340 
MR. J. J. THOMSON ON SOME APPLICATIONS OF 
so that though the term in <t> 3 and xfj 2 occur with opposite signs in T and T', the term 
involving the product <£>.*// occurs with the same sign in both. On this account we may 
apply Lagrange’s equation to the term 
)) j (Fu-fi G v + H w) dxdydz 
as it stands. 
If we apply Lagrange’s equation to the x coordinate we see that the force per unit 
volume parallel to x on the element conveying the current u, v, w is 
u 
dF 
dx 
L dG , 
+V T, +W 
dH 
dx 
(109) 
or 
f dG dF] fdF dH] , dF , dF , 
v \ , , +« + v , + 
L dx dl J 
dz dx J 
dx 
dy 
IV 
dF 
dz 
dF dF dF 
this differs from Maxwell’s expression for the force by the term u -\-v — -\-iv~. 
1 J dx dy dz 
This term vanishes when integrated over a closed circuit. There will be correspond¬ 
ing expressions for the forces parallel to y and z respectively. We have thus the forces 
given in Maxwell plus the forces 
dF dF dF 
u — -\-v — -\-iv — 
dy 
dx 
dG 
dz 
dG 
dG 
U Tx+ V ^+ W ~d. 
dH , dH , dH 
U--+V- -fwy 
dy dz 
dx 
parallel to the axes of x, y, z respectively. We can prove that if the circuits are 
closed these form a system of forces in equilibrium, so that the force on any closed 
circuit is the same as that given by Maxwell’s theory. 
If we apply Lagrange’s equations to the electrical coordinates we find that the 
electromotive forces parallel to the axes of x, y, z are respectively — dF/dt, — dG/dt , 
—dH/dt. And if we apply them to the magnetic coordinates yl, ym, yn, writing in 
the expression 
F« + Gr+H w)dxdydz 
the values of F, G, H given in equations (104) we find that the magnetic forces parallel 
to the axes of x, y, z respectively are 
