46 



VECTOR ANALYSIS. 



we might with equal right set the indefinite expression equal to other 

 definite expressions, and then be misled into supposing these definite 

 expressions to be equal to one another. It will be safe to say that 

 the above equations will hold, provided that the potential of u or w 

 has a definite value. It will be observed that whenever Potu or 

 Pot w has a definite value in general (i.e., with the possible exception 

 of certain points, lines, and surfaces),* the first members of all these 

 equations will have definite values in general, and therefore the 

 second members of the equation, being necessarily equal to the first 

 members, when these have definite values, will also have definite 

 values in general. 



94. Again, whenever Pot u has a definite value we may write 



J- / / / 



- 





where r stands for [p' p] . But 



whence 



M /y 



V Pot u = New u. 



Moreover, New u will in general have a definite value, if Pot u has. 



95. In like manner, whenever Pot w has a definite value, 



VxPot ft) = VX//T- dv' =fffVx~ dv' =fffV~x^ f dv'. 



*//t/ ft* //*/ M /*// /yi 



Substituting the value of V - given above we have 



Vx Pot w = Lap w. 



Lap w will have a definite value in general whenever Pot ft) has. 



96. Hence, with the aid of No. 93, we obtain 



VxLapw = Lap V; 

 V.Lap&) = 0, 



whenever Pot w has a definite value. 



97. By the method of No. 93 we obtain 



V. New u = 



u' dv =fffVuf. *f- dv'. 



To find the value of this integral, we may regard the point p, which 

 is constant in the integration, as the center of polar coordinates. 

 Then r becomes the radius vector of the point p', and we may set 



* Whenever it is said that a function of position in space has a definite value in 

 general, this phrase is to be understood as explained above. The term definite ia 

 intended to exclude both indeterminate and infinite values. 



