VECTOE ANALYSIS. 47 



where r z dq is the element of a spherical surface having center at p 

 and radius r. We may also set 



wr 

 We thus obtain 



V. New u fff-^- dqdr = 4>7rf-r- dr = 4nru r=00 4?ri// r =o > 



where u denotes the average value of u in a spherical surface of 

 radius r about the point p as center. 



Now if Pot u has in general a definite value, we must have u' = 

 for r = oo. Also, V. New u will have in general a definite value. For 

 r = 0, the value of u' is evidently u. We have, therefore, 



= 4nru, 



98. If Pot CD has in general a definite value, 



V. V Pot = V. V Pot [ui + vj -f wk] 



= V. V Pot ui + V. V Pot vj + V. V Pot wk 



= 47Tft). 



Hence, by No. 71, 



Vx VxPot w- VV.Pot o> = 47TO). 



That is, Lap V x ft> New V. o> = 47Tft). 



If we set 1 T _ 1 XT r? 



! = j Lap Vxft>, ft) 2 == ~r NewV.ft), 



we have ft) = ft) 1 H-ft) 2 , 



where ft) x and ft) 2 are such functions of position that V.ft) 1 = 0, and 

 Vxft) 2 = 0. This is expressed by saying that ft^ is solenoidal, and ft) 2 

 irrotational. Potft^ and Potft) 2 , like Pot ft), will have in general 

 definite values. 



It is worth while to notice that there is only one way in which a 

 vector function of position in space having a definite potential can 

 be thus divided into solenoidal and irrotational parts having definite 

 potentials. For if ft) x + e, ft) 2 e are two other such parts, 



V.e = and Vxe = 0. 

 Moreover, Pot e has in general a definite value, and therefore 



6 = 7 LapVxe i NewV.e = 0. Q.E.D. 



4?r r 4-7T 



* Better thus : V.V Potu=fff^V.Vudv=fffV.fau}dv - 

 = -ffuV-.dff= - 4ira. [MS. note by author.] 



