490
THE THEORY OF ANOMALOUS DISPERSION
[421
This notation is so different from Maxwell's, that it may be well to exhibit explicitly the correspondence of symbols.
k
III
y
X |
<rpz
m
0
Helmholtz . . .
Maxwell ......
When there is no dissipation (R = 0, 72 = 0), these interchanges harmonize the two pairs of equations. The terms involving respectively R and 72 follow different laws.
Similarly Helmholtz's results
2k en
a an2 /3V
/34 mn- — a2 — /32 a2*"2 (wms - a2 - /32)2 + 1
.(11)
oPn (mna-a?-{Py + ryW identify themselves with Maxwell's,, when we omit R and <f and make a2= 0. In order to examine the effect of a2, we see that when 7 = 0, (11) becomes J. fUi fj" tivtlr ft~ t-t c\\
n n '~"i)o o o ~/D9 J •*••••••»•.........• • \ *-*/
c a Gi-fir inn2 — a2 — p-or in terms of v*- (= c02/ca),
o T P" "^ ^ /'^" fTA\
l/-=l---------;;-----^---—.........................(I*;
If now in (14) we suppose n = 0, or X = oo, we find that v = oo , unless a2 = 0. If a2 = 0, we get, in harmony with (6),
^=1____^ -t .........................(15)
which is finite, unless ^wi2 = /32. It is singular that Helmholtz makes precisely opposite statements!:—"Wenn a = 0, wird k = Q und l/c = oo; sonstwerden beide Werthe endlich sein."
The same conclusion may be deduced immediately from the original equations (9), (10). For if the frequency be zero and the velocity of propagation in the medium finite, all the differential coefficients maybe omitted; so that (9) requires as — % = 0 and (10) then gives a2 = 0.
WiillnerJ, retaining a.2 in Helmholtz's equation, writes (14) in the form
s - A2'
.(16)
[* The result (12) is so given by Helmholtz; but the first "-" should be " + ", involving some further corrections in Helmholtz's paper.
t Helmholtz, however, supposes y =t= 0, and on that supposition his statements appear to be correct. They cannot, however, legitimately be deduced, as appears to be assumed by Helmholtz, from the equations which in his paper immediately precede those statements, since those equations are obtained on the understanding that the ratio of the right-hand side of (12) to that of (11) is zero when n = 0, which is not the case when a absolutely = 0. W. F. S.]
$ Wied. Ann. xvn. p. 580; xxni. p. 306. 2x — 2a cos 4*},...............(24)