454 MOSSOTTI ON THE FORCES WHICH REGULATE 
dq ak dG, d Gage. dGy 
1 kA a=— 4 — 4+ i Re eS a = ; 
(1) Gy dy a dy + dy + dy + dy hte 
digi dP ind Gy -d GpyeGsy dG 
tae Tide" de Pde hide ET ail 
which lead directly to the complete integral 
(IIL) kq=C-—F+G+4+6,4+ G,...... G, + ete.; 
C being an arbitrary constant. 
In order to determine, by means of this equation, the density g, we 
must substitute for F, G, 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 formule 
=r sin 6 cos p y=rsin 6sin p z=r cos 6 
z'=7' sin 6’ cos W' y' =r! sin sin yp’ z'=7' cos 0! 
the expression for F’ takes the form (see the additions to the Con- 
naissance des Temps for the year 1829, p. 356) 
(IV) eed bees te, “fade! )P, sins as ay" 
ok: SSS CS, £o)Pasintl ae ay | 
The coefficient P,, being given by the formula 
a clad A 2n—1) 
Ere ii Brean 
np 2(n-1) 9 2(m—1)(n—2)(n—8)_ Bh 
4. S@naf"* TS? ‘ete, b 
in which 
p =cos 6 cos 6! + sin § sin 6’ cos(p — W’ ), 
and the limits of the integrals relative to 6’ 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 G, let us in like manner put 
£ = psin w cos ¢, y= psinwsing, ¢=pcosw 
and represent by IT,, the function P;, when 7’, §’, p', are therein changed 
into p, w, ¢. 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) a ee pe GY gue't2dp \Mysinududs ] 
