76 DEFINITE INTEGRALS. [INT. III. 



fixed point or pole 0. Then the level surfaces are spheres, and the 

 parameter is 



and since h r = l, R = } 



drawn toward 0. ( 16.) 



Consider the surface integral of the normal component of R 

 directed into the volume bounded by a closed surface S not 

 containing 0, or as we shall call it, the flux of R into S, 



( i ) (JR cos (Rn) dS = - ffi cos (rn) dS. 



FIG. 19. 



The latter geometrical integral was reduced by Gauss. If to each 

 point in the boundary of an element dS we draw a radius and thus 

 get an infinitesimal cone with vertex 0, and call the part of the 

 surface of a sphere of radius r cut by this cone d%, dZ is the pro- 

 jection of dS on the sphere, and as the normal to the sphere is 

 in the direction of r, we have 



d% = dS cos (rn), 



the upper sign, for r cutting in, the lower for r cutting out. If 

 now we draw about a sphere of radius 1, whose area is 4-Tr, and 

 call the portion of its area cut by the above-mentioned cone dco, 

 we have from the similarity of the right sections of the cone 



The ratio da) is called the solid angle subtended by the infinitesimal 

 cone. 



