the use of Integral forms i aes 
will be ineffective. If for a point-charge e, a disturbance emitted 
at time and place (¢,7,4,2,) reaches a place (xyz) where the vectors 
are sought at time ¢, then with 
r2=(a—-uay, h=t—n7/V, 
and Vs = Vi, — 2 (@ — 2) vse e eee ph (11), 
0, (m) is defined as 
= | (u,da + v,dy + w,dz — Vdt)/s. 
But Vds =Vdr, — Su,dx + Yu2dt, — = (# — a) dt, 
= V°dt — Suda — {V?—Su2+ > (@— a) 4} dt, 
and therefore 
O, (m) = — e oe 
i — Due+ > (@— 4%) ty, 
dt,; 
Vs 
the effective section of which is 
| V?—Su?Z+>(#@—a”,)% 
Vr, — = (#@ — %) 
Forming the derivative ©, (m) we have 
(dt, 0x, Ot, ONG o Ot; ONG Ot, OX, 
X=a (5 ot ot nae pees | dz Oz ae 
— 0, (m) = — dt, =~ Ve, [yndty...(12). 
as values of the ea For the force on a charge e, at (tisY2Z.), 
b 
Bs ot, dy, dt, a 
f= een de dl On)’ 
dt, 0x, Otdy.\ — 
SE,u= eee RE Sun, Saran 
and y EoUs = 612 (5 aE BE x (14), 
do fd) fd) 0 
where Tae ee a a 
This force is derivable from a kinetic potential — eé; = , as 
is readily verified. The reason this modification of kinetic potential 
is possible is that Fu.+ Gv, + Hw, — Vy differs from — Ve,x%, = 
af 
by a quantity es which being a complete time-rate of a 
function yields no force. 
