GENERAL PRELIMINARY RESULTS. 21 



and consequently 



B 



Hence, throughout the interior of the mass 



of which, the equation = 8F for any point exterior to the body 

 is a particular case, seeing that, here p = 0. 



Let now q be any line terminating in the point p, supposed 



without the body, then \-j- ) = the force tending to impel a 



\ CL\j / 



particle of positive electricity in the direction of q, and tending 

 to increase it. This is evident, because each of the elements of 



F substituted for F in (-7 ) , will give the force arising from 

 this element in the direction tending to increase , and conse- 

 quently, ( -T ) will give the sum of all the forces due to every 



element of F, or the total force acting on p in the same direc- 

 tion. In order to show that this will still hold good, although 

 the point p be within the body ; conceive the value of F to be 

 divided into two parts as before, and moreover let p be at the 

 surface of the small sphere or b = a, then the force exerted by 

 this small sphere will be expressed by 



f dd> 



da being the increment of the radius a, corresponding to the 

 increment dq of q, which force evidently vanishes when a = 0: 

 we need therefore have regard only to the part due to the mass 

 exterior to the sphere, and this is evidently equal to 



T7- 4-7T 2 



V--a*p. 



But as the first differentials of this quantity are the same as 

 those of Fwhen a is made to vanish, it is clear, that whether 

 the point p be within or without the mass, the force acting upon 



it in the direction of q increasing, is always given by 



