( ^0 ) 



integral of pX over the boundary and tlie (/> -f- 1) fold integral of 

 /'"^' F over the bounded Spj^\. 



We can also imagine the scalar values of I'X set off along the 

 normal-AS,j_;/s. As such the integral over an ai'bitrary curved bilateral 

 closed Sn—p can be reduced to an {n — ^j -f- l)-diraensional vector 

 over a curved Sn—p+\, bounded by Sn—p- If again we set off the 

 scalar values of that vector along its normal-AS^-i, the vector p—'^Z 

 appears, which we shall call the second derivative of pX. For the 

 component vectors of i'~'^Z we find: 



^ The particularity may appear that one of the derivatives becomes 0. 

 If the first derivative of an "'A'' is zero we shall speak ofan,„_iA'', 

 if the second is zero of an ,„+iA'. 



Theorem 2. The first derivative of a /'A is a ^ pX, the second a 



'' pX ; in other words the process of the first derivation as well as 

 that of the second applied twice in succession gives zero. 



The demonstration is simple analytically, but also geometrically the 

 theorem is proved as follows : 



Find the integral of the first derivative of rX over a closed aS'^,^^ i, 

 then we can substitute for the addition given by an Sfj4.\ element 

 the integral of pX along the bounding Sp of that element. Along 

 the entire >S/, + i each element of those Sj, boundaries is counted twice 

 with op[»osite indicatrix, so that the integral must vanish. 



The analogous jtroperty for the second derivative is apparent, when 

 we evaluate the integral of the normal vector over a closed /S„_^,^-i. 



By total derivative we shall understand the sum of the first and 

 second derivatives and we shall represent the operation of total 

 derivation by v- 



h=n 



Theorem 3. v' = ~^^ ^r~~ 



A=l 



Proof. In the first |)lace it is clear from theorem 2 that the 

 vector v" is again a pX. Let us find its component A'i2....p- 

 The first derivative supplies the following terms 



