Next, put i = 2, and substitute for P, from the last Article.

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Hence the function a+bu? stands as follows, when arranged in terms of Laplace's Functions,

(a + b) + 6(xo

} ),

and consists of two Functions, of the order 0 and 2 respectively. The above is a long process to arrive at this result. It might have been so arranged at a glance. But the calculation has been given as an example of the use of the formula, which in most cases is the only means of obtaining the desired result.

Ex. 2. Arrange 49+30° +34 + N1-(40+72p) cos (w—a) +24 (1 – u?) cos 2(w-a) in terms of Laplace's Functions.

The result is 50 + {304 + 40 V1 – u* cos (w – a)} + {3pe – 1+ 724 N1 ?cos (w – a) + 24 (1 – pe?) cos 2(w-a)}, consisting of three functions of the orders 0, 1, 2.

Ex. 3. Let the function be

1+N2 – 24" cos (w + a) + (1 – ") cos 2 (6 + a).

The first term is a Laplace's Function of the order 0, and the second and third terms taken together are one of the second order.

Ex. 4. Let 1-(1-) cos w be the function. The arrangement is 2 1 1

COS 3 2


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40. As the Earth and other bodies of the Solar System are nearly spherical, and yet may not be precisely of the spheroidal form, it is found necessary in questions of Physical Astronomy to calculate the attraction of bodies nearly spherical. It is in these calculations that the value of the Functions we have been considering in the last Chapter is seen.

If r'o'w' be the co-ordinates to any element of the attracting mass, p be its density, and cos d' = r', then the mass of this element

= p'dr'r'de'r' sin O'dw' =- p'r'dr'du'dw', and the reciprocal of the distance being R, by Art. 18 and 24, the potential V


+P +


=SI1** (P.*+ P + ...) dr'dy'das'; SLS" (Px' +Pyr+P. +...+Peta +...

.) dr'du'da',


according as r, the distance of the attracted point from the origin, is greater or less than r'. We shall proceed soon to use these formulæ ; but we must first find the value of V for a perfect sphere.

Prop. To calculate the value of V for a homogeneous sphere.

41. Let the centre of the sphere be the origin of the polar co-ordinates (r'u'm') to any element of its mass, and the line through the attracted point be that from which the angles are

measured, and p the density. Then - pr" dr'du' dw' is the mass of the element: its distance from the attracted point

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Hence, a being the radius of the sphere,

pre dr'du'da
Nga 2 + y2 – 2rr''


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when the attracted point is without, and + when it is within the shell,

zo" dr'



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when the point is without the sphere.

When the point is within the sphere, the part of V for the shells which enclose the point

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and the part of V for the other shells of the sphere

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Hence V=


for an external particle, 3r

V= 27pa – pro for an internal particle.

Prop. To find the attraction of a homogeneous body, differing little from a sphere in form, on a particle without it.

42. Since the attracted particle is without the attracting mass, we must expand V in a descending series of powers of r, and shall therefore use the first of the expressions for V in Art. 40. Let the mean radius of the body = a; and let a (1 + y') be the variable radius, y' being a function of u and w, and its square being neglected.

Then, for the excess of the attracting mass over the sphere of which the radius = a, effecting the integration with respect to r' from r' = a to v' = a (1+y), the value of V



But if y, the same function of u and w that y' is of p' and w', be expanded in a series of Laplace's Functions, viz.

Y, + Y, + + Yi+..., then the theorems of Art. 26 and 32 show that

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a, is

Hence the value of V for the excess over the sphere becomes 4пра

a {Y. + Y, +...+

Yi+...}; 3r

(21 +1) gui and the part of V for the sphere, rad. = a,


3r Hence for the whole mass 4πρα, 4πρα

a' V=

+ { Y.+YA + ... + Y; +...}. 3r


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This is the first example in which we see the great value of the properties of Laplace's Functions; they here give us at

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