+Pd(P )P{P mJm(P)}, and changing in into -in, we find Jp-m (p) _ (- 2) npn-"'d (PPl P}PmJ-m (P) }HARMONICS 57 In a similar manner it can be proved that Jm p(P) =2PPT "`d(pz)P}Pm.Jm(P}• From the definition of Y,,,(p), and applying the above analysis, wt, prove that Ym+r(P)(-2)r'P''' PZ)p{P mYm(P)} and Ym-,,(p) = 2ppp-md(d;) p{PmYm(P) }.