221 223] ELECTROMAGNETISM. 433 



These are the fundamental equations of electromagnetism. In the 

 Gaussian system, we must introduce the factor A on the left. By 

 these equations the solenoidal vector q is expressed as the curl of the 

 magnetic force H. The magnetic forces cannot be derived from a 

 potential except where there is no current, but must be found by 

 integration of the partial differential equations (2). In order to 

 show how this may always be accomplished, we shall prove a 

 general theorem. 



223. Vector Potentials. Helmholtz's Theorem. Any 



uniform, continuous, vector point-function vanishing at infinity 

 may be expressed as the sum of a lamellar and a solenoidal part, 

 and the solenoidal part may be expressed as the curl of a vector 

 point-function. A vector point-function is completely determined 

 if its divergence and curl are everywhere given. 



Let R be the given vector, with components X, F, Z. Let us 

 suppose it possible to express it as the sum of the vector parameter 

 of a scalar function < and the curl of a vector-function Q, whose 

 components are U, F, W. Then 



Y _84 , dW_dV 



^ ^ "i > 

 ox o oz 



7 _. 

 ~dy + dz 



d<t> dV dU 



^ = 3 +^ -- -0~' 



oz doc oy 

 Finding first the divergence of JR, 



,. -p dX 8F dZ 

 div R = 3- + _ + _ = A<, 

 dx dy dz 



for the curl of any vector is solenoidal, 35. 



But by 85 (18) we know that if $ and its first derivatives 

 are everywhere finite and continuous, we have 



Since R is continuous by hypothesis, div R is finite, so that 



9F dZ dr 



w. E. 28 



