454' MOSSOTTI ON THE FORCES WHICH REGULATE 



,,., jdg dF . dG .(IG, . dG. , dG, , 



,^ = _^/ + ^ + ^.+f^. +^' + ctc., 



dz dz dz dz dz dz 



which lead directly to the complete integral 



(III) kq=:C-F-{-G-\-G, + G^ Gv + etc; 



C being an arbitrary constant. 



In order to determine, by means of this equation, the density q, we 



must substitute for F, G, G„ (?,,, G^, &c. the integrals which 



they represent. If the rectangular co-ordinates are changed into polar 

 co-ordinates by means of the known formulae 



X = r sin 9 cos v|/ i/ = r sin d sin 4' z ■=. r cos 9 



x' = r' sin 9' cos vj>' y' = r' sin 9' sin r|/' 2' = r' cos 9' 



the expression for F takes the form (see the additions to the Con- 

 naissance des Temps for the year 1 829, p. 356) 



(IV) F= ^'^[;^rijy(fJfg'r'-+W^p^sinQ'd&'d4;''] 



The coefficient P^ being given by the formula 



p _ 1.3.5. .2w — 1) 

 1.2.3 n 



\f 2{2n~lf ^ 2. 3(2«-l)(2«-3)^ -reic. j. 

 in which 



p = cos 9 cos 6' -f- sin 9 sin 9' cos (\^ — v/;' ), 



and the limits of the integrals relative to 9' and ^' should be such that 

 the value of F may take in the whole space, except the small portions 

 occupied by the material molecules. 



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



5 = f sin w cos <p, ')==/' sin w sin ^, i ^ P cos w 



and represent by n„ the function Pn, when r', 9', vj/', are therein changed 

 into p, w, (p. 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) 



