Equilibrium ellipsoid of three unequal axes.

Digression

harmonics.

of c can be assigned which will make these values of o2 equal Then we must show that the value of w2 thus found is positive. giving a real value of w. If we put, as in § 522,

0=(a3—b3) [ 。 √ (a2+4)(b3+4) (c° + Y)

2

1

, or

(a2+4)(b2+4)+a2b2 cì.

a2=b3 gives the ellipsoid of revolution already treated. But the equation may be satisfied without assuming a2= b2, since the factor in brackets, in the integral, may be written in the form

(c3a2+c3b2—a2b3)¥+c3¥3

a2b2 (a2+4)(b2+4) (c2+4)

Now if e be

of which the numerator alone can change sign.
greater than the greatest of a and b, the integral is positive; if
c be very small, it is evidently negative. Hence for any finite
values of a and b whatever the integral may be made to vanish
by properly assigning c. With this value of e the integral con-
tains an equal amount of positive and negative elements. But
it can have no negative elements unless ca+c2¿a — a2b2 is
negative, i.e., unless c is less than the least of a and b.

M

do

d(c2) "d(a2)}

dy

(a2-e2)

= ? } ] 。 √ (a2 + 4) (b2 + 4) (c2 + 4) (a2+4) (c2+4)

which is positive, as we have shown that c is less than a; and gives the required angular velocity when c has been found from (5).

с

779. A few words of explanation, and some graphic illustra on spherical tions, of the character of spherical surface harmonies may promote the clear understanding not only of the potential and hydrostatic applications of Laplace's analysis, which will occupy us presently, but of much more important applications to l made in Vol. II, when waves and vibrations in spherical fal or elastic solid masses will be treated. To avoid circund

on spherical

spheroid.

ations, we shall designate by the term harmonic spheroid, or Digression herical harmonic undulation, a surface whose radius to any harmonics. oint differs from that of a sphere by an infinitely small length Harmonic arying as the value of a surface harmonic function of the osition of this point on the spherical surface. The definitions of spherical Solid and Surface Harmonics [App. B. (a), (b), (c)] how that the harmonic spheroid of the second order is a surface of the second degree subject only to the condition of being approximately spherical: that is to say, it may be any elliptic spheroid (or ellipsoid with approximately equal axes). Generally a harmonic spheroid of any order i exceeding 2 is a surface of algebraic degree i more restricted than merely to being approximately spherical.

Let S be a surface harmonic of order i with coefficient of leading term so chosen as to make the greatest maximum value of the function unity. Then if a be the radius of the mean sphere, and c the greatest deviation from it, the polar equation of a harmonic spheroid of order i will be

a

if S is regarded as a function of polar angular co-ordinates, 0, 4.
Considering that is infinitely small, we may reduce this to an
equation in rectangular co-ordinates of degree i, thus :-Squaring
each member of (1); and putting for

с

from which it

cri -) ai+1

a

differs by an infinitely small quantity of the second order, we

have

2c
r2=a2+ riSi
a2

(2).

This, reduced to rectangular co-ordinates, is of algebraic degree i

nodal cone

780. The line of no deviation from the mean spherical Harmonic surface is called the nodal line, or the nodes of the harmonic and line. spheroid. It is the line in which the spherical surface is cut by the harmonic nodal cone; a certain cone with vertex at the centre of the sphere, and of algebraic degree equal to the order of the harmonic. An important property of the harmonic nodal line, indicated by an interesting hydrodynamic theorem due to Rankine, is that when self-cutting at any point or points, the different branches make equal angles with one another round each point of section.

"Summary of the Properties of certain Stream-Lines." Phil. Mag., Oct. 1864.

Digression on spherical harmonics.

Theorem regarding

nodal cone.

Cases of

solid har

monics re

solvable

B. (a)] a homogeneous function of degree i, we may write

V1=H12+H12-1+H12-2+H2-+etc.

12

where Ho is a constant, and H1, H2, H3, etc., denote integral homogeneous functions of x, y of degrees 1, 2, 3, etc.; and the the condition V2 Vi=0 [App. B. (a)] gives

VH,+i(i-1)H。=0, V3H2+(i—1) (i—2) H2 =0,
V2H2+(i-s+2) (i−8+1) Hi-8+2=0

which express all the conditions binding on H., H1, H1, etc.
Now suppose the nodal cone to be autotomic, and, for brevity
and simplicity, take OZ along a line of intersection. Then z=a
makes (3) the equation in x, y, of a curve lying in the tangent
plane to the spherical surface at a double or multiple point of the
nodal line, and touching both or all its branches in this point.
The condition that the curve in the tangent plane may have a
double or multiple point at the origin of its co-ordinates is, whet
(4) is put for Vi,

H.=0; and, for all values of x, y; H1=0.

we have A+B=0. This shows that the two branches cut one
another at right angles.

If the origin be a triple, or n-multiple point, we must have
H=0, H1=0...H-1=0,

H„=A{(x+y√−1)*+(x−y√=1)"}+B√=1{(x+y√—1)*—(x—y√—I,•},

or, if x=pcos, y=psin &,

Hn=2p"(A cos no+Bsinno)

which shows that the n branches cut one another at equal angles round the origin.

i

781. The harmonic nodal cone may, in a great variety of cases [V; resolvable into factors], be composed of others of lower into factors. degrees. Thus (the only class of cases yet worked out) each of the 2i+1 elementary polar harmonics [as we may conveniently call those expressed by (36) or (37) of App. B., with any on alone of the 2i+1 coefficients A,, B.] has for its nodes circles

Polar harmonics.

on spherical

Zonal and

sectorial harmonies

defined.

the spherical surface. These circles, for each such harmonic Digression ement, are either (1.) all in parallel planes (as circles of lati- harmonics. ide on a globe), and cut the spherical surface into zones, in hich case the harmonic is called zonal; or (2.) they are all in lanes through one diameter (as meridians on a globe), and cut e surface into equal sectors, in which case the harmonic is alled sectorial; or (3.) some of them are in parallel planes, nd the others in planes through the diameter perpendicular to hose planes, so that they divide the surface into rectangular uadrilaterals, and (next the poles) triangular segments, as reas on a globe bounded by parallels of latitude, and meridians t equal successive differences of longitude.

With a given diameter as axis of symmetry there are, for complete harmonics [App. B. (c), (d)], just one zonal harmonic of each order and two sectorial. The zonal harmonic is a function

π

of latitude alone (—— 0, according to the notation of App. B.);

(0)

2

sin Ocosip, and sin'@sin ip

being the given by putting s = 0 in App. B. (38). The sectorial harmonics of order i, being given by the same with si, are (1). The general polar harmonic element of order i, being the coss and sins of B. (38), with any value of s from 0 to i, has for its nodes i-s circles in parallel planes, and s great circles intersecting one another at equal angles round their poles; and the variation from maximum to minimum. along the equator, or any parallel circle, is according to the simple harmonic law. It is easily proved (as the mathematical student may find for himself) that the law of variation is approximately simple harmonic along lengths of each meridian. cutting but a small number of the nodal circles of latitude, and not too near either pole, for any polar harmonic element of high order having a large number of such nodes (that is, any one Tesseral for which is is a large number). The law of variation along surface by a meridian in the neighbourhood of either pole, for polar har- polar harmonic elements of high orders, will be carefully examined and illustrated in Vol. II., when we shall be occupied with vibrations and waves of water in a circular vessel, and of a circular stretched membrane.

division of

nodes of a

monic.

Digression on spherical harmonics.

Murphy's

analytical invention

of the zonal

harmonics.

782. The following simple and beautiful investigation the zonal harmonic due to Murphy1 may be acceptable to ti analytical student; but (§ 453) we give it as leading to a r ful formula, with expansions deduced from it, differing fr any of those investigated above in App. B. :

"PROP. I.

:

"To find a rational and entire function of given dimensions "with respect to any variable, such that when multiplied by any rational and entire function of lower dimensions, ti integral of the product taken between the limits 0 and 1 shall always vanish.

"Let f(t) be the required function of n dimensions with respect "to the variable t; then the proposed condition will evidently re"quire the following equations to be separately true; namely, “(a)..............ff(t)dt=0, £f(t).tdt=0, [f(t).t3dt=0,......[f(t),t®−1dt=6° "each integral being taken between the given limits.

"Let the indefinite integral of f(t), commencing when t=0, be represented by fi(t); the indefinite integral of fi(t), commencing "also when t=0, by f(t); and so on, until we arrive at the "function f(t), which is evidently of 2n dimensions. Then the "method of integrating by parts will give, generally,

"ff(t).tdt=tf1(t)—xt-1ƒ(t)+x. (x − 1).t22. ƒ,(t)-etc. "Let us now put t=1, and substitute for x the values 1, 2, 3. "......(i-1) successively; then in virtue of the equations (h we get,

“ (b).......................f1(t)=0, fa(t)=0, fa(t)=0,.........fi(t)=0.

"Hence, the function fi(t) and its (i-1) successive differential "coefficients vanish, both when t=0, and when t=1; therefore “ți and (1—t)' are each factors of fi(t); and since this function is "of 2i dimensions, it admits of no other factor but a constant c. 'Putting 1-t=t', we thus obtain

"Corollary. If we suppose the first term of f), when arranged according to the powers of t, to be unity, we evidently have