106 Electrical Vibrations associated with Conducting Rods. 
enclosed in a coaxal sheath of radius 7,*. If in Maxwell’s 
notation P, Q, R be the components of electromotive inten- 
sity; a, b, ¢ those of magnetic induction; V the velocity of 
light; we have 
Cet 
P, Q, R=cos pé . cos mz (32) zt 0 
7 
’ 4 
a, 0. c= Cos pt . Cos mz tee = 0); 
, ar 
in which z is measured along the axis and m=p/V. These 
expressions are independent of 7; and 7, and thus the nodal 
distance corresponding to a given frequency of vibration is 
the same whatever may be the diameters of the cylinders. 
But although the relation of m to p is unchanged, as 7 Is 
supposed to be reduced, the corresponding energies crease 
and that without limit. Apart from the first two factors, 
the value of (e. 7.) the resultant electromotive intensity varies 
as 1/7; so that the integrated square is proportional to 
log (7,/7;).. We see that when 7, is reduced without limit, 
the phenomenon is ultimately dominated by the infinite 
energies associated with the immediate neighbourhood of the 
attenuated corductor. Moreover, when 7 is already infini- 
tesimal, its further reduction in a finite ratio causes only a 
vanishing relative change in the value of the energy of 
vibration. 
In the problem for which the solution is above analytically 
expressed the section of the rod is circular and uniform, but 
the considerations already advanced point to the conclusion 
that when the section is infinitesimal neither the circularity 
nor the uniformity are essential. So long, at any rate, as all 
diameters, whether of the same section or of different sections, 
are in finite ratios to one another, the relation of nodal in- 
terval to frequency remains undisturbed. 
The same line of argument further indicates that the con- 
clusion may be extended to a terminated rod of infinitesimal 
section. or the infinite energy associated with the neigh- 
bourhood of the conductor is unaffected except at points 
infinitely near the ends. It appears therefore that the 
wave-length of the electrical vibration associated with a 
straight terminated rod of infinitesimal section, is equal to 
twice the length of the rod, whether the shape be cylindrical 
so that the radius is constant, or ellipsoidal so that the radius 
varies in a finite ratio at different points of the length. And 
* See, for example, Phil. Mag. xliv. p. 199 (1897): Scientific Papers 
iv. p. 327. 
