PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 271 
lines of magnetization, every magnet of finite dimensions may be divided into an 
infinite number of longitudinally magnetized infinitely thin bars or rings, any dis- 
tribution of magnetism which is not solenoidal might be called a complex solenoidal 
distribution ; but no advantage is obtained by the use of this expression, which is 
only alluded to here, on account of the analogy with the subject of the preceding 
definition. 
71. Prop. — The action of a magnetic solenoid is the same as if a quantity of positive 
or northern imaginary magnetic matter numerically equal to its magnetic strength, 
were placed at one end, and an equal absolute quantity of negative or southern matter 
at the other end. 
The truth of this proposition follows at once from the investigation of Chap. III. 
§§ 36, 37 , 38. 
Cor. 1. — The action of a magnetic solenoid is independent of its form, and depends 
solely on its strength and the positions of its extremities. 
Cor. 2 . — A closed solenoid exerts no action on any other magnet. 
Cor. 3. — The resultant force” (defined in Chap. IV. § 49) at any point in the sub- 
stance of a closed magnetic solenoid vanishes. 
72. Prop. — If i he the intensity of magnetization, and u the area of the normal section 
at any point P, at a distance s from one extremity of a complex solenoid, and if [i<y] 
and {io;} denote the values of the product of these quantities at the extremity from 
which s is measured, and at the other extremity respectively ; the magnetic action will 
he the same as if there were a distribution of imaginary magnetic matter, through the 
length of the har of which the quantity in an infinitely small portion ds, q/* the length 
at the point P, would be — ^^^ds, and accumulations of quantities equal to — {\co] and 
{\a} respectively at the two extremities. 
The truth of this proposition follows immediately from the conclusions of Chap. III. 
§ 38. 
73. Prop. — The potential of a magnetic shell at any point is equal to the solid angle 
which it subtends at that point multiplied by its magnetic strength*. 
Let d^ denote the area of an infinitely small element of the shell, A the distance 
of this element from the point P, at which the potential is considered, and & the 
angle between this line, and a normal to the shell drawn through the north polar 
ide of c?S. Then if X denote the magnetic strength of the shell, the magnetic moment 
of the element dS will be X <fS, and (§ 54.) the potential due to it at P will be 
xc?S . cos 5 
3 ? 
* This theorem is due to Gauss (see his paper “ On the General Theory of Terrestrial Magnetism,” § 38 ; 
of which a translation is published in Taylor’s Scientific Memoirs, vol. ii.). Ampere’s well-known theorem, 
referred to by Gauss, that a closed galvanic circuit produces the same magnetic effect as a magnetic shell of 
any form having the circuit for its edge, implies obviously the truth of the first part of Cor. 2 below. 
