VECTOR ANALYSIS. 51 



If the values of Lap Lap <*>, New Max o>, and Max New u are in general 

 definite, we may add 



47r Pot ft> = Lap Lap o> New Max o>, 

 4-7T Pot u = Max New u. 

 In other words : The Maxwellian is the divergence of the potential, 



- and V are inverse operators for scalars and irrotational vectors, 



4-7T j 



for vectors in general j V Max is an operator which separates the 

 irrotational from the solenoidal part. For scalars and irrotational 

 vectors, - A Max New and - - New Max give the potential, for sole- 



47T -i 47T 



noidal vectors j Lap Lap gives the potential, for vectors in general 



1 1 



New Max gives the potential of the irrotational part, and j Lap 



Lap the potential of the solenoidal part. 



103. Def. The following double volume-integrals are of frequent 

 occurrence in physical problems. They are all scalar quantities, and 

 none of them functions of position in space, as are the single volume- 

 integrals which we have been considering. The integrations extend 

 over all space, or as far as the expression to be integrated has values 

 other than zero. 



The mutual potential, or potential product, of two scalar functions 

 of position in space is defined by the equation 



Pot (u, w) =ffffff dv dv' =fffu Pot w dv =fffw Pot u dv. 



In the double volume-integral, r is the distance between the two 

 elements of volume, and u relates td dv as w' to dv'. 



The mutual potential, or potential product, of two vector functions 

 of position in space is defined by the equation 



Pot <*, ) =ffffff^~- dv dv' =fff$ . Pot dv =///, . Pot * dv. 



The mutual Laplacian, or Laplacian product, of two vector 

 functions of position in space is defined by the equation 



, ) =ffffff" .^xfdv dv' 



. Lap dv =fff<t> . Lap w dv. 



The Newtonian product of a scalar and a vector function of position 

 in space is defined by the equation 



/ 

 New (u, <*)) =f/ffif (30 . ? u' dv dv' =fff<*> . New u dv. 



