354 Prof. J. J. Thomson on 
electricity at its ends and positive at its sides. A little closer 
consideration will show, however, that such an effect is not 
probable, for suppose the negative particles are describing 
closed circular orbits, then the aver age effect will be the same 
as if the charge were uniformly distributed round the orbit; 
thus the average effect will be that due to a uniformly elec- 
trified ring rotating about an axis through its centre at right 
angles to its plane, and in this case the “electric force due to 
the motion obviously vanishes. 
Electromotive Force due to Induction.—Take the case of a 
particle fixed by the coordinates 2, 4, Z1y in the presence of 
any number of charged particles moving about in any 
manner. Let us fix the position of these particles by coordi- 
nates &, 7, ¢. Then we see from the preceding example 
that T, the kinetic energy, can. be written as follows : 
T= ss (uy? +o? +?) +(u4F +v,G+w,H) +T,, 
where F, G, H are linear functions of the velocities E 7 x! t. 
and do not involve Uy, Vy, Wr} Ty is a quadratic function of the 
velocities Es Ns oe The momentum of the first particle 
parallel to « is 
Mu,+eF. 
We have seen that F is the vector potential due to the 
remaining particles at the first particle. 
By Lagran ge’s equation we have 
oa dT, 
@ (Men +eF)— Ean + Gv, + Hw,) — 7, = force tending 
1 to increase 2. 
Now e 
oy = i dE ak 
“le fy= é = " de, =U Vy dy, + Wy, if 
and 
} bi d¥ IG dH. 
ete (uF G+ wH) =e( 1m Te ae +g + w, — ae = 
hence 
@ po 2 en (22) on, (2 a) 
qi lu= ings | aig da, dy, "1 dz, da, 
au = + external force tending to increase 2. 
| 
Thus, in consequence of the variation in the external field 
