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 theor}'. As Huy- 

 ghens had constructed sucli 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 

 Huvghens's method did not effect, even in the simple case which he 

 considered — he showed also a necessary connection between tlie two 

 surfaces • they were in fact not two, but portions of one surface — parts 

 of the geometi-ical representation of the same algebraic equation, or, 

 in the h^nfuao'e of mathematicians, "a curve surface of two sheets." 

 Thus Frcsnel's theory showed not only the laws by which each ray 

 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 calc 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, fc. page 48. 

 London, 1841. 



The mathematical investigation has since called forth much eluci- 

 dation, especially in supplying the suppressed processes of Fresnel, ia 

 which the analysis of Mr. A. Smith, as well as those of Sir J. Lub- 

 bock, Professor Sylvester, Sir W. K. 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 reiva.ct\on."— Translator. 



