KBCENT PROGRESS IN THEORKTICAL PHYSICS. 207 



and, for the second case, 



d^v d^r cpv __ 



dz^ dy^ dz^ ~~ ^ 



The ordinary magnetic and electric forces are derived from these 

 potentials by the application of Hamilton's operator, 



V =«■ — + - — 4-;- A. 



dz di/ dz ' 



that is, to find the magnetic or electrical attraction or repulsion 

 along a line, we take the differential coefficient of the potential 

 (magnetic or electrical) with reference to the direction of that line. 

 For ordinary magnets the potential V, is single-valued for any 

 given point of space; for electro-magnets Fis many-valued like 



^^ ^ having, in fact, an infinite series of values at any given 

 point; these values differing by ^n'^i where z is the intensity, or 

 strength of the electric current in the electro-magnetic wire. 



Carefully to be distinguished from V is another quantity, which, 

 in the case of solerioidul distributions of magnetism at least, also 

 fulfills Laplace's Equation. This quantity may be designated by I. 

 I is a quantity so related to the magnetization that, calling the 

 components of the latter in three directions at right angles to each 

 other, A, B, and C, we have 



dx'' dy ^ dz' 



I, which determines the magnetization (not the magnetic force) 

 at any point, is called the Potential of Magnetization. 

 \i But, besides the scalar (or non-directed) potential, V. and the 

 Potential of Magnetization, I, mentioned abore, we have, when con- 

 sidering not only the magnetic ybrce but likewise also the magnetic 

 induction^ a vector (or directed) potential. The magnetic induction 

 is derived from this vector potential in a precisely similar man- 

 ner to the derivation of the magnetic force from the scalar po- 

 tential, namely: by the application of Hamilton's operator V. 



If three quantities F. G, and H, be regarded as the components, 

 in three directions, at right angles to each other, of the scalar 



*Owing to the want of Greek type the printer has placed this letter n to represent 3.1416 

 the ratio of the circumference of a circle to its diameter. 



