ON OPTICAL THEORIES. 255 
If it be necessary that J should vanish, then & or = must be zero. 
According to Helmholtz, however, J is not necessarily zero, and the 
equation to determine it is— 
phK OS we BA 2 Tol at Heal, aust sgl) 
dt? 
so that J, and therefore , is propagated through the medium as a 
wave of normal disturbance with the velocity 
/e 
kK 
On Helmholtz’s theory there may therefore be a normal wave in addition to 
the transverse wave. Helmholtz’s theory becomes Maxwell’s if we put 
® = 0, and then unless the value i = is admissible J = 0, and there 
is no normal wave. If & = 0 there will still be no normal wave, for its 
velocity will be infinite. 
When we consider the problem of double refraction, we can show that 
all the possible directions of vibration L, M, N corresponding to a given 
wave normal /, m, are given by the equation— 
£24 2@-m+ 2@_iao. . (18) 
There are therefore, in general, an infinite number of such directions. 
Tf, however, we are to assume that there are only two, and those the two 
Given by Fresnel’s theory, we must have JL +mM+x2N=0. Thus 
ell’s solenoidal condition, 
df , dg , dh 
cae Gh we . . . . (19) 
is a necessary and sufficient condition to give Fresnel’s construction. 
Chapter III.—Duispersion, ere. 
According to the theory as left by Maxwell, waves of all lengths 
travel at the same rate. Dispersion does not come into consideration. 
This question has been dealt with by Willard Gibbs! and H. A. Lorentz.2 
§ 1. According to Gibbs’s views the displacements of which we are 
cognisant in the phenomena of light are the average displacements taken 
through a space which is small in comparison with the wave length, but 
contains many molecules of the body. The real displacement at each 
point of such an elementary space probably differs considerably from the 
average value, and a complete theory should take into account the two. 
This is done in Gibbs’s paper. The average displacements being &, n, Z, 
the complete displacement is taken as £ + é, &e. &, n’,Z' are denoted as 
‘the irregular parts of the displacements. It is shown that t’, n’, ¢ are 
linear functions of £, n, ¢; they are therefore of the same period, and the 
phase of the irregular displacement throughout the element Dv is the 
| 1 J. W. Gibbs, American Journal of Science, vol, xxii. April, 1882. 
* H. A. Lorentz, Wied. Ann. t. ix.; Schlimilch. Zeitschrift, t. xxiii. 
