352 VIII. NEWTONIAN POTENTIAL FUNCTION. 



These results could have been obtained as direct results of the 

 divergence theorem. For the tubes of the vector function E are 

 cones with vertex 0. If is outside 8, E is continuous in every 

 point within S, and since the area of any two spheres cut out by a 

 cone are proportional to the squares of the radii of the spheres, we 

 have the normal flux of 



equal for all spherical caps. Consequently E is solenoidal, and the 

 flux through any closed surface is zero. If is within S, E is 

 solenoidal in the space between 8 and any sphere with centre 

 lying entirely within S, and the flux through 8 is the same as the 

 flux through the sphere, which is evidently 4 it. 



The fact that E is solenoidal and V harmonic may be directly 

 shown by diiferentiation. If the coordinates of are a, b, c, 



44) r 2 = (x - a) 2 + (y - 6) 2 + (* - c) 2 , 



AK \ cr x a Or y b dr z c 



45) 7j- = -- 9 ~ = - - ? TT- = -- ; 



ex r oy r oz r 





- _ 



dx\r r*dx~ r s ' 



_ _ I , 3 (a?-) ll' = 3(a?-a) 2 -r 8 



f ~ f * ~ 5 



8 . (I) g.(I 

 48) ^ (i) = W + 



\r/ dx* dy z 



and is harmonic, except where r = 0. 



119. Definition and fundamental Properties of Potential. 



We have seen in 28, 34) that if we have any number of material 

 particles m repelling or attracting according to the Newtonian Law 

 of the inverse square of the distance, the function 



= - y ^ + ?p + ... + 



'i '2 



