Page:Scientific Memoirs, Vol. 1 (1837).djvu/466

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454
MOSSOTTI ON THE FORCES WHICH REGULATE.
`
(I)'

which lead directly to the complete integral

(III)

being an arbitrary constant.

In order to determine, by means of this equation, the density , we must substitute for , , , , … , &c. the integrals which they represent. If the rectangular co-ordinates are changed into polar co-ordinates by means of the known formulæ


the expression for takes the form (see the additions to the Connaissance des Temps for the year 1829, p. 356)

(IV)


The coefficient being given by the formula



in which


and the limits of the integrals relative to and should be such that the value of may take in the whole space, except the small portions occupied by the material molecules.

In order to have the expression for , let us in like manner put


and represent by the function , when , , , are therein changed into , , . Then, if we suppose the origin of the co-ordinates to be taken in the interior of the molecule, we shall have (see Connaissance des Temps for the year 1829, p. 357)

(V)