42] 



Gauss' Theorem 



35 



seven, or any odd number of times. Thus a cone through a small element of 

 area dco on a unit sphere about P^ may cut the closed surface any odd number 

 of times. However many times it cuts, the first small area cut off will con- 

 tribute Bide* to / 1 NidS, the second and third small areas if they occur will 



FIG. 8. 



contribute e-^dw and +0^0) respectively, the fourth and fifth if they occur 

 will contribute e^co and + e$w respectively, and so on. The total contri- 

 bution from the cone surrounding dco is, in every case, + e^dw. Summing 



FIG. 9. 



over all cones which can be drawn in this way through P l we obtain the 

 whole value of I IN^S, which is thus seen to be simply ^ multiplied by the 



total surface area of the unit sphere round P lt and therefore Aire^ 



32 



