444 Prof A. Gray on the Dynamical 



which is the equation of motion we should have if T^ were 

 the potential energy of the system. 



Though molecular dynamics, notwithstanding its great ad- 

 vances, may never succeed in giving us a perfect knowledge 

 of the coordinates, which are thus ignored in the sense that 

 the energy is expressed without them, still in some cases, as 

 in the general dynamical treatment of electric currents intro- 

 duced by Maxwell, it adds greatly to clearness of statement 

 and view to adopt the idea that the intrinsic energy of the 

 system, or that part of it alone concerned in electromagnetic 

 action, is really kinetic energy. Thus, in using the expression 

 for this electrokinetic energy in the Lagrangian equations, 

 we find the electromotive forces of induction, and the electro- 

 magnetic forces between the parts of the system. 



It has been shown by Maxwell (Electricity and Magnetism, 

 vol. ii. chap, vi.), and the proof need not be repeated here, that 

 the electrokinetic energy of a system of current-carrying 

 conductors may be written as a homogeneous quadratic func- 

 tion of the current-strengths, which are regarded as velocities 

 corresponding to generalized coordinates y lf y 2 , an d specifying 

 the different circuits. Adopting the notation y lf y 2l &c. for 

 these velocities, and putting T for the electrokinetic energy, 

 we have 



•T = i(L I ^ + 2M 122 / 1 ^ + ... + L^ 2 2 + 2M 23 2/ 2 2/3 + •..); ■ (1) 

 where L x , M 12 , &c, L 2 , M 23 , <fcc. are coefficients involving only 

 the ordinary position coordinates of the material system, and 

 fulfilling the relation M. pq = M. qp . L l5 L 2 , &c. are in fact the 

 coefficients of self-induction for the different circuits M 12 , M l3 , 

 and M 23 &c. the coefficients of mutual induction for the pairs 

 of circuits indicated by the suffixes. 



If we denote the magnetic inductions through the different 

 circuits by Nj, N 2 , . . . N*, . . . , we have 



dT 



N 1= -TT- =L l2 / 1 + M 122 / 2 + . . . + M. lk y k + . . . 



dT 



dT 

 N)k== a% =Mk ^ 1 + M ^2/2 + . • . + !*& + . . . ; 



(2) 



so that (1) becomes 



T = i(N0i+N 22 / 2 + ... + N^ + ...) (3) 



