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J0(p)

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Originally appearing in Volume V25, Page 661 of the 1911 Encyclopedia Britannica.

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J0(p)

J0(p) =;. 2f ° o See also:

sin (p cosh 4,)d4 Yo(p) =-f o See also:

cos (p cosh 4,)dq,. 25. See also:

BESSEL, FRIEDRICH WILHELM (1784-1846)

Bessel's Functions with Imaginary See also:

ARGUMENT

Argument.—The functions with purely imaginary argument are of such importance in connexion with certain See also:

DIFFERENTIAL

differential equations of physics that a See also:

SPECIAL

special notation ; .~I sin (54_0 I1z8P 12 .32.5 3(8P2)3 1-* — V . .. when m is an integer, Km(r)=(-1),, / e_' 1+4m212+(4m212)(4}2-32) +... 2r I.8r I.2(8r2 I",(r)= I e (j I-4m2-12+(4m2-12)(4m2-32) i~2ar ( - 1.8r 1.2(802 27. The See also:

BASSET, or BASSETTE

Basset's functions of degree See also:

HALF

half an 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)

odd integer are of special importance in connexion with the differential equations of physics. The two equations at = kV2u, ata = k'v2u, are reducible by means of the substitutions u=e-k'v, u=e,"v to the See also:

FORM (Lat. forma)

form V2v+v=o. If we suppose v to be a See also:

FUNCTION

function of r only, this last differential See also:

EQUATION (from Lat. aequatio, aequare, to equalize)

equation takes the form dl (sr) dra {-vro, so that v has the values sin r/r, cos tit.; in See also:

order to obtain more See also:

general solutions of the equation pav+v=o, we may operate on sin r/r, cos r/r with the operator y'' ( a a a 1 ax' ay' az ' where Y,, (x, y, z) is any spherical solid See also:

HARMONIC

harmonic of degree n. The result of the operation may be at once obtained by taking, Y„ (x, y, z) for Mx, y, z) in the theorem (7'), we thus find as solutions, of viv+v=o, the expressions d^ sin r ~, d" cos r Y"(xy z)d(ra)" , "(x, z y )d(ra)m r By recurring to the See also:

DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

definition of the function Jm(r), we see that 2p i — See also:

R2(rt-s2)

r2 + r4 2 sin r Ji(r)=~ir 2.3 2.3.45—... =~~ thus r iJi(r)='V sin 7 r Using the relation between Bessel's functions whose orders differ by an integer, we have Li++(r ) = 2) "r"+id(rad)"Ji(Jrr) ( 2)nN/xr 2"+id(dra)sn sinr g (— =— .

It may be shown at once thatthat the function K"(z) has no real zeros unless n=2k+i where k is an integer, when it has one real negative zero; and that K"(z) has no purely imaginary zeros, and no zero whose real See also:

PART

part is See also:

POSITIVE (or PORTABLE) ORGAN

positive, other than those at infinity. When i> n> o, K"(z) has no zeros other than those at infinity, when 2>n> i, it has one zero whose real part is negative, and when m+I>-n> m where m is .an integer, there are m zeros whose real parts are negative. When n is an integer, K"(z) has n zeros with negative real parts. 29. Spheroidal Harmonics.—For potential problems in which the boundary is an See also:

ELLIPSOID

ellipsoid of revolution, the co-ordinates to be used are r, 0, o where in the See also:

case of a prolate See also:

SPHEROID (Gr. c4aipa-etbis, like a sphere)

spheroid x=cJra—i sin9cos¢, y=cJr- -i sin0sin0, z=crcos0, the surfaces r=ro, B=Bo, 4,=0o are confocal prolate spheroids, confocal hyperboloids of revolution, and planes passing through the See also:

AXIS (Lat. for " axle ")

axis of revolution. We may suppose r to- range from i to so , 0 from o to r, and d, from o to 2,r, every point in space has then unique co-ordinates r, 0, O. For oblate spheroids, the corresponding co-ordinates are r, 0, ¢ given by x=cJr2-} isin0cos0, y=cJr2+isin0sin0, z=crcos0, where O <r <so, o <B <1r, o <_O._<2ir; these may be obtained from those for the prolate spheroid by changing c into — °c, and r into or. Taking the case of the prolate spheroid, See also:

LAPLACE, PIERRE SIMON, MARQUIS DE (1749—1827)

Laplace's equation becomes 8 S aV 1 8( avl r2—~os20 82V _ See also:

art (1'2— i)ar +sing aBlsin aBl +(r2—i)siniO 42 —o' and it will be found that the normal solutions are Pn (r) ? P;,"(cos 0) ? cos Qn (r) Q (cos B) sin m4,. For the space inside a bounding spheroid the appropriate normal forms are Pn (r)P;,"(cos0)s IImo, where n, m are positive integers, and for the See also:

EXTERNAL

external space Qn (r)Pn (cos0)sinmo' For the case of an oblate spheroid, P, (°r), Q,'(or), take the See also:

PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place of P;, (r), Qn (r)• 30. Toroidal Functions.—For potential problems connected ith the See also:

ANCHOR (from the Greek ayrcvpa, which Vossius considers is from 6yi17 , a crook or hook)

anchor-See also:

RING (O.E. hring; a word common to Teutonic languages; and probably cognate with the Lat. circus, Gr. KtpKOS or KpLKOS, Skt'. chakra, wheel, circle, cf. also " harangue "); in art, a band of circular shape of varying sizes, made of any material and used f

ring, the following co-ordinates are appropriate:' If A, B are points at the extremities of a See also:

DIAMETER (from the Cr. &a, through, µErpov, measure)

diameter of a fixed circle, and P is any point in the See also:

PLANE

plane PAB which is perpendicular to the plane of the fixed circle, let P=See also:

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

log(AP/BP), 0= L APB, and let ¢ be the See also:

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle the plane APB makes with a fixed plane through the axis of the circle. Let 0 be restricted to See also:

lie between -7 and a, a discontinuity in its value arising as we pass through the circle,. so that within the circumference 0 is r on the upper See also:

side of the circle; and —a on the See also:

LOWER

lower side; 0 is zero in the plane of the circle outside the circumference; p may have any value between —co and co . and 4, any value between o and err.

The position of a point is then uniquely represented by the co-ordinates p, 0, 0, which See also:

ate the parameters of a See also:

SYSTEM

system of totes with the fixed circle as limiting circle, a system of See also:

BOWLS

bowls with the fixed circle as See also:

COMMON

common rim, and a system of planes through the axis of the totes. If x, y, z are the co-ordinates of a point referred to axes, two of which x, y are in the plane of the circle and the third along its axis, we find that a sinh p a sinh p sin ¢, z= Q. sin 0 x=cosh p—cos BCOS 4,,y=cosh p—cos B cosh p—cos 0' where a is the See also:

RADIUS

radius of the fixed circle. " Laplace's equation reduces to a€ sinh p aV a 5 sinh p aV) i a2V ap Pa ap + ae P2 ae +1'2 sinh p 04,2 —o' when P denotes cosh p—cos 0). It can be shown that this equation is satisfied by J (cosh p—cos 0)P,, }(cosh p) cos cos (cosh Q"_}(cosh p) sin sin the functions P i(cosh p), Qn_}(cosh p) required for the potential problems, are associated See also:

Legendre's functions of degree n—2, half an odd integer, of integral order in, and of argument real and greater than unity; these are known as toroidal functions. For the space external to a boundary tore the function Qn_}(cosh p) must be used, and for the See also:

INTERNAL

internal space Pn_i(cosh p). d" cos r d(r2)" r is a second See also:

SOLUTION (from Lat. solvere, to loosen, dissolve)

solution of Bessel's equation of order n+;; thus the differential equation v0v+v=o is satisfied by the expression Y"(x, y, z)Jr and by the corresponding expression with a second solution of Bessel's equation instead of J,,+ i (r) ; if S,;(u, 4,) denotes a See also:

SURFACE

surface harmonic of degree n, the expression Sn(si, 0)y..J.+i(r) is a solution of the equation p2v+v=o. The Bessel's functions of degree half an odd integer are the only ones which are expressible in a closed form involving no transcendental functions other than circular functions. It will be observed that in this case the semi-convergent See also:

SERIES (a Latin word from serere, to join)

series for J," becomes a finite one as the expressions P, Q then break off after a finite number of terms. 28. The Zeros of Bessel's Functions.—The determination of the position of the zeros of the Bessel's functions, and the values of the argument at which they occur, have been investigated by Hurwitz (Math. See also:

Ann. vol. xxxiii.), and more completely by H. M.

See also:

Macdonald (Prot. Lond. Math. See also:

Soc. vols. See also:

XXIX

xxix., See also:

xxx.). It has been shown that the zeros of Js(z)/z" are all real and associated with the singular point at infinity when n is real and > — i, and that all the real zeros of J"(z)/z" when n is real and <-i, and not an integer, are associated with the essential singularity at infinity. When n is. a negative integer — m, J"(z)/z" has, in addition„ 2m real zeros co-incident at the origin. When n = —m —v, in being a positive integer, and i>v>o, J"(z)/z" has a finite number 2m of zeros which are not associated with the essential singularity. If n is real, and starts with any positive value, the zeros nearest the origin approach it as n diminishes, two of them reaching it when n= --,i, and two more reach it whenever n passes through a negative integral value; these zeros then become complex for values of n not integral. The zeros of J"(z)/z" are separated by those of J,,+i(P)/z", one zero of the latter, and one only, lies between two consecutive zeros of J"(z),/z". When n is real and > —i, all the zeros of J"(z)/z" are given by a See also:

formula due to See also:

Stokes; the m°' positive zero in order of magnitude is given by a=4i- i 4(4+22 — i) (28n0 -3i) &c., 8a 3. (8a) 2 where a=}ir(2n+4m—i). It. has been. shown by Macdonald The following expressions may be given for the toroidal functions: P " (cosh p) = (— II (n- z) "A cos in4) "-} a II(n- m— J)See also:

J5(z)

J5 (cosh p+sinh p cos ,p)-,do' = n(n+m- a) (' (cosh p + sinh p cos ¢)" mspdsp.

7, II(n—) Jo 2 p cosh n4 P,.; (Cosh p) _ ~I o v2 cosh p — 2 cosh st, -sinh p cosh w) cosh mwdw -A See also:

net, =(-I)'"See also:

A2(z)

a2"'II(m-z)n(-z) sinh n'PJ cos d¢. o (2 cosh p - 2 cos The relations between functions for three consecutive values of the degree or the order are 2n cosh pP,7 1 (cosh p) — (n —m+z)P,+ (cosh p) - (n+m - 2) P'" i (cosh p) = c P (cosh p) + 2(m + i) coth pP+;[(cosh p) -(n-m- )(n+m+I)P:L(cosh p) =0, with relations identical in form for the functions Q. , (cosh p). The function Q,,_ (cosh p) is expansible in the form ti(n) which is useful for calculation of the function when p is not small. P,. (cosh p) can also be expressed in terms of e.° by a somewhat complicated. formula. 31. Ellipsoidal Harmonics.—In order to treat potential problems in which the boundary surface is an ellipsoid, Lame took as co-ordinates the parameters p, p, v of systems of confocal ellipsoids, hyperboloids of one See also:

SHEET

sheet, and of two sheets; these co-ordinates are three roots of the equation x2 y2 „2 t"-+t2—ly2'12_0=I (k>h); we thence find that pm, 1p2—h2Jp2-h2vh2_v2 dp?-k'LJk2-p2A/le-v2 x= 5= hvk2-h2 ' z= Jk2_Iz2 where co > p2 >— h2 k2 p2 h"-, and k2 > y2 > o. We find from these values of x, y, z (dx)2 + (dy)2 + (dz) 2 (p2h2) (P2 k2)) (dp)2 + 22 -15)) (dp) 2 (P2 v2) (p2 p2) + (2 — p2) (k2 - y2 j (di') z, and on applying the general transformation of Laplace's equation that equation becomes (p2 —P2) N a'YV ,32V I .+. (y,”. _- p2) ay + (P2 ^ p2} -0, where 4, n, { are defined by the formulae It can now be shown that .Laplace's equation is satisfied by the product E(p)E(p)E(v), where E(o) satisfies the differential equation d2E(P)-{n(n+I)p2-(h2+k2)See also:

piE(p) o; de and E(p), E(v) satisfy the equations d2E +{11(11-+ 1)p2-p(h2+k2)1E(p) =0, d2E (v) -In (n+ I) p2 ,.p (h2+k2)1 E (v) = o, where n and p are arbitrary constants. 7 On substituting the valuesof the parameters I;, n, in terms of p,.p, v, we find that the equation satisfied by E(p) becomes (p2 1t2)(PZ—k2) 1.(+P(2P2—h2k2) dE (P) dp +{(h2+k2)p-n(n-f I)P2}E(p) -o, and E(p), E(v) satisfy equations in p, v respectively of identically the same form; this equation is known as Lame's equation.

If n be taken to be a positive integer, it can be shown that it is possible in 2nd-I ways so to determine p that the equation in E(p) is satisfied by an algebraical function of degree n, rational in p, 1/(p2-h22), v(p2-k'2). The functions so determined are called Lame's functions, and the 2ndi functions of degree n are of one of the four forms. K(p) =aop"+See also:

alp"~+. . L(p) = J p2.-h2(ao' 1"-1 + + . . . ), M(p)=1Ipi._k2(ao"+a"1P" +. . ), N(p) p2—k2 12—h2(a'P"-2+ai pn-4+...) These are the four classes of See also:

Lime's functions of degree n; of the functions IK there are r+2n, or }(n+i), according as n is even or odd; of each of the functions L, M, there are sn, or 2(n-I), and of the functions N, there are In, or i(n+i). The normal forms of . solution of Laplace's equation, applicable to the space inside the ellipsoid, are the 211+1 products E (p) E(p) E(v). It can be shown that the 2n+1 values of p are real and unequal. It can be shown that, subject to certain restrictions, a function of p and v, arbitrarily given over the surface of the. ellipsoid p=po can be expressed as the sum of products of Lame's functions of p and v, in the form 00 211+t I 2=I the potential function for the space inside the ellipsoid, which has the arbitrarily given value over the surface of the ellipsoid, ie consequently` E En(P)EnI(pp) En(v) c" En Wl) It can be shown that a second solution of Lame's equation is F"(P) where cosh p a d,y o J 1 - k12 sin2t,ti = fP dp Cm. d is PZ —h2 P2 — ks — J p2 ha k2 _ 2' _ lc)" di' -J o v 112 v2 which are See also:

EQUIVALENT

equivalent to memoir, Theorie See also:

des attractions des spheroides et de la figure des pkdn(k4, ki), p=kdn(K—kn, ki), v=ksn(kE, ki'), planktes, published by the See also:

Paris See also:

Academy in Y785. Laplace was where See also:

KIT (1) (probably an adaptation of the Middle Dutch kitie, a wooden tub, usually with a lid and handles; in modern Dutch kit means a tankard)

kit, kl'2 denote the quantities t -h2/0, h2/k2 and K denotes the first to consider the functions of two angles, which functions the See also:

COMPLETE

complete elliptic integral , have consequently been known as Laplace's functions; his investi- gations on these functions are given in the Mecanique See also:

CELESTE, MADAME (1815–1882)

celeste, tome ii. livre iii., tome v. livre xi., and in the supplement to vol. v. The notation'P'n) was introduced by Dirichlet (see Crelle's See also:

Journal, vol. xvii., " sur See also:

les series dont In terme general depend de deux angles" &c.

; see also his memoir, " Ueber einenneuen Ausdruck zur Eestimmung der Dichtigkeit einer unendlich dfinnen Kugelschale," in the Abhandlungen of the See also:

Berlin Academy, 185o). The name " Kugelfunctionen " was introduced by See also:

GAUSS, KARL FRIEDRICH (1777-1855)

Gauss (see Collected' See also:

WORKS

Works, vi. 648). A See also:

DIRECT

direct investigation of the expression for the reciprocal of the distance between two points in spherical surface harmonics was given by See also:

Jacobi (Crelle's Journal, vol. See also:

XXVI

xxvi., see also vol. xxxii.). The functions of the second See also:

KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind were first introduced by Reine (see his " Theorie der Anziehung eines Ellipsoides," Crelle's Journal, vol. xlii., 1851). The above-mentioned investigators employed almost entirely polar co-ordinates; the use of Cartesian co-ordinates for the expression of spherical harmonics was introduced by See also:

KELVIN, WILLIAM THOMSON, BARON (1824-1907)

Kelvin in his theory of the See also:

EQUILIBRIUM (from the Lat. aequus, equal, and libra, a balance)

equilibrium of an elastic spherical See also:

shell (see F"(p) (2n+I)En(P)j~ dp p{ (P)}Zdp2-h2p-ki this f See also:

UNCTION (Lat. unctio, anointing, ungere, unguere, to smear with ointment, to anoint; cf. " ointment," O.Fr. oignement, from oigner, mod. oindre, to anoint)

unction F"(p) vanishesat infinity as p-^-', and is therefore adapted to the space outside the bounding ellipsoid. The external potential which has at the surface p = pi, the value EcnEi(p)E,I(v) isE EcnF(P)E"(p)En(v)• 32. See also:

HISTORY

History and Literature.—The first investigator in the subject was Legendre, who introduced the functions known by his name, and at See also:

PRESENT

present also called zonal surface harmonics; he applied them to the determination of the attractions of solids of revolution: Legendre's investigations are contained in a memoir of the Paris Academy, Sur'l'attraction des spheroides, published in 1785, and in a memoir published by the Academy in 1787, Recherches sur la' figure des planktes; his investigations are collected in his Exercices, and in his Traite des functions elliptiques. The potetial function was introduced by Laplace, who also first obtained the equation which i bears his name; he applied spherical surface harmonics to the determination of the potential of a nearly spherical solid, in his Phil. Trans. See also:

ROY, WILLIAM (1726-1790)

Roy. Soc., 1862), and also independently by Clebsch (see his See also:

paper, " Ueber See also:

die Reflexion an einer Kugelflache," Crelle's Journal, vol. lxi., 063).

The general theory of spherical harmonics of unrestricted degree, order and argument has been treated by Hob-son (Phil. Trans., 1896) ; see also a paper by See also:

Barnes in the Quar. Journ. Math. 39, p. 97. The functions which See also:

bear the name of Bessel were first introduced by See also:

Fourier in his investigations on the See also:

CONDUCTION, ELECTRIC

conduction of See also:

HEAT (0. E. haktu, which like " hot," Old Eng. hat, is from the Teutonic type haita, hit, to be hot; cf. Ger. hitze, heiss; Dutch, hitte, heet, &c.)

heat (see his Theorie analytique de la chaleur, 1822); they were employed by Bessel in the theory of planetary See also:

motion (see the Abhandlungen of the Berlin Academy, 1824). The functions which are now known as Bessel's functions of degree half an odd integer were employed by See also:

POISSON, SIMEON DENIS (1781-1840)

Poisson in the theory of the conduction of heat in a solid spherical See also:

BODY

body (see the Journ. de l'ecole polyt., 1823, cah. 19). The toroidal functions were introduced by C. See also:

Neumann (Theorie der Elektricitats- and Warmevertheilung in einem Ringe, See also:

Halle, 1864), and independently by See also:

Hicks (Phil. Trans.

Roy. Soc., 1881). The ellipsoidal harmonics were first investigated by Lame in connection with the stationary motion of heat in an ellipsoidal body (see Liouville's Journal, 1839, pt. iv. The external ellipsoidal harmonics were introduced by Liouville and See also:

HEINE, HEINRICH, (1797-1856)

Heine (see Liouville's Journal, vol. x., and Crelle's Journal, vol. xxix.). The ellipsoidal harmonics have been considered as expressed in Cartesian co-ordinates by See also:

Green (see Collected Works), by See also:

Ferrers (see his See also:

TREATISE

treatise), and by W. D. Niven (Phil. Trans. Roy. Soc., 1892). A method of representing ellipsoidal harmonics in a form adapted for actual use in certain See also:

PHYSICAL

physical problems has been See also:

DEVELOPED

developed by G. H.

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