for the whole surface. Adding the corresponding terms for y a constant

and z, and remarking that

gives a theorem of Green's.

where B and C denote the inclinations of the outward normal through dS to lines drawn through dS in the positive directions parallel to y and z respectively, we perceive the truth of (1).

(b) Again, let U and U' denote two functions of x, y, z, which have equal values at every point of S, and of which the first fulfils the equation

d(a2 dt) d (a db) a (a dl)

dy

dz

= 0........

dz

·(2),

Equation of the conduction of heat.

For the first member is equal identically to the second member

dxdydz,

of which each term vanishes; the first, or the double integral, because, by hypothesis, u is equal to nothing at every point of S, and the second, or the triple integral, because of (2).

solution

S.

(c) The second term of the second member of (3) is essentially Property of positive, provided a has a real value, whether positive, zero, or in negative, for every point (x, y, z) within S. Hence the first given over member of (3) necessarily exceeds the first term of the second member. But the sole characteristic of U is that it satisfies (2). Solution Hence U' cannot also satisfy (2). That is to say, U being any proved to

be determi

nate;

proved to be possible.

one solution of (2), there can be no other solution agreeing with it at every point of S, but differing from it for some part of the space within S.

(d) One solution of (2) exists, satisfying the condition that U has an arbitrary value for every point of the surface S. For let U denote any function whatever which has the given arbitrary value at each point of S; let u be any function whatever which is equal to nothing at each point of S, and which is of any real finite or infinitely small value, of the same sign as the value of

at every

at each internal point, and therefore, of course, equal to nothing internal point, if any, for which the value of this expression is nothing; and let U' = U + Ou, where @ denotes any constant. Then, using the formulæ of (b), modified to suit the altered circumstances, and taking Q and Q' for brevity to denote

dU

sss {(alb)" + (aly ) * + (alle)" } dxdydz,

dy

dz

and the corresponding integral for U', we have

dU

Q = Q-2055su{ d (ao dl) + d (a dl) + d (a dl) } dxdydz

dx

dy

dz dz

+ 0°ƒƒƒ{(adu)" + (adu)* + (adu)" } dxdydz.

dx

The coefficient of - 20 here is essentially positive, in consequence of the condition under which u is chosen, unless (2) is satisfied, in which case it is nothing; and the coefficient of 02 is essentially positive, if not zero, because all the quantities involved are real. Hence the equation may be written thus:

where m and n are each positive. This shows that if any positive value less than n is assigned to 0, Q' is made smaller than Q; that is to say, unless (2) is satisfied, a function, having the same value at S as U, may be found which shall make the Q integral smaller than for U. In other words, a function U, which, having any prescribed value over the surface S, makes the integral for the interior as small as possible, must satisfy equation (2). But the Q integral is essentially positive, and therefore there is a limit than which it cannot be made smaller.

Hence there is a solution of (2) subject to the prescribed surface Solution condition.

proved possible.

(e) We have seen (c) that there is, if one, only one, solution of (2) subject to the prescribed surface condition, and now we see that there is one. To recapitulate, we conclude that, if the value of U be given arbitrarily at every point of any closed a constant surface, the equation

de (ade) + d (ad) + d (ad) = 0

dx

dy dz dz

determines its value without ambiguity for every point within that surface. That this important proposition holds also for the whole infinite space without the surface S, follows from the preceding demonstration, with only the precaution, that the different functions dealt with must be so taken as to render all the triple integrals convergent. S need not be merely a single closed surface, but it may be any number of surfaces enclosing isolated portions of space. The extreme case, too, of S, or any detached part of S, an open shell, that is a finite unclosed surface, is clearly included. Or lastly, S, or any detached part of S, may be an infinitely extended surface, provided the value of U arbitrarily assigned over it be so assigned as to render the triple and double integrals involved all convergent.

gives Green's theorem."

B.-SPHERICAL HARMONIC ANALYSIS.

spherical

analysis.

The mathematical method which has been commonly referred object of to by English writers as that of "Laplace's Coefficients," but harmonic which is here called spherical harmonic analysis, has for its object the expression of an arbitrary periodic function of two independent variables in the proper form for a large class of physical problems involving arbitrary data over a spherical surface, and the deduction of solutions for every point of space.

(a) A spherical harmonic function is defined as a homogeneous Definition function, V, of x, y, z, which satisfies the equation

Its degree may be any positive or negative integer; or it may

of spherical harmonic functions.

be fractional; or it may be imaginary.

Examples of spherical harmonics.

EXAMPLES. The functions written below are spherical harmonics of the degrees noted; r representing (x2+y2+z2)1:—

if V, denote any harmonic of degree 0: for instance, group III.

Vo

below.

III.

IV.

where V, denotes any spherical harmonic of integral degree, j,

and 8, -,-, homogeneous integral functions of

d d d

dx' dy' dz'

of degrees n and n-j-1 respectively: for instance, some of group II. above, and groups V. and VI. below.

d

dz

position of z constant, and differentiation with reference to

on supposition of x and y constant.

sin

Degree -i-1, or +i, and type H{z, √(x2 + y2)}

пф.

COS

H denoting a homogeneous function; n any integer; and i any positive integer.

0

Let U and denote functions yielded by V. and VI. preceding. The following are the two distinct functions of the degrees and types now sought, and found in virtue of (g) (15) below: