The following expressions may be given for the toroidal functions: —
P^j(coshp)
ir H(»— m— j)J a (cosh p + sinh p cos .
cos m(j>
JTP^
I n(re+m — J) f , i , -i ,n„_i ,j,
= — jtH r~- (coshp -f sinhp cos 0)" 5 cos m$d<t>.
w ii\n~i) Jo
P^.(coshp)=2J^_
cosh n<t>
=d<t>.
0V2 cosh p — 2 cosh <£
Q^j(coshp) = (-1)" n („_|) j .. ■( cosh P
■— sinh p cosh w)"~3 cosh mwdw
^(-^TUm-mi-i) -nh " P f; (2 cosh ;°_ S f cos0) ^^.
The relations between functions for three consecutive values of the degree or the order are
2»t cosh pP ? ^,(cosh p) — (« — m + 1) P™ , (cosh p) — (n-\-m — l)P^_i (cosh p) = o. P^?(cosh p) + 2(m + 1) coth pP^Hcosh p) — (n.— »»;— j)(» + »»4-5)P^_, (cosh p) = 0,
with relations identical in form for the functions Q^, (cosh p). The function Q„_;(cosh p) is expansible in the form
3^g2^W+npF(i, M + h -n+.i, «-«p),
which is useful for calculation of the function when p is not small. P»_i(cosh p) can also be expressed in terms of tr* 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 sheet, and of two sheets; these co-ordinates are three roots of the equation
-+
pTp-h^P-U
+ «
j = l, (k>h);
we thence find that
_ppv Vp'-^VM^-gVtf-.- 2 _Vp 2 -£ 2 Vfe 2 -p. 2 V'& 2 -y 2
where 00 >p 2 >h 2 , W- < p. 2 < h- , and £ 2 >f 2 >o.
We find from these values of *, y, z
(p 2 -^)(m 2 -, 2 )
and on applying the general transformation of Laplace's equation that equation becomes
(M 2 -^)-g|2 +(p'-^)y^ -Hp 2 -/* 2 )-^ =0, where £, if, f are defined by the formulae
t ff d» r)=C* d »
- -A
<z*
^
of the parameters £, jj, f in terms of p, p, 1/, ^e find that the equation satisfied by E(p) becomes
(p 2 -^)(p 2 -* 2 )^+p(V-ft 2 -* 2 )^
+\(h 2 +k 2 )p-n(n + l)p 2 }E(p) =0,
and E(p.), E(i>) satisfy equations in p, v respectively of identically the same form ; this equation is known as Lamp's equation.
If n be taken to be a positive integer, it can be shown that it is possible in 2« + i ways so to determine p that the equation in E(p) is satisfied by an algebraical function of degree n, rational in p, V (p 2 — h 2 ), V(p 2 — k 2 ). The functions so determined are called Lamp's functions, and the 2» + I functions of degree n are of one of the four forms. -
K(p) = flo p" + a ip"^ + ■• ■> L(p) = Vp 2 -A 2 (oo' p"" 1 + o'lp"" 3 + ...), M(p) = V 7^F (o o"p"- 1 + a"i p»- 3 +...), N(p) = yl7^k\^7^¥{<"p n ^+a["p"- i + . . .).
These are the four classes of Lame's functions qf degree, n ; of the functions K there are l+|», or i(n-\-l), according as n is even or odd ; of each of the functions L, M, there are in, or i(n — 1), and of the functions N, there are -|», or i(n + i).
The normal forms of. solution of Laplace's equation, applicable to the space inside the ellipsoid, are the 2m + i products E (p) Kip.) E(v). It can be shown that the 2ra + i values of p a,re 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 = pi, can be expressed as the sum of products of Lame's functions of p and c, in the form
00 2M+I
2 2 C jEjl(p.)E;(x); i s-i
the potential function for the space inside the ellipsoid, which has the arbitrarily given value over the surface of the ellipsoid, is consequently
22
, e^(p)e;( m )e„w EJKpO
Wtf-vVF-?'
which are equivalent to
p^kdn{k£, ki),p~=kdn(K~kr), ki), v-ksn(k(, hi), where ki 2 , k\ n denote the quantities i—h 2 jk', VjV and K denotes the complete elliptic integral
Vi-fe ! sinV
It can now be shown that Laplace's equation is satisfied by the product E(p)E(p,)E(>'), where E(o) satisfies the differential equation
^^-M«+l)p 2 -(ft 2 + fe 2 )£!E(p)=o ;
and E(jj), E(v) satisfy the equations
^|^+[»(n4 i) M t-.0(A»+tf)]EGi)«O,__
,^|^~ [n{n+ iy -p{W+V)]E{S) =0, where n and p are arbitrary constants. 'On substituting the values
It can be shown that a second solution of Lame's equation it, F„(p) Where
F„( P ) = (2B+I
)En{ P )j' i
dp
plE„(p)l 2 Vp 2 -^-Vp 2 -fe 2 '
this function F„(p) vanishes'at infinity as p~" -1 , and is therefore adapted tq the space outside the bounding ellipsoid. The external potential which has at the surface p = pi, the value
22^ E *) E »« is S2 c "
'F« P i)
El(p)El(v).
32. History and Literature. — The first investigator in the subject was Legendre, who introduced the functions known by his name, and at present also called zonal surface harmonics; he applied them to the determination bf the attractions Of solids of revolution. Legendre's investigations are contained in a memoir Of the Paris Academy, Sur V 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 potential function was introduced by Laplace, who also first obtained the equation' which tears his name; he applied spherical surface harmonics to the
j determination of the potential of a nearly spherical solid, in his memoir, ThSorie des attractions des spheroides el de la figure des planUes, published by the Paris Academy in ^785. Laplace was the first to consider the functions of two angles, which Junctions have consequently been known as Laplace's functions; his investi-
' gationson these functions are given in the Mecanique celeste, tome ii.
j livre iii., tome v. livre xi., and in the supplement to vol. v. The notation P'") was introduced by Dirichlet (see Crelle's Journal, vol. xvii., " sur les series dont le terme general depend de deux angles " &c. ; see also his memoir, " Ueber einen neuen Ausdruck zur Bestimmung der Dichtigkeit einer uhendlich diinnen Kugelschale," in the Abhandlungen of the Berlin Academy, 1850). The name " Kugel-functionen " was introduced by Gauss (see Collected Works, yi. 648). A direct investigation of the expression for the reciprocal of the distance between two points in spherical surface harmonics was given by Jacobi (Crelle's journal, vol. xxvi., see also vol. xxxii.). The functions of the second kind were first introduced by Heine (see his " Theorie der Anziehung eines Ellipsoides," Cretle'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 Kelvin iri' his. theory of the equilibrium of an elastic •spherical shell (see
