223, 224] ELECTROMAGNETISM. 435 



is uniquely determined by its divergence and curl. This theorem 

 was given by Helmholtz in his celebrated paper on Vortex 

 Motion*. 



224. Symbolic Formulae. These relations may be con- 

 cisely expressed by means of Hamilton's and Gibbs's symbols V 

 and Pot ( 78). In words we may say that any solenoidal vector 

 is the curl of the vector potential belonging to it, which is the 

 vector potential of l/4nr times its curl. 



By virtue of the definition of Hamilton's operator we have the 

 vector equation 



(9) B = V 



so that we may call the sum of the scalar <f> and the vector Q 

 the quaternion potential belonging to R, from which R is derived 

 by the single vector operation V. Inserting the values of < 

 andQ, 



(10) R = V - (_ Pot divJ + Pot curl R) 



so that the operator (Pot curl - Pot div)/4?r is the inverse of V, 

 when applied to a vector-function. 



For a lamellar vector we have 



and for a solenoidal vector 



(12) divE = 0, E = ~V Pot curl R = ^- curl Pot curl R. 



4nr 4?r 



Taking the curl of o>, we find in like manner 



curl o> = curl 2 R = &R, 

 (R being solenoidal) so that 



(13) R = ^- Pot curl 2 R. 



4-7T 



In fact since the operations of definite integration and partial 

 differentiation are commutative, the operations Pot and curl 

 must be. 



* Helmholtz. "Ueber Integrale der hydrodynamischen Gleichungen, welche 

 den Wirbelbewegungen entsprechen," Crelle's Journal, Bd. 55, 1858, p. 25. Wiss. 

 Abh. Bd. i. p. 101. 



282 



