STATICS. [ 2 8j. 



284. Differentiating the equation of the equipotential surface, 

 and dividing by ds and by the attraction R whose components 

 are X, Y, Z, we find 



dF dV 87 



dx d,r dy dy dz dz 

 ~R'ds + ^R"ds Jr ~R'lds = ' 



The first factors in each term are the direction cosines of R, 

 the second factors are those of a tangent to the surface ; the 

 equation expresses, therefore, the fact that the attraction R at 

 any point of an equipotential surface is normal to the surface. 



The attraction at any point P of an equipotential surface 

 is, therefore, equal to dVj ' dn, where dn is the element of the 

 normal at P between this and the next equipotential surface. 

 Consequently, the attraction is inversely proportional to the 

 distance between the successive equipotential surfaces. 



Let the normal of an equipotential surface at any point P inter- 

 sect the next equipotential surface at a point P' ; let the normal 

 at P 1 intersect the next surface at P" ; and so on. The elements 

 PP' t P'P", etc., will form a curve which is at every point normal 

 to the equipotential surface passing through that point. Such 

 a curve is called a line of force, since its tangent at any point 

 indicates the direction of the resultant attraction at that point. 

 The lines of force cut the equipotential surfaces orthogonally. 



285. Potential of a Homogeneous Spherical Shell. (a) For an 

 internal point, we may proceed similarly as in Art. 272. The element 

 of mass cut out at A (Fig. 81) by the small cone whose solid angle is 

 d<* is again p-rVw/cosK if ^.PAC= "4.PA'C = a ; the corresponding 

 potential at P is Kpn/co/cos a : similarly, the potential due to the mass at 

 A is K/arVco/cos . Their sum is 



' 3 



ait) = 2 K 



cos a 

 since r -f r* = 2 a cos . 



As J c/w = 2 TT, we find V\ = 



i.e. the potential has the same constant value for all points within the 

 hollow of the shell. It follows that the attraction is zero, as found in 

 Art. 272. 



