120 



MATHEMATICAL AND PHYSICAL SCIENCE. 



[Diss. VI. 



main sources of information on this subject, and on phy- 

 sical optics generally, not only in this country but on 

 the Continent. It is remarkable that in France, which 

 possesses so many admirable scientific books, there 

 should not exist a single good treatise on optics. 

 Had not Mr Airy's attention been necessarily with- 

 drawn from optics to astronomy, it is very evident that 

 the theory of light would have received from him 

 many farther important additions. 



(550.) "Whilst an impulse was thus given to the mathe- 

 Sir\Vm ma tj ca l theory of light in the University of Cam- 

 ton and Dr bridge, a similar progress was being made in the 

 Lloyd. sister University of Dublin, where three of her most 

 eminent professors, Sir WILLIAM ROWAN HAMILTON 

 and Professors LLOYD and MACCULLAGH devoted 

 themselves energetically to its improvement and veri- 

 fication. 



(551.) To the two former of these we owe the prediction and 

 Conical re- ocular demonstration of the most singular and critical 



biaSl n iU of a11 the results of F resnel ' s theory. Sir William 

 crystals. Hamilton, a geometer of the first order, having un- 

 dertaken the more complete discussion of the wave 

 surface ofFresnel (see Section Third of this Chapter), 

 to the equation of which he gave a more elegant 

 form than heretofore, 1 ascertained the exact nature 

 of that surface, and consequently the exact direction 

 of refracted rays in the neighbourhood of the " optic 

 axes." It had been shown by Fresnel that, in the case 

 of crystals with two axes, a plane section in a certain 

 direction cuts the two sheets of the wave surface in 

 a circle and in an ellipse, which necessarily intersect 

 each other in four places. (See the annexed figure.) 

 In the lines joining these four points with the centre 

 of the figure the velocity of the two rays is equal. 

 Now the cusps or sharp inflections of the wave surface 

 in these particular directions, occur not only in the 

 particular plane of section which we have considered, 

 but in any section of the wave surface passing through 

 these lines of equal ray- velocity. In the figure, there- 

 fore, of the compound sheet there is not a furrow, as 

 Fresnel had supposed, but a pit or dimple, with 

 arched sides something like the flower of a convol- 

 vulus, and the surfaces meet at the bottom of the pit 

 at a definite angle. Let the circle and ellipse, 

 in the annexed figure, represent the section of the 

 wave surface we have described ; then P is the line 

 of uniform propagation, and P is the bottom of the 



conoidal pit M P N. Now 

 suppose a slender ray of light 

 to move through the crystal in 

 the line P, and to emerge 

 into air at a surface of the 

 crystal cut perpendicular or 

 nearly so to the direction of 

 single ray- velocity P. If we 

 confine our attention at first to 

 the plane of the figure only, 

 that ray having intersecting 

 tangents both proper to the wave surface, would 

 give rise, on Huygens' construction (art. 475), to two 

 emergent rays inclined at an angle. But since this 

 is the case, not only in the plane of the figure, but 

 (as has been stated) in any plane passing through 

 the ray in question, the emergent light must form a 

 conical luminous sheet, the angle of the cone being 

 determined by the refractive properties of the crystal. 

 This beautiful and unexpected result was verified with 

 great skill and address by Dr Lloyd in the case of 

 Arragonite, which is a biaxal crystal, and he found 

 the position, dimensions, and conditions of polariza- 

 tion of the emerging cone of light to be exactly such 

 as theory assigns. When all the necessary correc- 

 tions are attended to, the angle of the cone of light 

 is about 3. There is another case of conical (or it 

 might be called cylindrical) refraction, which occurs 

 nearly in the same portion of a crystal, which was 

 predicted and discovered in like manner, but which 

 we will not stop to particularize. 2 The observations 

 of Dr Lloyd have been extended by M. Haidinger to 

 the case of Diopside, a crystal also having two optic 

 axes. 



Every one capable of appreciating such evidence, (552.) 

 will feel the irresistible impression which so curious Other 

 an anticipation, so accurately fulfilled, gives us of the^. or ^ s r * 

 positive truth of a theory admitting of such veri- m jj ton ' an , 

 fications. The names of Sir W. Hamilton and Dr Dr Lloyd. 

 Lloyd will be handed down to posterity in connec- 

 tion with this admirable discovery. But they have 

 also other claims to our respect, to which we can 

 here only refer in the most general terms. The for- 

 mer has generalized the most complicated cases of 

 common geometrical optics by a peculiar analysis de- 

 veloped in his essays on " Systems of Rays" (Irish 

 Academy Transactions, vols. xv.-xvii.) 3 To Dr Lloyd 



Quater- 

 nions. 



i y _i_ 



1 The equation is ZajT^gj. 2_ 2" 3 . 2 , 8 7^ + 2 , 3 , 2 _ 



2 It was shown by Sir W. Hamilton that the tangent plane M N touches the wave surface, not in two points merely, but in a 

 circle of contact ; consequently, the perpendicular to this tangent plane, OM, is the direction of one of the optic axes (or the velo- 

 city is the same for both portions of the compound wave). Hence a ray incident externally so as to be refracted along this perpen- 

 dicular, will at entrance spread into a hollow cone interior to the crystal, and on emergence at a parallel face each portion of 

 the ray recovers a direction parallel to its primitive direction, and a luminous hollow cylinder is the result. See Dr Lloyd, in 

 the Irish. Academy Transactions, vol. xvii., and Sir W. R. Hamilton's third supplement to his " Systems of Rays" in the same vol. 



3 Sir W. R. Hamilton is also a discoverer in pure analysis and its connection with geometry. Following up the ideas of Mr 

 Warren on the geometrical significance of the symbol V~l, as indicative of direction, Sir W. Hamilton has developed the theory 

 of a new class of imaginary quantities, which he terms quaternions, by means of which he contrives to express simultaneously 

 the direction in space and magnitude of a line or form ; and this calculus he has applied to the solutions of problems of geometry 

 and physical astronomy. The quaternion appears to express something even beyond this ; and this redundancy has been consi- 

 dered as a difficulty by some mathematicians. The superfluous number is considered by Sir W. Hamilton as representing time 

 in mechanical problems. 



