or (-1)), by the symbol P(u), 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 adopt

Heine also represents the series

The several expressions, T), ), ), P., P., are connected together as follows:

8. It has been already remarked that the roots of the equation P= 0 are all real. It follows also that those of the dP. dᏢ αμπ

equations

άμ

=

0... are real also, Hence we may

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

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 σ=0, we obtain - small circles, situated as before, and a great circles, determined by the equation cosσ = 0, or sin op = 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

π

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 Yr, such that Y shall be the product of a function of μ, and of a function of 4, which functions we will denote by the symbols M,,,, respectively. The differential equation, to which this will lead, is

Now this will be satisfied, if we make M, and Þ, satisfy the following two equations:

And, taking as an integer, positive or negative, the

σ

d\ito

former is satisfied by M,= T(~”), i.e. (1— μ3)3 (11)**° (1 — μ”)',

do+2P.

do+1p

αμστε

αμστι

αμσ

(1-μ) 1 - 2 (0+1) μ duo +1 - (0+1)

do+1

+ (i − o) (i + o + 1) (1 − μ3) dze + o2 (1 − μ3å ¬d2Ð ̧

άμα

αμσ

Hence the equation above given for M, is satisfied by M1 = T), and the equation in Y, is satisfied by

Y1 = CT, cos σ$+ C'T) sin op.

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, P.Pdu=0. This is a particular case of the general theorem that if Y, Y be two surface harmonics of the degrees i and m respectively,

m

For, let V, V be the corresponding solid harmonics, so that V=Y, VY. Then, by the fundamental property of potential functions, we have at every point at which no attracting matter is situated,

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

[[[(V.、aV„— V„VaV.) dx dy dz = 0.

SSS

But also, when the integration extends over all space comprised within any closed surface, we have

[[[(V♡aV„− V„V1V) dx dy dz= [[(V.

т

m

dV

Vi an

m

dV

m dn

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,

and, the integration being extended all over the surface of the sphere, the limits of μ are - 1 and 1, those of 4,0 and 2π. Hence