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CHAPTER II.

LAPLACE'S COEFFICIENTS AND FUNCTIONS.

17. In the present Chapter we shall develop the properties of those remarkable quantities which have received the name of their great discoverer, under the designation of LAPLACE'S COEFFICIENTS AND FUNCTIONS. To do this it will be necessary to anticipate the subject of the following Chapter, and to bring in here a Proposition which should properly stand at the head of that division of this treatise.

Prop. To obtain formulo for the calculation of the attraction of a heterogeneous mass upon any particle.

18. Let p be the density of the body at the point (ayz); fgh the co-ordinates of the attracted particle; and, as before, suppose that A, B, C are the attractions parallel to the axes x, y, z.

Then

(fx) dx

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the limits being determined by the equation to the surface of the body.

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dV
dy

dV
.:. A

C=-
)
df

dh 19. It follows, then, that the calculation of the attractions A, B, C depends upon that of V. This function cannot be calculated except when expanded into a series. It is a function of great importance in Physics: and, for the sake of a name, has been denominated the Potential of the attracting mass, as upon its value the amount of the attractive force of the body depends.

20. As the axes and origin of co-ordinates in the previous Article are altogether arbitrary, it follows that if r be the distance of the attracted point from any fixed point in the attracting body, then the attraction in the line of r, towards

dV the origin of r,

dr

d? V d d V PROP. To prove that

+ dfat dg

= 0, or 47p', ac

dh? cording as the attracted particle is not or is part of the mass itself; p being the density of the attracted particle in the

latter case.

21. By differentiating V, we have V

-p(f,x) dx dy dz df d

{2 (f - )– (9 - y)2 (h – z)} dx dy dz df?

{(f - x)+ (9- y)2 + (h — 2)}}!

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In the same manner we shall have

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d V

p{2
dgo

{(f - 2)2 + (9- y)+ (h — 2)}!
d
dh?

{(f - 2)? + (9-y)2 + (h — 2)?}!
do V dvd V

0 x dz +

+ df? dg*

dh

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When the attracted particle is not a portion of the attracting mass itself, then xyz will never equal fgh respectively,

= 0.

in that case,

dh?

and consequently the expression under the signs of integration vanishes for every particle of the mass :

d? V dV dV

+ af dg *

dh This equation was first given by Laplace: and Poisson was the first who showed that it is not true when the attracted particle is part of the attracting mass. In that case the denominator of the fraction under the signs of integration vanishes,

0 and the fraction becomes ó, 9

when x=f, y=9, z=h.

d V d d V To determine the value of + +

dfat dg suppose a sphere described in the body, so that it shall include the attracted particle; and let V = U+ U', U referring to the sphere, and U' to the excess of the body over the sphere. Then, by what is already proved,

d'U' dU'

d'U'
+ + = 0;

dg dh?
d? V d’V d? V d’U d’U d’U
+

+
df?
dg* * Ahdf?

dg dk

dh? The centre of the sphere may be chosen as near the attracted particle as we please ; and therefore the radius of the sphere may be taken so small that its density may be considered uniform and equal to that at the point (fgh), which we shall call p'.

Let f'gh' be the co-ordinates to the centre of the sphere; then the attractions of the sphere on the attracted point parallel to the axes are, by Art. 3,

4πρ

' 3

3

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d’U MU dU

= - 4πρ';
dfa + dga + dhe
dV d V dV
+

4πρ',
df* dg* dh
when the attracted particle is within the attracting mass.

22. It may be shown by precisely the same process as in the previous Article, that

d'R MR MR

dfr * dg + dh? where R={(f-2) + (9- y) + (h – x)"}}, the reciprocal of the distance of any point of the body from the attracted particle.

PROP. To transform the partial differential equation in R into polar co-ordinates.

09

23. Let row be the co-ordinates of (fgh), and r'o'w' of (ayz), the angles 8 and O' being measured from the axis of z; w and w' being the angles which the planes on which 0 and 0 are measured make with the plane zác. Then

f=r sin 8 cos w, g=r sin 0 sin w, h=r cos e,
x = r sin cosw', g'=r' sin O' sin w', h'=r' cos O'.

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= .... (1) ;

These are the same as

h

tan w= gue = f' +g* + ', cos 0 =

9
fa+g2 +7
dR dR dr dR do dR dw

+

+
df dr af de af* do af'
d'R d ddr d ddd dR da

+

+
df - af dr af* af de af* af aw df

dR dr dR de dR doo
+

dr dfat do df do df P. A.

2

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d'R dr d'R do dR da?
+

+
dr* df do? dfdw? df?
d'R dr do d’R dr do d’R do do
+ 2

+ 2
dr df df+ 2

drdo df df do dw df df
dd ddddo
+

+
dr dfde df dw df"

dR d'R The expressions for and

are of the same form.

dg dh* These three must be added together and equated to zero. When this is effected the formulæ (1) make

d2R dr2 dr2 dr the coefficient of

+ + dra dfdg

dh d'R do do do the coefficient of

+ + d° df

df" dgt "dh

d2R dw2 dw2 dw2 1 the coefficient of

dw? df?dg*

df* dg*dk på sin'0'

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dar

=

r

dR dr d2r

2 the coefficient of

dr dfa+dge +dh

dddthe coefficient of

ddfdg2 * dh* = pole sin '

cos A

dR do do do the coefficient of

dw dfa* do* + dh

0.

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