266 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
§§ 63 and 64*. On the Expression of Mutual Action between two Magnets by means of 
the Differential Coeffcients of a Function of their relative Position. 
63. By a simple application of the theory of the potential, it may be shown that 
the amount of mechanical work spent or gained in any motion of a permanent mag- 
net, effected under the action of another permanent magnet in a fixed position, depends 
solely on the initial and final positions, and not at all upon the positions successively 
occupied by the magnet in passing from one to the other. Hence the amount of work 
requisite to bring a given magnet from being infinitely distant from all magnetic 
bodies, into a certain position in the neighbourhood of a given fixed magnet, depends 
solely upon the distributions of magnetism in the two, and on the relative position 
which they have acquired. Denoting this amount by Q, we may consider Q as a 
function of coordinates which fix the relative position of the two magnets ; and the 
variation which Q experiences when this is altered in any way will be the amount of 
work spent or lost, as the case may be, in effecting the alteration. This enables us 
to express completely the mutual action between the two magnets; by means of dif- 
ferential coefficients of Q, in the following manner : — 
If we suppose one of the magnets to remain fixed during the alterations of relative 
position conceived to take place, the quantity Q will be a function of the linear and an- 
gular coordinates by which the variable position of the other is expressed. Without 
specifying any particular system of coordinates to be adopted, we may denote by 
the augmentation of Q when the moveable magnet is pushed through an infinitely 
small space in any given direction, and by df^ the augmentation of Q when it is 
turned round any given axis, through an infinitely small angle d<p. Then, if F denote 
the force upon the magnet in the direction of and L the moment round the fixed 
axis of all the forces acting upon it (or the component, round the fixed axis, of the 
resultant couple obtained when all the forces on the different parts of the magnet are 
transferred to any point on this axis), we shall have 
— Ydl=df^, and —Ed(p=df^, 
since a force equal to — F is overcome through the space d^ in the first case, and a 
couple, of which the moment is equal to — L, is overcome through an angle d(p in the 
second case of motion. Hence we have 
F= 
L= 
d(pGt 
d<p ‘ 
64. It only remains to show how the function Q may be determined when the distri- 
butions of magnetism in the two magnets and the relative positions of the bodies are 
* Communicated June 20, 1850. 
