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the study of the statistics of the fluctuating gravitional force in infinite homogeneous systems was pionered by chandrasekhar & von neumann in two classical papers ( chandrasekhar & von neumann 1942 , 1943 hereafter cn43 ) .
the analysis of the fluctuating gravitional field , developed by the quoted authors , was formulated by means of a statistical treatment in the case of uniform systems , and with no correlations .
two distributions are fundamental for the description of the fluctuating gravitional field :    * \1 ) @xmath1 which gives the probability that a test star is subject to a force @xmath2 in the range @xmath2 , @xmath2 + d@xmath2 ; * \2 ) @xmath3 which gives the joint probability that the star experiences a force * f * and a rate of change * f * , where @xmath4 .    from a pure theorethical ground we expect that inhomogeneity affects all the aspects of the fluctuating gravitational field ( antonuccio & colafrancesco 1994 ; del popolo 1996a , b ; del popolo & gambera 1998 ) .      assuming that the system density is described by a powe law of index @xmath6 , the expression of @xmath3 is given following markoff s method by ( cn43 ) : @xmath7 } \right\ } d{\bf k } d{\bf \sigma } \label{eq : set}\end{aligned}\ ] ] a lenghty calculation leads us ( see del popolo & gambera 1998 for a derivation and the meaning of simbols ) to find the function @xmath8 : @xmath9 \cdot   [ q({\bf \sigma } ) + k r({\bf \sigma } ) ] \ }   \label{eq : trense}\end{aligned}\ ] ] this last equation introduced into eq .
( [ eq : set ] ) solves the problem of finding the distribution @xmath10 and makes it possible to find the moments of @xmath11 that give information regarding the dynamical friction . using this last expression for @xmath8 and performing a calculation similar to that by cn43 the first moment of @xmath12
is given by : @xmath13 \label{eq : quaruno}\end{aligned}\ ] ] where @xmath14 \left [ \frac { \sin{x}}{x^{(2 - p)/2 } } - \frac{\cos{x}}{x^{p/2 } } \right ] dx \nonumber \\             &    & - \ ; 2 \int_{0}^{\infty }             \left [ e^{(x/\beta)^{\frac{(3 - p)}{2 } } }             \right ] \cdot \frac{\sin{x}}{x^{(p - 2)/2 } } dx \label{eq : quardu}\end{aligned}\ ] ] as shown by eq .
( [ eq : quaruno ] ) , in a inhomogeneous system , differently from homogeneous ones , @xmath12 is a function of the inhomogeneity parameter @xmath6 .    at this point
we may show how dynamical friction changes due to inhomogeneity . from eq .
( [ eq : quaruno ] ) we see that @xmath15 differs from that obtained in homogeneus system only for the presence of a dependence on the inhomogeneity parameter @xmath16 .
if we divide eq .
( [ eq : quaruno ] ) for the correspondent of cn43 we obtain : @xmath17 this last equation is an increasing function of @xmath6 .
this means that for increasing values of @xmath16 the star suffers an even greater amount of acceleration in the direction @xmath18 ( when @xmath19 ) than in the direction @xmath20 ( when @xmath21 ) , with respect to the homogeneous case .
this is due to the fact that the difference between the amplitude of the decelerating impulses and the accelerating ones is , as in homogeneous systems , statistically negative , but now larger , being the scale factor greater .
this finally means that , for a given value of @xmath22 , the dynamical friction increases with increasing inhomogeneity in the space distribution of stars .
in addition , by increasing @xmath22 the dynamical friction increases , just like in the homogeneous systems , but the increase is larger than the linear increase observed in homegeneous ones .    antonuccio - delogu v. , colafrancesco s. , 1994 , apj 427 , 72 chandrasekhar s. , von neumann j. , 1942 , apj 95 , 489 chandrasekhar s. , von neumann j. , 1943 , apj 97 , 1 ( cn43 ) del popolo a. , 1996a , a&a 305 , 999 del popolo a. , 1996b , a&a 311 , 715 del popolo a. , gambera m. , 1998 , a&a , 342 , 34
in this paper we extend chandrasekhar and von neumann s analysis of the statistics of the gravitational field to systems in which particles ( _ e.g. _ stars , galaxies ) are not homogeneously distributed . 
 we derive a distribution function @xmath0 giving the joint probability that a test particle is subject to a force * f * and an associated rate of change of * f * given by d*f*/dt . 
 we calculate the first moment of d*f*/dt to study the effects of inhomogenity on dynamical friction .