importance in connexion with the differential equations of physics. The two equations
are reducible by means of the substitutions u = e~ k 'v, u = e lk 'v to the form v*»+»=o. If we suppose v to be a function of r only, this last differential equation takes the form
0,
so that v has the values
sin r/r, cos rjr; in order to obtain more general solutions of the equation v*»+d=o, we may operate on
sin rjr, cos rjr with the operator
Y f±. ± -±) In Va*' dy' dzj '
where Y„ (x, y, z) is any spherical solid harmonic of degree n. The result of the operation may be at once obtained by taking Y„ {x, y, z) for /„(*, y, z) in the theorem (7'), we thus find as solutions, of V*v+f = o, the expressions
Y.(x, y. z)j^ By recurring to the definition of the function J m (r), we see that
sin r v , s d" cosy
-7—, '»ft y< z -><Z( r 2)» r '
j*wV£i'-
r 2 r
2.3 2-3-
4-5
-â– M
2 sin r
thus
r-ijl^
, /2 sin r
Using the relation between Bessel's functions whose orders differ by an integer, we have
Mr).(-2)^,^-(-.)VV' d " Sinr
<Z(r 2 )"
It may be shown at once that
that the function K„(z) has no real zeros unless «=2fe + l where k is an integer, when it has one real negative zero; and that K n (z) has no purely imaginary zeros, and no zero whose real part is positive, other than those at infinity. When l>re>0, K„(z) has no zeros other than those at infinity, when 2>n>l,it has one zero whose real part is negative, and when m+i>-n>m where m is ah integer, there are to 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 ellipsoid of revolution, the co-ordinates to be used are r, 0, <t> where in the case of a prolate spheroid
- = c \V 2 — I sin cos <f>, y = c-Jr* — l sin sin <p, z = cr cos 0,
the surfaces r = r , 0=0o, 4>=<t><s are confocal prolate spheroids, confocal hyperboloids of revolution, and planes passing through the axis of revolution. We may suppose r to range from 1 to 00 , 8 from o to ir, and <j> from o to 2ir, every point in space has then unique co-ordinates r, 8, <j>.
For oblate spheroids, the corresponding co-ordinates are r, given by
8, 4,
»+i
d n cos r
d{r*)» r
a second solution of 4>) denotes a surface
x = c^r z + i sin 8cos<t>, y = c\V 2 + l sin 8 sin <t>, z = cr cos 8,
where
0< r < w , o < < ff, o < tfr < 2jt ;
these may be obtained from those for the prolate spheroid by chang- ing c into — ic, and r into ir.
Taking the case of the prolate spheroid, Laplace's equation becomes
3V\ , r 2 -cos'0 a 2 v
d { ,„ ,aV) . 1 a / . „av\ .
m<t>.
dr S ' sin 8 08 V*" " 68/ " r (^-i)sin»9 dtf and it will be found that the normal solutions are
p»)p;T(cos0);
Q:(r))Q:(cos8)
For the space inside a bounding spheroid the appropriate normal
m are positive integers, and for the external space
forms are P"(r)Vâ„¢ (cos 8)^?*m<t>, where n,
is a second solution of Bessel's equation of order n-f-j; thus the differential equation y*v+v = o is satisfied by the expression
y.(*. y, 3) J -^tV
and by the corresponding expression with Bessel's equation instead of J^+i (r) ; if S„(/j, harmonic of degree n, the expression
S.O., *)^ w l(r)
is a solution of the equation v 2 ti-{-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 trans- cendental functions other than circular functions. It will be observed that in this case the semi-convergent series for J m 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. Ann. vol. xxxiii.j, and more completely by H. M. Macdonald (Proc. Lond. Math. Soc. vols. xxix.,xxx.). It has been shown that the zeros of J»(z)/z* are all real and associated with the singular point at infinity when n is real and > — 1 , 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 — to, Jr.(z)/z" has, in addition,, 2w real zeros co- incident at the origin. When n — — m— v, m being a positive integer, and i>f>o, LCzWz" has a finite number 2tn 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-,i, and t\vo 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(z)/z n , one zero of the latter, and one only, lies between two consecutive zeros of Jn(z),/z". When n is real and >— I, all the zeros of Jn(z)/z" are givenby a formula due to Stokes; the m th positive zero in order of magnitude is given by
q:wp„-(cos0)^>*.
5£r
For the case of an oblate spheroid, P"('f), Q7("')i take the place
of K(r), QT(r).
30. Toroidal Functions. — For potential problems connected with the anchor-ring, the following co-ordinates are appropriate: If A, B are points at the extremities of a diameter of a fixed circle, and P is any point in the plane PAB which is perpendicular to the plane of the fixed circle, let P = log(AP/BP), 0=/_APB, and let <t> be the angle the plane APB makes with a fixed plane through the axis of the circle. Let be restricted to lie between — ir and ir, a discontinuity in its value arising as we pass through the circle, so that within the circumference is ir on the upper side of the circle, and — ir on the lower side ; 8 is zero in the plane of the circle outside the circumference; p may have any value between —00 and co . and <j> any value between o and 2ir. The position of a point is then uniquely represented by the co-ordinates p, 8, <t>, which ate the parameters of a system of tores with the fixed circle as limiting circle, a system of bowls with the fixed circle as common rim, and a system of planes through the axis of the tores. If x, y r 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 . , _ ,0. sin
,sin<p, Z =
cosh p— cos 8
4n»— 1 4(4^— i)(28w 8 — 31)
— &C.,
I
sinh p dtf " can be shown
that this
a sinh p x ~ cosh p - cos COS *' y = cosh p - cos C
where a is the radius of the fixed circle. ' Laplace's equation reduces to
_a ( sinhp aV ) _a \ sinhp 3V dp\ P 2 dp ) + d9\ P 2 d8 )
when P denotes V(cosh p— cos 0). It equation is satisfied by
1 1 u . P»_i(cosh p) cos „ cos ,
V (cosh p-cos ^Q-^cosh p) sin "" sin m *'
the functions P"_ } (cosh p), Qr_|(cosh p) required for the potential problems, are associated Legendre's functions of degree n — |, half an odd integer, of integral order «, and of argument real and greater than unity; these are known as toroidal functions. For the space
external to a boundary tore the function Qâ„¢_ i (cosh p) must be
where (i = iir(2»+4»»-i). It has been, shown by Macdonald | used > al » d for the internal space P'^cosh p).
