254 - -REPoRT—1]1862. 
sometimes, on the other hand, the grand advance made by Fresnel is depre- 
ciated on account of his theory not being everywhere perfectly rigorous. If 
we reflect on the state of the subject as Fresnel found it, and as he left it, the 
wonder is, not that he failed to give a rigorous dynamical theory, but that a 
single mind was capable of effecting so much. 
The first deduction of the laws of double refraction, or at least of an ap- 
proximation to the true laws, from a rigorous theory is due to Cauchy*, 
though Neumann? independently, and almost simultaneously, arrived at the 
same results. In the theory of Cauchy and Neumann the ether is supposed 
to consist of distinct particles, regarded as material points, acting on one 
another by forces in the line joining them which vary as some function of 
the distances, and the arrangement of these particles is supposed to be dif- 
ferent in different directions. The medium is further supposed to possess 
three rectangular planes of symmetry, the double refraction of crystals, so far 
as has been observed, being symmetrical with respect to three such planes. 
The equations of motion of the medium are deduced by a method similar to 
that employed by Navier in the case of an isotropic medium. The equations 
arrived at by Cauchy, the medium being referred to planes of symmetry, 
contain nine arbitrary constants, three of which express the pressures in the 
principal directions in the state of equilibrium. Those employed by Neumann 
contain only six such constants, the medium in its natural state being sup- 
posed free from pressure. 
In the theory of double refraction, whatever be the particular dynamical 
conditions assumed, everything is reduced to the determination of the velocity 
of propagation of a plane wave propagated in any given direction, and the 
mode of vibration of the particles in such a wave which must exist in order 
that the wave may be propagated with a unique velocity. In the theory of 
Cauchy now under consideration, the direction of vibration and the reciprocal 
of the velocity of propagation are given in direction and magnitude respec- 
tively by the principal axes of a certain ellipsoid, the equation of which con- 
tains the nine arbitrary constants, and likewise the direction-cosines of the 
wave-normal. Cauchy adduces reasons for supposing that the three constants 
G, H, I, which express the pressures in the state of equilibrium, vanish, 
which leaves only six constants. For waves perpendicular to the principal 
axes, the squared velocities of propagation and the corresponding directions 
of vibration are given by the following Table :— 
Waveanormal 0. au ary. a we y Zz 
we L R Q 
Direction of vibra- y R M P 
Zz Q P N 
For waves in these directions, then, the vibrations are either wholly normal 
or wholly transversal. The latter are those with which we have to deal in 
the theory of light. Now, according to observation, in any one of the prin- 
cipal planes of a doubly refracting crystal, that ray which is polarized in the 
principal plane obeys the ordinary law of refraction. In order therefore that 
the conclusions of this theory should at all agree with observation, we must 
* Mémoires de I’ Académie, tom. x. p. 293- 
+ Poggendorff’s Annalen, vol. xxv. p. 418 (1832). 
