756 REPORT — 1890. 
Pp] 
them, namely, eee &c., we must assume them connected with a current 
intensity at each point u, v, w, by equations of the form 
= | | [te Sut dxdydz, 
r 
when w=, cos ¢ is assumed as the particular case of an harmonic solution. From 
this we can see that 
w= [208 C=) dedyde, 
where e=e, cos ¢ is the varying electrical charge at any point, will satisfy the 
conditions 
awat¥ and — ay 4 afi, dG oH 
dé dt dee ideas 
We have thus the means of calculating at any point the electric force 
dF dv 
ae 
: dG dH. - pr e3: 
and the magnetic force a= Aa, a if we know the distribution of electricity 
Gs 4 
and electric currents in a neighbouring conductor. It is sometimes more conve- 
nient for calculation to apply this method than that of assuming a knowledge of the 
distribution of electric and magnetic force in the neighbourhood of the conductor. 
For example, in the case of a small Hertzian vibrator we get at once that if we 
calculate a quantity I= ences dts) then the function Y= = being here due to 
Tr as 
two equal and opposite charges at a distance e apart, while all the current being 
w, we get nae so that Hertz’s S is Maxwell’s vector potential, for Hertz’s 
equations for the electric and magnetic forces are those derived from Maxwell's in 
the way above described. 
If we apply this method to calculate the forces due to an harmonic distribution 
of electrification and current on a line we require to evaluate integrals of the 
form 
sin 2 sin7 
U= | . dx 
ze 
where 
=a +p, 
If we suppose 
sin 7 
—— =A + Ay, Cosa + A, Cos 2a one 
5 
we can evaluate A,,and observing that the function satisfies the differential 
equation 
a oe: a =U, 
dp» p dp dz 
so that 
PAy ,1 das 
dp’ p ap 
the solution of which is the Bessel function 
An =J, (iv /1+7°*). 
If we wish to apply Hertz’s method we get the same equations, but we must first 
see how to build up a large body with given currents and electrifications out of a 
—(1+n2)A, =0, 
