456 MOSSOTTI ON THE FORCES WHICH REGULATE 
and it being observed that 
CF  @F , @F 2G ,_@&G4 , &G 
dat bids dete gc3. Wgecies? 1 
with respect to which see the third volume of the Bulletin de la Société 
Philomatique, p. 388. 
If in this equation we change the differentials taken relatively to the 
rectangular co-ordinates into differentials taken relatively to the polar 
co-ordinates, we have 
ear) drg 
(1) Orgy 1 d°rq 
eat <a r2sin?§ dw2 ae 
Let us suppose that 7 ¢ is developed in aseries of integer and rational 
functions of the spherical co-ordinates, so that we may have 
(2) rq=Q+O.+Q.------- + Q;-+ ete. ; 
in which any one of the quantities Qi renders identical the equation 
a(sin 8 i) } cd? Q} yore: ns 
8) ~sinbdd + peak Ut) ao 
On this supposition the equation (1) will be satisfied by taking in 
general 
(4) Aj eee a Qit=ans Q. 
In order to integrate this differential equation of the second order * 
let us take 
and consequently 
(1) (1) (1) 
dQ _ Q , 1dQ@_ 1:2, 
Pe 72 r dr i dr? 
(1) (1) (1) (1) 
#Q; 5 Qi 2dQ, LEQ 14; 
dr? 3 ow dry r dr? 4 dF a 
* The integration of this equation with the second member negative has also 
exercised the ingenuity of the two illustrious geometers Plana and Paoli. See 
the Memoirs of the Academy of Turin, vol. xxvi., and those of the Italian 
Society, vol. xx. 
