544 Mr. C. G. Darwin on the 



The double summations are for each pair of particles counted 

 once only. 



Finally, re-writing (11) without the vector notation 



L = 2im 1 r 1 2 + 2-^m 1 c/-S^ 1 + 2^r 1 A 1 co SXl -22^ 2 



+ tt ^§ ^ (cos x/r 12 - cos 6 X 2 cos 2 ') , . (12) 



where g. 1? w x , Uj are the charge, ma:-s, and velocity of the 



first particle ; 

 0! and A x are the scalar and vector potentials at e t 



due to external fields ; 

 %! is the angle between the line of motion of e x and 



the vector potential ; 

 r h2 is the distance between e Y and e 2 ; 

 \jr 12 is the angle between their lines of motion ; 

 #] 2 is the angle between the line of motion of e\ and 



the line joining it to e 2 . 



There is a certain interest in knowing how far wrong the 

 approximation (11) will be according to the classical theory. 

 This is done by calculating the next term for t in (8) and 

 evaluating the corresponding terms in (9). These are then 

 substituted in (5). The force on e x from e 2 is found to be 



J -ijp r' 2 . Thus the total force on e 1 is .y^Se/iV The 



summation will include e±, as well as the rest, as this term 

 is the reactive force of an electron's radiation on itself. 

 From the point of view of generalized coordinates we have 



-T7. = s— j so that the equations of motion can be put in the 

 B<7 oq 



dF 1 



form ^XL = ^777' where F — 0/ „ (Sei'r\) 2 . If F is neglected 

 H oq oKj 6 



altogether, it is easy to see that the ratio of terms omitted to 

 those included is of the order v/C 



5. When the Lagrangian for any problem has been found, 

 the transition to the Hamiltonian follows in the usual way. 



We find the momenta p= =— and solve for the o-'s in terms 



of them. The Hamiltonian is then iL = ^pq — L expressed in 

 q's and ^>'s, and the equations of motion have the canonical 



form p = — s— , <7= ^- . Thus if p 7 be the momentum 

 Oq - dp 





