117, 118] 



GAUSS'S THEOREM. 



351 



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 d2!, 



dZl is the projection of dS 



on the sphere, Fig. 125, and 



as the normal to the sphere 



is in the direction of r, we have 



Fig. 125. 



dScos(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#, and call the portion of its 



area cut by the above-mentioned cone do, we have from the similarity 



of the right sections of the cone 



da> 



The ratio d& is called the solid angle subtended by the infinitesimal cone. 

 Accordingly 



dScoB (nr) 



41) 



. 7 



= + - r -+- do. 

 - r 2 



Now for every element dm, where r cuts into S, there is another 

 equal one, do, where r cuts out, and the two annul each other. 

 Hence for outside S, 



42) .. 



If on the contrary, lies inside S, the integral I I dot is to be 



taken over the whole of the unit sphere with the same sign, and 

 consequently gives the area 4 jr. Hence for within S, 



43) 



rfcos( 



JJ-1 



These two results are known as Gauss's theorem, and the integral 

 will be called Gauss's integral. 1 ) 



1) Gauss, Theoria Attractions Corporum Sphaeroidicorum Ellipticorum 

 homogeneorum Methodo nova tractata. Werke, Bd. Y, p. 9. 



