142 . REPORT—1863. 
product of the length of the conductor, into the strength of the current, the in- 
tensity of the magnetic field, and the sine of the angle between the lines of 
force and the direction of the current. This may be more concisely expressed 
by saying, that if a conductor carrying a current is moved in a magnetic field, 
the work done on the conductor by the electromagnetic forces is equal to the 
product of the strength of the current into the nwmber of lines of force which 
it cuts during its motion. 
Hence we arrive at the following general law, for determining the mecha- 
nical action on a closed conductor carrying a current and placed in a magnetic 
field :— 
Draw the lines of magnetic force. Count the number which pass through 
the circuit of the conductor, then any motion which increases this number 
will be aided by the electromagnetic forces, so that the work done during the 
motion will be the product of the strength of the current and the number of 
additional lines of force. 
For instance, let the lines of force be due to a single magnetic pole of 
strength m. These are 47m in number, and are in this case straight lines 
radiating equally in all directions from the pole. Describe a sphere about 
the pole, and project the circuit on its surface by lines drawn to the pole. 
The surface of the area so described on the sphere will measure the solid 
angle subtended by the circuit at the pole. Let this solid angle =w, then 
the number of lines passing through the closed surface will be mw; and if C 
be the strength of the current, the amount of work done by bringing the 
magnet and circuit from an infinite distance to their present position will be 
Cmw. This shows that the magnetic potential of a closed circuit carrying a 
unit current with respect to a unit magnetic pole placed at any point is equal 
to the solid angle which the circuit subtends at that point. 
By considering at what points the circuit subtends equal solid angles, we 
may form an idea of the surfaces of equal potential. They form a series of 
sheets, all intersecting each other in the circuit itself, which forms the boun- 
dary of every sheet. The number of sheets is 42 C, where C is the strength 
of the current. The lines of magnetic force intersect these surfaces at right 
angles, and therefore form a system of rings, encircling every point of the 
circuit. When we have studied the general form of the lines of force, we can 
form some idea of the electromagnetic action of that current, after which the 
difficulties of numerical calculation arise entirely from the imperfection of our 
mathematical skill. 
24. General Law of the Mechanical Action between Electric Currents and 
other Electric Currents or Magnets.—Draw the lines of magnetic force due to 
all the currents, magnets, &c., in the field, supposing the strength of each 
current or magnet to be reduced from its actual value to unity. Call the 
number of lines of force due to a circuit or magnet, which pass through 
another circuit, the potential coefficient between the one and the other. This 
number is to be reckoned positive when the lines of force pass through the 
circuit in the same direction as those due to a current in that circuit, and 
negative when they pass in the opposite direction. 
If we now ascertain the change of the potential coefficient due to any dis- 
placement, this increment multiplied by the product of the strengths of the 
currents or magnets will be the amount of work done by the mutual action of 
these two bodies during the displacement. The determination of the actual 
value of the potential coefficient of two things, in various cases, is an import- 
ant part of mathematics as applied to electricity. (See the mathematical dis- 
cussion of the experiments, Appendix D.) 
