138 REPORT—1885. 
de df__ PR ay 
K dt dé dzxdt’ 
; 7) eae 
but, since dor a v7F, 
Dp iea tte ed 
eas Fl i 
pK ™ dt? a 
Now, since 7*#=0, a particular solution of this differential equation 
will be 
we see that 
dF , do 
-_.+ _' =0 
dt % daz ‘ 
while the general solution will be the sum of this solution and the 
general solution of 
1 _o @F 
are r= —— 
nes de 
The particular solution is propagated at the same rate as ¢, while the other 
part of the solution represents a wave travelling with the velocity 1//uK. 
Since the part of the solution which travels at an infinite rate satisfies 
the equation 
dF , do 
zee i) 
dt i dz 
or iv 
we see that the electromotive force due to the change in the vector 
potential just balances the electrostatic electromotive force, so that until 
the part of the vector potential which travels at the rate 1/./»K comes 
up the resultant electromotive force vanishes. This explains how the 
electromotive force on Maxwell’s theory travels at a different rate from 
the potential, and a similar explanation will apply to Helmholtz’s theory. 
Helmholtz’s equations for a conductor are 
bie ode 8 | os t7, sot 
ov?u= (14479) 4A oe v7 + (1+ 47$—hk) A or 
where o is the specific resistance of the conductor ; on Maxwell’s theory 
the equations are 
du 
oyu = Arp ai 
These equations differ by terms involving the unknown constant k; but 
v. Helmholtz’s! investigations on the motion of electricity along thin 
conducting wires show that there is not much hope of distinguishing be- 
tween the theories by experiments on conductors. We have seen that 
we can make certain equations which occur in Helmholtz’s theory 
coincide with the corresponding ones in Maxwell’s by giving par- 
ticular values to certain constants. The difference in Helmholtz’s and 
Maxwell’s views as to the continuity of the currents is too serious to let 
us expect that we should ever get a complete agreement between the 
1 Ueber die Benegungsgleichungen der Elektricitat fiir ruhende leitende Kérper. 
Gesammelte Werke, vol. i. p. 603. 
