Study of Multipole Expressions

The expansion Eq. (5), taken in its entirety, is evidently valid regardless of the origin or the orientation of the chosen axis. But it is essential to keep in mind that the various integrals depend on the choice of the reference system. To see this we perform for instance the translation defined by  

gxdv becomes Jgx dw = jgxdv — X gdv, and similar formulas hold fory and z. Thus, if the total charge jgdv does not vanish, the dipole components gxdv,, are not invariant upon translation. 

The same kind of conclusion holds for the quadmpole terms  

In general, only the components of the first non-vanishing multipole are invariant. This indicates that it is always necessary to specify the choice of the reference system used in the calculations. 

This lack of invariance may seem at first to represent a serious difficulty if the exact expression of the potential must be replaced by a limited expansion. Actually, it can be taken advantage of by choosing the coordinate system so as to simplify the expansion as far as possible. We consider a few specific cases  

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