39] DEFINITE INTEGKALS. 77 



Accordingly 



dSc< 

 r * 



d8cos(rn)_ d% _ 

 ( 2 ) ^ - ^ ~ 



and 



Now for every element da), where r cuts into S, there is 

 another equal one, day, where r cuts out, and the two annul 

 each other. Hence for outside S, 



(4) 



If on the contrary, lies inside S, the integral is to be taken over 

 the whole of the unit sphere with the same sign, and consequently 

 gives the area 4?r. Hence for within 8, 



(5) 



These two results are known as Gauss's theorem, and the integral (3) 

 will be called Gauss's integral*. 



These results could have been obtained as direct results of the 

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

 cones with vertex 0. If is outside 8 t R is continuous in every 

 point within 8, 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 R is solenoidal, and the 

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

 solenoidal in the space between S and any sphere with center 

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

 flux through the sphere, which is evidently 4-Tr. 



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

 shown by differentiation. If the coordinates of are a, 6, c, 



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



* Gauss, Theoria Attractionis Corporum Sphaeroidicorum Ellipticorum homo- 

 geneorum Methodo nova tractata. Werke, Bd. v., p. 9. 



