538 Mr. C. G. Darwin on the 



will not have the simple connexion with the kinetic and 

 potential energy that it has in ordinary dynamics. 



It appears to me most probable that a great part of the 

 theorems of genernl dynamics here given are already known, 

 for the work of Sommerfeld and the later work of Bohr 

 would be naturally based on them. But in the cases dis- 

 cussed by these writers the Hamiltonian form can be Very 

 quickly derived from first principles, and they make no 

 mention of an} r general method of formulation, so that it 

 seemed to me that it might be worth while to exhibit such a 

 method. In developing this we can start either from Least 

 Action or from modified Newtonian equations of motion. 

 Least Action was shown by Maxwell to be applicable to the 

 aether, and we should therefore only require to prove that 

 the electric and magnetic forces in free space could be made 

 ignorable by the addition of suitable terms for the particles. 

 But this method involves distinctly more advanced dynamical 

 principles, and so, in spite of its superior elegance, I have 

 preferred to proceed by starting from the equations of 

 motion, and have followed methods similar to those by 

 which Lagrange's equations are introduced in dynamical 

 text-books. In this way the problem is kept throughout as 

 a problem of particles, and I hope it will be thereby made 

 more accessible to those unfamiliar with the later developments 

 of dynamics. 



2. The problem is really one of relativistic dynamics, but 

 no direct use will be made of the relativity transformations. 

 If the mass of each particle is made the proper function of 

 its velocity and if the electromagnetic equations are used in 

 Lorentz's form *, then the motions described will be invariant 

 for such transformations, and there is no need to go beyond 

 a set of axes fixed in space and a fixed time-scale. This 

 saves us from rather complicated considerations about rela- 

 tive velocity and acceleration. 



When several particles are free to move, the difficulty of 

 the problem lies in the fact that the force exerted by one 

 of them on another depends on its position and motion at a 

 certain previous time. In other words, we have to work 

 with retarded potentials, and it will be seen that the effect of 

 the retardation is of an order that is not negligible. It can 

 only be calculated by approximation, and so it will be neces- 

 sary to limit ourselves to motions where the velocities of the 



* For the general principles of electromagnetic theory here used 

 reference may be made to fl. A. Lorentz, ' The Theory of Electrons," 

 but I use ordinary electrostatic units and not the Heaviside type. 



