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: - 1)* fusion
(1–0) (i-o – 1) (0 -0 =2)(1–0–3) wi-o-...},
Heine represents the expression
(-1) ), by the symbol Pot (re), and calls these expressions by the name Zugeordnete Functionen Erster Art (Handbuch der Kugelfunctionen, pp. 117, 118) which Todhunter translates by the term “Associated Functions of the First Kind," which we shall adopte Heine also represents the series (8-0) (-0-1)
2 (2i – 1)
(-o) (i-0-1)(-5-2) (-0-3)
Hi-o-, 2.4 (2i - 1) (21-3) by the symbol P: (u), (p. 117).
The several expressions, T6), olo), 9), Pó, P., are connected together as follows: 126.96.36.199...
T (O) = 0,02 2i (2i – 1)...(i-o+1)
) 2-0 i (i-1)...(0+1) i ) o+22i 8. It has been already remarked that the roots of the equation P,= 0 are all real. It follows also that those of the
dạP equations = 0, = 0... are real also, Hence we may
du du2 arrive at the following conclusions, concerning the curves, traced on a sphere, which result from our putting any one of these series of spherical harmonics = 0.
By putting a zonal harmonic=0, we obtain i small circles, whose planes are parallel to one another, perpendicular to
(i+o+1)(1+0+2)...279)=(-1)* P= (1 -myÝ P..
the axis of the zonal harmonic, and symmetrically situated with respect to the diametral plane, perpendicular to this axis. If i be an odd number this diametral plane itself becomes one of the series.
By putting the tesseral harmonic of the order o=0, we obtain i-o small circles, situated as before, and o great circles, determined by the equation cos o=0, or sin o = 0, as the case may be, their planes all intersecting in the axis of the system of harmonics, the angle between the planes of any two consecutive great circles being
By putting the sectorial harmonic = 0, we obtain i great circles, whose planes all intersect in the axis of the system, the angle between any two consecutive planes being
9. The tesseral harmonic may be regarded from another point of view. Suppose it is required to determine a solid harmonic of the degree i, and of the form Ycnt, such that Y, shall be the product of a function of y, and of a function of o, which functions we will denote by the symbols M,,,, respectively. The differential equation, to which this will lead, is
M, doo, ) +
du 1-p? d Now this will be satisfied, if we make M, and , satisfy the following two equations :
IM o2 )
i (i+1) M,+ amor (1 –w)
Φ = 0 coς σφ+ O' sin σφ.
former is satisfied by M,= T(«, i.e. (1–2)* (2) *** (1 – *)',
as we proceed to prove.
Differentiate o times, and we get
do+1p; (1-13 -2(0+1) u
do +1 P.
= :0, duo
(1 – 19)
duoti + (i–a)(i+o+1) and, multiplying by (1 – 12)
o Now, putting (1-4°)?
TO 1- μ
Hence the equation above given for M, is satisfied by M,= T (0), and the equation in Y, is satisfied by
Y,= CT, cos o$ + C'T) sin o$. 10. In Chap. II. Art. 10 we have established the fundamental property of Zonal Harmonics, that if i and m be two unequal
positive integers, L., P. Pudu=0. This is a particular case
of the general theorem that if Y., Yo be two surface harmonics of the degrees i and m respectively,
For, let Vu Vm be the corresponding solid harmonics, so that Vi=pky. Vm=gYm. Then, by the fundamental pro
Y perty of potential functions, we have at every point at which no attracting matter is situated, MV, MV, TV, dV
+ + = 0,
Ꮴ , Ꮴ , ]
dy or, in accordance with our notation, V.VV. - V.,°V,= 0.
Now, integrate this expression throughout the whole space comprised within a sphere whose centre is the origin and radius a, a being so chosen that this sphere contains no attracting matter. We then have
Sljivv*- 7.7°V) dx dy da=0.
Sliv.v* V. - V„v*V) d.odyda = J\(v.
But also, when the integration extends over all space comprised within any closed surface, we have
d dS denoting an element of the bounding surface, and
dn differentiation in the direction of the normal at any point.
Now, in the present case, the bounding surface being a sphere of radius a, and V., V. homogeneous functions of the degrees i, m, respectively,
ау. dS = a dudo,
= mam. Ym, dn
dn and, the integration being extended all over the surface of the sphere, the limits of u are – 1 and 1, those of $,0 and 27. Hence dV