PHYSICS: H. BATEMAN 143 
. e v(a) , e c 
A = - , $ = - • 
4:ir v 47T y 
We shall be interested chiefly in the effect of an operation analogous to differ- 
entiation. Let us suppose that the co-ordinates of the point charge at time 
a depend on a parameter /3 as well as a and let b denote the vector with com- 
ponents d£/d/3, drj/dft d£/dft then if f is a function of a and (3 it is easy to 
prove that 
_d 
d/3 
as 
When / = dco (a, /3)/da this formula may be written in a more convenient 
form by making use of the relations 
*(i)+*(A+i(L) + *n) = o, 
dx \ v / d;y \ *> / ds \ v ) dt \v J 
fc'A + 4- f'J* 4- A = A 
* ^ ^dj ^ dz d* da 
where the operators in the last equation are supposed to act on a function of 
a and /3. 
We thus obtain the equation 
1 M = A p frfafl l-u-fr [1 sfa^ l + A P b fa>^ l+ d P d(co > a) 1 
r da/ dx d (ft a) J dv d (ft a) J dz d (ft a) J d/ L" d (ft «) J 
In particular we have 
Aif^-iTAfi/ 'M-V^YLAfi/V^- A^l 
d/3 4tt Ldv V V d/3 * d/3/J dz \ A ^ dft/ J d* V ¥/J 
d/3 , 4r Ldx \v d/3 J dy d/3 J dz (.r dft 
Writing p = eb/47r, we find that for an electric doublet of moment 47rp the 
electric potentials are 
A = r ot{l(vXp)}-|{lp},*= C div{ip}. 
This of course is a simple generalisation of the well known result due to Hertz 
and Righi. 
It is well known and easy to verify that the same electromagnetic field 
may be derived from the magnetic potentials. 
B = 
crot {^}"^l€ (vx 4 o=div € (vx 4 
