4^56 MOSSOTTI ON THE FORCES WHICH REGULATE 



and it being observed that 



dx^ "^dy^ "^rfz« *'r/.j', dx^"^ dy^ "^t/z- ~° 



with respect to which see the third volume of the Bulletin de la Societd 

 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 



r d (sin fl i^ ) 1 



dr"- r'^sinfl dQ r'^sin-9 d4/- \ ■' ' 



Let us suppose that ;• q is developed in a series of integer and rational 

 functions of the spherical co-ordinates, so that we may have 



(2) r9 = Qo + QL + Q. +Q, + etc.; 



in which any one of the quantities Q i renders identical the equation 



,/sin6l^'\ 



On this supposition the equation (1) will be satisfied by taking in 

 general 



In order to integrate this differential equation of the second order * 

 let us take 



(1) (1) 



Q_Q}_ L ^Q» 



^' r i dr , 



and consequently 



(1) (1) (1) 



dr r* r dr i dr'^ 



(1) (1) (1) (1) 



dr- 7-3 7-* dr r dr'^ i dr* 



• 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. 



