918 MATHEMATICAL AND PHYSICAL SCIENCE. 
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 
conoidal pit MPN. Now 
suppose a slender ray of light 
to move through the crystal in 
the line OP, and to emerge f 
into air at a surface of the ™ 
crystal cut perpendicular or 
nearly so to the direction of 
single ray-velocity O P. If we 
many farther important additions. confine our attention at first to 
_(550.) Whilst an impulse was thus given to the mathe- the plane of the figure only, oat 
pu mij. Matical theory of light in the University of Cam- that ray having intersecting 
ton and Dr bridge, a similar progress was being made in the 
Lloyd. 
(551.) 
sister University of Dublin, where three of her most 
eminent professors, Sir Wirt1am Rowan Hamittron 
and Professors Liroyp and Maccurtacu devoted 
themselves energetically to its improvement and veri- 
fication, 
Tothe two former of these we owe the prediction and 
Conical re- ocular demonstration of the most singular and critical 
fraction in 
biaxal 
crystals. 
of all the results of Fresnel’s theory. Sir William 
Hamilton, a geometer of the first order, having un- 
dertaken the more complete discussion of the wave 
surface of Fresnel (see Section Third of this Chapter), 
to the equation of which he gave a more elegant 
form than heretofore,! 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. 
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 oceurs 
nearly in the same portion of a erystal, which was 
predicted and discovered in like manner, but which 
we will not stop to particularize.? The observations 
of Dr Lloyd have been extended by M. Haidinger to 
the case of Diopside, a crystal also having two optic 
axes, 
Now the cusps or sharp inflections of the wave surface Every one capable of appreciating such evidence, (552,) 
in these particular directions, occur not only in the will feel the irresistible impression which so curious Other 
particular plane of section which we have considered, an anticipation, so accurately fulfilled, gives us of the ae oe ye 
but in any section of the wave surface passing through positive truth of a theory admitting of such veri-jijton and 
these lines of equal ray-velocity. In the figure, there- fications. The names of Sir W. Hamilton and Dr Dr Lloyd, 
fore, of the compound sheet there is nota furrow, as Lloyd will be handed down to posterity in connec- 
Fresnel had supposed, but a pit or dimple, with tion with this admirable discovery. But they have 
arched sides something like the flower of aconvol- also other claims to our respect, to which we can 
vulus, and the surfaces meet at the bottom of the pit here only refer in the most general terms. The for- 
at a definite angle, Let the circle and ellipse, mer has generalized the most complicated cases of 
in the annexed figure, represent the section of the common geometrical optics by a peculiar analysis de- 
wave surface we have described ; then O P is the line veloped in his essays on “ Systems of Rays” (Irish 
of uniform propagation, and P is the bottom of the Academy Transactions, vols, xv.-xvii,)® To Dr Lloyd 
ax? b2y2 E 422 ah 
SpyP+e—a® Pe yP+P—P wpe ~ 
® 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 4/—1, as indicative of direction, Sir W. Hamilton has developed the theory 
Quater- of a new class of imaginary quantities, which he terms quaternions, by means of which he contrives to express simultaneously 
anes 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, 
1 The equation is— 
