Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/268

617.] We may therefore adopt, as a definition of ${\mathfrak {A}}$, that it is the vector-potential of the electric current, standing in the same relation to the electric current that the scalar potential stands to the matter of which it is the potential, and obtained by a similar process of integration, which may be thus described.—

From a given point let a vector be drawn, representing in magnitude and direction a given element of an electric current, divided by the numerical value of the distance of the element from the given point. Let this be done for every element of the electric current. The resultant of all the vectors thus found is the potential of the whole current. Since the current is a vector quantity, its potential is also a vector. See Art. 422.

When the distribution of electric currents is given, there is one, and only one, distribution of the values of ${\mathfrak {A}}$, such that ${\mathfrak {A}}$, is every where finite and continuous, and satisfies the equations

$\nabla ^{2}{\mathfrak {A}}=4\pi \mu {\mathfrak {C}},\quad S.\nabla {\mathfrak {A}}=0,$

and vanishes at an infinite distance from the electric system. This value is that given by equations (5), which may be written

${\mathfrak {A}}={\frac {1}{\mu }}\iiint {{\frac {\mathfrak {C}}{r}}\,dx\,dy\,dz}.$

Quaternion Expressions for the Electromagnetic Equations.

618.] In this treatise we have endeavoured to avoid any process demanding from the reader a knowledge of the Calculus of Quaternions. At the same time we have not scrupled to introduce the idea of a vector when it was necessary to do so. When we have had occasion to denote a vector by a symbol, we have used a German letter, the number of different vectors being so great that Hamilton's favourite symbols would have been exhausted at once. Whenever therefore, a German letter is used it denotes a Hamiltonian vector, and indicates not only its magnitude but its direction. The constituents of a vector are denoted by Roman or Greek letters.

The principal vectors which we have to consider are:—

	Symbol of Vector.	Constituents.
The radius vector of a point	ρ	x y z
The electromagnetic momentum at a point	${\mathfrak {A}}$	F G H
The magnetic induction	${\mathfrak {B}}$	a b c
The (total) electric current	${\mathfrak {C}}$	u v w
The electric displacement	${\mathfrak {D}}$	f g h
The electromotive force	${\mathfrak {E}}$	P Q R
The mechanical force	${\mathfrak {F}}$	X Y Z
The velocity of a point	${\mathfrak {G}}$ or ${\dot {\rho }}$	${\dot {x}}{\dot {y}}{\dot {z}}$
The magnetic force	${\mathfrak {H}}$	α β γ
The intensity of magnetization	${\mathfrak {J}}$	A B C
The current of conduction	${\mathfrak {K}}$	p q r