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Laplace's equation in generalized co-ordinates.

we find for the symbol y' in terms of the generalized co-ordinates &, &', $",

1 sd 1 [R'R” sin' a d p =

+R(cos a cos a' – cos a") QRRR d Q R d's

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d dỹ'

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where for Q, its value by (8) in terms of a, a', a" is to be used, and a, a', a", R, R', Rare all known functions of $, &', É" when the system of co-ordinates is completely defined.

($) For the case of rectangular co-ordinates whether plane or curved a=a' = a" = A = A' = A" = 90° and Q=1, and therefore

Case of rectangular co-ordi. nates, curved or plane.

we have

(

(Fe) ...(1

1 Id /R'R'' d d R"Rd d RR d

+

+ RRR" Is

...(13), R

R ds

" ( R'' di which is the formula originally given by Lamé for expressing in terms of his orthogonal curved co-ordinate system the Fourier equations of the conduction of heat. The proof of the more general formula (12) given above is an extension, in purely analytical form, of a demonstration of Lamé's formula (13) which was given in terms relating to thermal conduction in an article “On the equations of Motion of Heat referred to curvilinear co-ordinates" in the Cambridge Mathematical Journal (1843).

(9) For the particular case of polar co-ordinates, r, 0, 0, considering the rectangular parallelepiped corresponding to år, 80, 8$ we see in a moment that the lengths of its edges are år, r8, r sin 686. Hence in the preceding notation R=1, R' = r, R"= r sin 0, and Lamé's formula (13) gives

1
sin dd)

... (14).

(b) Again let the co-ordinates be of the kind which has Laplace's

equation in been called “columnar”; that is to say, distance from an columnar axis (r), angle from a plane of reference through this axis to dinates. a plane through the axis and the specified point (®), and distance from a plane of reference perpendicular to the axis (z). The co-ordinate surfaces here are

+

transformation

coaxal circular cylinders (r = const.),
planes through the axis ( =const.),

planes perpendicular to the axis (z = const.).
The three edges of the infinitesimal rectangular parallelepiped
are now dr, rdo, and dz. Hence R=1, R'=r, R"=1, and
Lamé's formula gives

ld d d =

..(15), dr rdd dz which is very useful for many physical problems, such as the conduction of heat in a solid circular column, the magnetization of a round bar or wire, the vibrations of air in a closed circular cylinder, the vibrations of a vortex column, &c. &c.

(i) For plane rectangular co-ordinates we have R = R' = R'' ; Algebraic so in this case (13) becomes (with x, y, z for Ś, Š, ''),

from plane do d đ

.(16), dx dy't dz

ordinates. which is Laplace's and Fourier's original form.

(j) Suppose now it be desired to pass from plane rectangular co-ordinates to the generalized co-ordinates.

Let x, y, z be expressed as functions of $, &,"; then putting for brevity

dz
dx

dx X,

= X', &c.; de

=X", &c. ... (17); αξ

d}" we have

dx = X8% + X'8' + X"85",

&°X8€) dy=Y8+ Y'DE' + Y"SE",

...(18); dz = ZOE + Z'SÉ + Z" SE", whence R= N(X + Y? + Z'), R' = 1/(X" + Y" + Z'),

R" = (X" + Y"+ Z''?).................(19),

rectangular to generalized co

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+

dc de

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....(20).

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cos a'

Algebraic and the direction cosines of the three edges of the infinitesimal
transtor-
mation

parallelepiped corresponding to dě, , " are
from plane
rectangular

X Y Z X' Y' Z" X" Y" Z" to generalized coÃ' Ã'R R' R''

R'' R'' R'' ordinates.

Hence
X'X" + Y'Y" + Z'Z"

X"X+Y"Y + Z'Z
R'R'

RR
XX' + YY' + ZZ'
cos a" =

..(21).

RR
(k) It is important to remark that when these expressions
for cosa, cos a', cos a", R, R', R", in terms of X, &c. are used in
(8), Q' becomes a complete square, so that QRR'R" is a rational
homogeneous function of the 3rd degree of X, Y, Z, X', &c.

For the ordinary process of finding from the direction cosines
(20) of three lines, the sine of the angle between one of them and
the plane of the other two gives

|X, Y, Z
sin p= X', Y', Z'
RR'R" sin a......

(21);
X", Y", 2"
from this and (7) we see that QRR’R" is equal to the deter-

minant. From this and (8) we see that
Square of a (X’ + Y + Z')(X" + Y" + Z') (X"? + Y" + Z'')
determi-
-(X+YR+Z)(X'X"+Y'Y"+Z'Z")-(X"+Y"+Z“)(X"X+Y"Y+Z" 2)

- (X"? + Y"+ Z")(XX' + YY' + ZZ")
+2 (X'X"+Y'Y" + Z'Z")(X"X+Y"Y+Z"Z)(XX'+YY'+ZZ')

X, Y, Z,
X', Y', Z',

.(22),
X", Y", Z",
an algebraic identity which may be verified by expanding both
members and comparing.

(1) Denoting now by T the complete determinant, we have

T

:(23),

RR'R and using this for Q in (12) we have a formula for p in which only rational functions of X, Y, Z, X', &c. appear, and which

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1

is readily verified by comparing with the following derived from Algebraio (16) by direct transformation.

from plane

rectangular (m) Go back to (18) and resolve for 8, 88, 8". We find

lized coL M N ds

dy +

8z, SÉ = &c., 8" = &c., T T T

transformation

to genera

da +

ordinates.

where
L=Y'Z'-Y'Z', M=Z'X"-2"X', N=X'Y"-X"Y',
L'=Y"Z-YZ", M'=2"X-ZX", N'=X"Y-XY", ..... (24).
L"=YZ'-Y'Z, M"=ZX'-Z'X, N"=XY-X'Y,
Hence

+

d
s dz

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+

+

T

d Ld L'd L" d d

&c., &c.,
dx
Ταξ Τ αξ'

T dy
and thus we have
Ld L'd L" d Md M'd M" d

+

+
I de+ T de TE

T de T T ds"
Nd N' d N" d

(25).
Τ' αξ" Τ αξ
T

d? đ Expanding this and comparing the coefficients of

dg' dede'' d

&c. with those of the corresponding terms of (12) with (21) dછું” and (23) we find the two formulas, (12) and (25), identical.

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A.-EXTENSION OF GREEN'S THEOREM.

It is convenient that we should here give the demonstration of a few theorems of pure analysis, of which we shall have many and most important applications, not only in the subject of spherical harmonics, which follows immediately, but in the general theories of attraction, of fluid motion, and of the conduction of heat, and in the most practical investigations regarding electricity, and magnetic and electro-magnetic force.

(a) Let U and U' denote two functions of three independent variables, x, y, z, which we may conveniently regard as rectangular co-ordinates of a point P, and let a denote a quantity which may be either constant, or any arbitrary function of the

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a constant gives a theorem of Green's.

variables. Let SS dxdydz denote integration throughout a finite singly continuous space bounded by a close surface S; let sids denote integration over the whole surface S; and let d, prefixed to any function, denote its rate of variation at any point of S, per unit of length in the direction perpendicular to Soutwards. Then

IU dU

dU DU" dUdU"
dx dx dy dy dz da

dU
a
dx
dy

dz
= Sjas. Ua®SU SSSU"

dædydz
dx
dy

dz
ed!
doc

dy
= Sjas. Va®SU' - SISU

dxdydz dac dy

dz

........(1). For, taking one term of the first member alone, and integrating " by parts,” we have

il d "

DU

dx dxdydz = SSU'a dydz - SSSU dadydz, dx dx

dx

dx

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the first integral being between limits corresponding to the surface S; that is to say, being from the negative to the positive end of the portion within S, or of each portion within S, of the line x through the point (0, y, z). Now if 4, and A, denote the inclination of the outward normal of the surface to this line, at points where it enters and emerges from S respectively, and if ds, and ds, denote the elements of the surface in which it is cut at these points by the rectangular prism standing on dydz, we

have

dydz=- cos A,AS, = cos A,AS,. Thus the first integral, between the proper limits, involves the

dU elements U'a

U° . cos Aças., and – U'QdUcos Aças,; the latter dac

dx
of which, as corresponding to the lower limit, is subtracted.
Hence, there being in the whole of S an element ds, for each
element ds, the first integral is simply

dU
SJU"a? cos A ds,

dx

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