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MR. J. J. THOMSON ON SOME APPLICATIONS OP 
ordinary way. Lagrange’s equations and Hamilton’s method of Varying Action are 
methods of this kind, and it is the object of this paper to apply these methods to 
study the transformations of some of the forms of energy, and to show how useful 
they are for coordinating results of very different kinds as well as for suggesting new 
phenomena. A good many of the results which we shall get have been or can be got 
by the use of the ordinary principle of Thermodynamics, and it is obvious that this 
principle must have close relations with any method based on considerations about 
energy. 
Lagrange’s equations were used with great success by Maxwell in his £ Treatise 
on Electricity and Magnetism,’ vol. ii., chaps. 6, 7, 8, to find the equations of the 
electromagnetic field. 
In order to confine this paper to a reasonable size I shall limit myself to the con¬ 
sideration of the relations existing between various phenomena in elasticity, heat, 
electricity and magnetism, but even with this limitation it will only be possible to 
consider a few of the more prominent out of the many phenomena to which the 
method can be applied. 
When we apply Lagrange’s or Hamilton’s methods to discuss the motion of any 
material system we have first to choose coordinates which can fix the configuration of 
the system, and then to find an expression for the kinetic energy in terms of these 
coordinates and their differential coefficients with respect to the time. Before apply¬ 
ing these methods therefore to any physical phenomenon we must have coordinates 
which are sufficient to fix the configuration of the system which takes part in the 
phenomenon we are considering. The notation which we shall use will be as follows :— 
To fix the geometrical configuration of the system, i.e., to fix the position in space 
of any bodies of finite size which may be in the system, we shall use coordinates 
denoted by the letters x v x 2 . . . , x n , and when we want to denote a geometrical 
coordinate generally without reference to any one in particular we shall use the letter x. 
To fix the configuration of the strains in the system we shall in any particular case 
use the ordinary strain components a, b, c, e, f g (Thomson and Tait’s ‘ Natural 
Philosophy,’ vol. ii., § 669), but as it will be convenient to have a letter typifying the 
coordinates generally we shall use the letter w for this purpose. 
To fix the electrical configuration of the system we shall use coordinates denoted 
by the letters y x , . . . , and y for the typical coordinate, where y in a dielectric is 
what Maxwell calls an electric displacement and in a conductor the time integral of 
a current. 
To fix the configuration of the magnetic field by coordinates in such a way that 
Lagrange’s equations can be used I have found it necessary, for reasons which will 
be given later, to introduce two coordinates to fix the magnitude of the intensity of 
magnetisation at a point: one of these is what we may call a kmosthenic* coordinate, 
* I am indebted to Professor J. P. Postg&te, Fellow of Trinity College, Cambridge, for this word. 
