Experimental Researches on Gravitation. 493 



Extending up to the limits, where it is possible, 



J'B +r -Hz 

 cLv\. 

 K-r X J 



The integral remaining in this expression is transcendent, 

 and one can only obtain its value by developing this 

 expression in series ; but this can be avoided by an opportune 

 artifice. Let us call dm not only the mass that is contained 

 in the point P, but all the mass of a spherical layer whose 

 radius is r and thickness dr, 



dm = 4tfrr 2 8 v dr, 



therefore 



_,-H<^ (i+R _,,)_ H(R2 _ ; , 2) j-^^. 



To obtain the value of the total flux that emerges from all 

 the points of the sphere, it is necessary to integrate this 

 expression from to R, and we have 



F = M v f * r dr [«-*<»-> (k + R + r \ 



= kirBS- 



2R 2 _2R 2_ _i_ 



H J riW-^l_ — *] 



We can carry out the double integration of the last term, 

 changing the order of integration ; but it is necessary to 

 change also suitably the limits of integration. Proceeding 

 in this way, and making p = RH, we have finally 



F =^ R3 g-i + ^(? + ^)]- • (3) 



The external action of gravitation can therefore be considered 

 as the effect of this flux. Inasmuch as k is the coefficient 



