44 VECTOR ANALYSIS. 



and, if the space is periphractic, that the surface-integral of uw 

 vanishes for each of the bounding surfaces. 

 The existence of the minimum requires that 



while &o is subject to the conditions that 



and that the tangential component of Sao in the bounding surface 

 vanishes. In virtue of these conditions we may set 



where Sq is an arbitrary infinitesimal scalar function of position, sub- 

 ject only to the condition that it is constant in each of the bounding 

 surfaces. (See No. 67.) By substitution of this value we obtain 



or integrating by parts (No. 76) 



ffu w.dcrSq -fffV. [u co]Sq dv = 0. 



Since Sq is arbitrary in the volume-integral, we have throughout the 



whole space 



V.[>ft>] = 0; 



and since Sq has an arbitrary constant value in each of the bounding 

 surfaces (if the boundary of the space consists of separate parts), we 

 have for each such part 



Potentials, Newtonians, Laplacians. 



91. Def. If u' is the scalar quantity of something situated at a 

 certain point p', the potential of u for any point p is a scalar function 

 of p, defined by the equation 



, u' 

 potu = r -, T~, 



IP -A 

 and the Newtonian of u' for any point p is a vector function of p 



defined by the equation 



/ 



new u' = p ^ u'. 



[p -pB 



Again, if &>' is the vector representing the quantity and direction of 

 something situated at the point p', the potential and the Laplacian of 

 ft)' for any point p are vector functions of p defined by the equations 



, ft)' 



P0tft)= f , 



