196 FRESNEL. 
Those physicists (I could here cite the names of some 
of the most celebrated) who have sought to include in a 
| 
Hence the necessity for a more comprehensive theory. As Huy- 
ghens had constructed such a theory by means of an independent 
sphere and spheroid, Fresnel not only generalized the construction by 
a method giving two curved surfaces of higher forms, but he did what 
Huyghens’s method did not effect, even in the simple case which he 
considered,—he showed also a necessary connection between the two 
surfaces; they were in fact not two, but portions of one surface—par 
of the geometrical representation of the same algebraic equation, or, 
in the language of mathematicians, “a curve surface of two sheets.” 
Thus Fresnel’s theory showed not only the laws by which each Tay 
was refracted, but also why there must be two rays. 
Of this more generalized mathematical investigation, the greater 
part of the steps were omitted by Fresnel in his memoir, as being of — 
too complicated and tedious a nature for the patience of his readers; 
he presents only the conclusions, which are derived from certain sup- 
positions with respect to the elasticity of the ether, as being different 
in different directions within the crystal, and ultimately lead to an 
algebraic equation, representing a curved surface of the fourth order, 
consisting of two sheets or portions, as the general form assumed by 
the waves, but which in certain cases, as in cale spar, is reducible to 
the simpler form of the sphere and spheroid of Huyghens. 
For a connected view of these investigations the reader is referred 
to Professor Powell’s Treatise on the Undulatory Theory, gc. page 48. 
London, 1841. 
The mathematical investigation has since called forth much eluci- 
dation, especially in supplying the suppressed processes of Fresnel, itt 
which the analysis of Mr. A. Smith, as well as those of Sir J. Lub- 
bock, Professor Sylvester, Sir W. R. Hamilton, and others, have been 
eminently successful; while the last-named mathematician pointed 
out the very curious consequence that this surface, mathematically 
speaking, presents, at the extremities of the axis, conoidal cusps,—that 
is, depressions of a pointed funnel shape,—which, physically inter- 
preted, would show that a ray passing along that direction ought to 
emerge no longer a single ray, but spread out in a conical surface 
whose surface would not be a point of light, but a ring with a dark 
central space. This extraordinary prediction, so wholly unlike any 
thing hitherto imagined, was, however, fully verified by the observa- 
tions of Dr. Lloyd on a crystal of aragonite; the phenomenon being 
known by the name of “ conical refraction.””— Translator. 
h 
