its ends, and not at all on its form, whether straight or curved, between these points.
Hence the ends of a solenoid may be called in a strict sense its poles.
If a solenoid forms a closed curve the potential due to it is zero at every point, so that such a solenoid can exert no magnetic action, nor can its magnetization be discovered without breaking it at some point and separating the ends.
If a magnet can be divided into solenoids, all of which either form closed curves or have their extremities in the outer surface of the magnet, the magnetization is said to be solenoidal, and, since the action of the magnet depends entirely upon that of the ends of the solenoids, the distribution of imaginary magnetic matter will be entirely superficial.
Hence the condition of the magnetization being solenoidal is
where A, B, C are the components of the magnetization at any point of the magnet.
408.] A longitudinally magnetized filament, of which the strength varies at different parts of its length, may be conceived to be made up of a bundle of solenoids of different lengths, the sum of the strengths of all the solenoids which pass through a given section being the magnetic strength of the filament at that section. Hence any longitudinally magnetized filament may be called a Complex Solenoid.
If the strength of a complex solenoid at any section is m, then the potential due to its action is
This shews that besides the action of the two ends, which may in this case be of different strengths, there is an action due to the distribution of imaginary magnetic matter along the filament with a linear density
Magnetic Shells.
409.] If a thin shell of magnetic matter is magnetized in a
