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200

CHAPTER

7

DENSITY INHOMOGENEITIES

I$.-..::::---=--=---'---r

center surjace

FIGURE 7.6: The powers rn (0 ~ r ~ 1)

7.6.4 Zero-Potential Densities

The solution (7-38) gives Xk

=

0 if the right-hand side of (7-37), b

=

Vnm , is zero.

This is the case of the homogeneous equation corresponding to (7-36),

L

N anmkXnmk = 0 k=O

or briefly, corresponding to (7-37),

(7-48)

(7-49)

which represents the case of a mass distribution that produces zero external potential.

These are the "zero-potential densities" (sec. 7.2), forming the kernel of the N ewto- nian operator, for the present case. It is very easy to find non-zero solutions of (7-48) or (7-49): eq. (7-49) means simply that the vector x is normal to the given vector a (in the usual Euclidean metric)! Thus any vector x in the plane normal to a is admissible.

Finally we mention that the set of solutions of (7-49), forming the vector Z(2) in (7-52) i8 "orthogonal" to the vector (7-38), denoted in (7-52) by X(l), if we again take P as metric tensor. This is geometrically evident and is also immediately verified by direct computation: using (7-38) in matrix notation, we have

x(1)T PX(2) = _b_ a T C PX(2) = 0

aTCa

since CP

= I

(unit matrix) and aTx(2)

=

0 by (7-49).

(7-50)

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