128, 129.] ANALYTICAL TREATMENT OF THE EQUATIONS. 151 





i /Y/T, , 



* = &TJJJ r *&quot; 



where the accents attached to 0, f , 77, f denote the values of these 

 quantities at the point (x, y, z) and r stands for the distance 



The integrations are supposed to include all parts of space at 

 which 0, g, 77, have values different from zero. 



We must now examine whether the above values of L t M, N 



really satisfy (8). Since ^- - =---, -, &c., &c., we have 

 J dx r dx r 



dL 



^ 

 dx dy dz 



_-, 



27rJJJV dx r dy r dz 



by (17), Art, 64. The volume-integral vanishes by (1), and 

 the surface-integral vanishes because by hypothesis we have 

 f t 77, f = at all points of the (infinite) surface over which it is 

 taken. Hence (8) is satisfied, and the values (6) of u, v, w satisfy 

 (3) and (4). They also evidently vanish at infinity. 



The above results hold even when 6, %, 77, f are discontinuous 

 functions, provided only that u t v, w be continuous. As regards 

 6 this is obvious ; but a discontinuity in f , 77, f will necessitate a 

 modification in (12). Let us suppose that as we cross a certain 

 surface X the values of f, 77, f change abruptly, and let us dis- 

 tino-uish the value on the two sides by suffixes. Two cases 



