(81.) Thus for each of the two points P, P' the line of vibration is parallel to one of the lines of curvature at ; and it is evident, from what precedes, that the other of these last lines has the direction of the corresponding Orthogonal (66.) at P or P': nor is there any danger of confusion.

(82.) As regards quaternion expressions, for the two vibrations on a given warefront, the sub-article, 410, (8.), with notations suitably modified, shows by its formulæ XIX. XXII. that we have here the equations,

and

CLXVII... 0 = Sμ εp v ̧ εp vi

= Sμ δρ νο Sν δρ + Sμδρ ν. Sιο δρα CXVIII. . . δρ | UV μ' - UVμνι,

if vo, v1 be, as in earlier formulæ of the present Series 422, the cyclic normals of the reciprocal ellipsoid, which are often called the Optic Axes of the Crystal.

(83.) And hence may be deduced the known construction, namely, that "for any given direction of wave-front, the two planes of polarization, perpendicular respectively to the two vibrations in Fresnel's theory, bisect the two supplementary and diedral angles, which the two optic axes subtend at the normal to the front :" or that these planes of polarization bisect, internally and externally, the angle between the two planes, μro and μνι.

(81.) It may not be irrelevant here to remark, that if u and μ, be any two index-vectors, which have (as in (76.)) the same direction, but not the same length, the equation LXIV. enables us to establish the two converse relations:

CLXIX... abcTμ, = (Sμoμ);

CLXIX'. . . abcTμ = (Sμ‚μ‚) .

(85.) Either by changing a, b, c, p, μ to a 2, b-2, c2, -1, p, or by treating the form LXIII., in (19.), of the Equation of the Wave, as we have just treated the form LXIV., of the equation of Index Surface, in the same sub-article (19.), we see that if and p. ρ be any two condirectional rays (Up, Up), then,

and

CLXX... (abe)-1Tp, = (Sp¢-1p) 3, or, abcTp,1= (Spp ̄1p)} ;
CLXX'... (abe) ̄1Tp = S (p, ̄1p,), or, abcTo-1 (Sp ̧¢ ̈1p,)1.

(86.) A somewhat interesting geometrical consequence may be deduced from these last formula, when combined with the equation LIX. of that variable ellipsoid, Spp 1p= h1, which cuts the wave in a line of vibration (h). For if we introduce this symbol h for Sop-1p, and write r, instead of Tp, to denote the length of the second ray p the first equation CLXX. will take this simple form,

CLXXI... r, = abch-2,

which shows at once that r, and h are together constant, or together variable; and therefore, that ". a Line of Vibration on one Sheet of the Wave is projected into as Orthogonal Trajectory to all such Lines on the other Sheet, and conversely the latter into the former, by the Vectors p of the Wave:" so that one of these two curves would appear to be superposed upon the other, to an eye placed at the Wave- Centre 0.

(87.) The visual cone, here conceived, is represented by the equation CLVI., with some constant value of r; and as being a surface of the second degree, it ought to cut the wave, which is one of the fourth, in some curve of the eighth degree; or in some system of curves, which have the product of their dimensions equal to eight.

CHAP. III.]

FRESNEL'S THEORY.

755

Accordingly we now see that the complete intersection, of the cone CLVI. with the wave, consists of two curves, each of the fourth degree; one of these being, as in (67.), a complete sphero-conic (r), and the other a complete line of vibration (h): a new geometrical connexion being thus established between these two quartic

curves.

(88.) As additional verifications, we may regard the three principal planes, as limits of the cutting cones; for then, in the plane (a) for instance, the circle (a) and the ellipse (a), which are (in a sense) projections of each other, and of which the latter has been seen to be a line of vibration, are represented respectively by the two equations,

CLXXII. . . r = a, and CLXXII'. . . bc = h2,

in agreement with CLXXI.; and similarly for the two other planes.

(89.) It was an early result of the quaternions, that an ellipsoid with its centre at the origin might be adequately represented by the equation (comp. 281, XXIX., or 282, XIX.),

CLXXIII... T (1p + pk) = k2 — 1o, if Ti>TK;

or, without any restriction on the two vector constants, , k, by this other equation, *

CLXXIII'... T (ip + pk)2 = (k2 – 12)2.

(90.) Comparing this with Sppp = 1, as the equation XXIX. of the Generating Ellipsoid, we see that we are to satisfy, independently of p, or as an identity, the relation (comp. 336):

CLXXIV. . . (k2 — 12)2 Sppp = (ip + px) (p + kp

= (12 + k2) p + 2Siрkp;

which is done by assuming (comp. again 336) this cyclic form for p,

CLXXV. . . (k2 = 12)2 pp = (12 + k2) p +2Vkpi

or as in (24.) comp. 359, III. IV.,

= (1 − K)2p + 21Sкp + 2kSip;

pp=gp + Vλp\', Spop=gp2 + Sλpλ'p = 1; LXXII. LXXIII.

*This equation, CLXXIII'. or CLXXII., which had been assigned by the author as a form of the equation of an ellipsoid, has been selected by his friend Professor Peter Guthrie Tait, now of Edinburgh, as the basis of an admirable Paper, entitled: "Quaternion Investigations connected with Fresnel's Wave-Surface," which appeared in the May number for 1865, of the Quarterly Journal of Pure and Applied Mathematics; and which the present writer can strongly recommend to the careful perusal of all quaternion students. Indeed, Professor Tait, who has already published tracts on other applications of Quaternions, mathematical and physical, including some on Electro-Dynamics, appears to the writer eminently fitted to carry on, happily and usefully, this new branch of mathematical science: and likely to become in it, if the expression may be allowed, one of the chief successors to its inventor.

with expressions for the constants g, A, A', which give, by LXXVI., the following values for the scalar semiaxes,*

(91.) Knowing thus the form CLXXV. of the function, which answers in the present case to the given equation CLXXIII. of the generating ellipsoid, there would be no difficulty in carrying on the calculations, so as to reproduce, in connexion with the two constants, K, all the preceding theorems and formulæ of the present Series, respecting the Wave and the Index-Surface. But it may be more useful to show briefly, before we conclude the Series, how we can pass from Quaternions to Cartesian Co-ordinates, in any question or formula, of the kind lately considered. (92.) The three italic letters, ijk, conceived to be connected by the four fundamental relations,

i2 = j2 = k2 = ijk = − 1,

(A), 183, were originally the only peculiar symbols of the present Calculus; and although they are not now so much used, as in the early practice of quaternions, because certain general signs of operation, such as S, V, T, U, K, have since been introduced, yet they (the symbols ijk) may be supposed to be still familiar to a student, as links between quaternions and co-ordinates.

(93.) We shall therefore merely write down here some leading expressions, of which the meaning and utility seem likely to be at once perceived, especially after the Calculations above performed in this Series.

(94.) The vector semiaxes of the generating ellipsoid being called a, ẞ, y (comp. (40.) (42.)), we may write,

CLXXVIII... a = ia, ẞ=jb, y=ke;

CLXXIX. . . ¢p = a ̄1Sa ̄1p + B-1Sẞ ̄1p+y ̄1Sy ̄1p = £a ̄1Sa ̄1p = − £iz-3r ; CLXXX... Spop = (Sa1p)2 = Ya ̄2x2;

CLXXXI... Spp-1p = £a2x2;

CLXXXII... (p + e) p = Σa (a ̄2 + e) Sa ̄1p;

The reader, at this stage, might perhaps usefully turn back to that Construction of the Ellipsoid, illustrated by Fig. 53 (p. 226), with the Remarks thereon, which were given in the few last Series of the Section II. i. 13, pages 223-233. It will be seen there that the three vectors, 1, K, 1-k, of which the lengths are expressed by CLXXVII., are the three sides, CB, CA, AB, of what may be called the Generating Triangle ABC in the Figure; and that the deduction CLXXVI., of the three semiaxes, abc, from the two vector constants, 1, K, with many connected results, can be very simply exhibited by Geometry. The whole subject, of the equation T (p+ px)=K2 of the ellipsoid, was very fully treated in the Lectures; and the calculations may be made more general, by the transformations assigned in the long but important Section III. ii. 6 of the present Elements, so that it seems unnecessary to dwell more on it in this place.

CHAP. III.] GEN. LAWS OF REFLEXION AND REFRACTION.

CLXXXIII... ( + e) ̄1p = Σa (a ̈2 + e)-1 Sa ̄1p;

CLXXXIV... if r2 = Tp2 = Ex2, then v=r2(p+r-2)-1p

757

and the Index-Surface may be treated similarly, or obtained from the Wave by changing abe to their reciprocals.

423. As an eighth specimen of physical application we shall investigate, by quaternions, Mac Cullagh's Theorem of the Polar Plane,* and some things therewith connected, for an important case of incidence of polarized light on a biaxal crystal: namely, for what was called by him the case of uniradial vibrations.

(1.) Let homogeneous light in air (or in a vacuum), with a velocity† taken for unity, fall on a plane face of a doubly refracting crystal, with such a polarization that only one refracted ray shall result; let p, p', p" denote the vectors of ray-velocity of the incident, refracted, and reflected lights respectively, p having the direction of the incident ray, prolonged within the crystal, but p" that of the reflected ray outside; and let u be the rector of wave-slowness, or the index-vector (comp. 422, (1.)), for the refracted light: these four vectors being all drawn from a given point of incidence o, and μ', like p', being within the crystal.

(2.) Then, by all‡ wave theories of light, translated into the present notation, we have the equations,

II.

I... p2 = Sμ'p' = p"2 = -1;

p"-vov, with II'... v = μ' - p,

where v is a normal to the face; whence also,

See pp. 39, 40 of the Paper by that great mathematical and physical philosopher, "On the Laws of Crystalline Reflexion and Refraction," already referred to in the Note to page 737 (Trans. R. I. A., Vol. XVIII., Part I.).

Of course, by a suitable choice of the units of time and space, the velocities and slownessses, here spoken of, may be represented by lines as short as may be thought convenient.

These equations may be deduced, for example, from the principles of Huyghens, as stated in his Tractatus de Lumine (Opera reliqua, Amst., 1728).

so that the three vectors, p, μ', p", terminate on one right line, which is perpendicular to the face of the crystal: and the bisector of the angle between the first and third of them, or between the incident and reflected rays, is the intersection ɩ of the plane of incidence with the same plane face.

(3.) Let T, T'," be the vectors of vibration for the three rays p, p', p”, conceived to be drawn from their respective extremities; then, by all* theories of tangential vibration, we have the equations,

VI... Spr = 0;

VII... Sμ'T'=0;

VIII... Sp"r"= 0;

to which Mac Cullagh adds the supposition (a), that the vibration in the crystal is perpendicular to the refracted ray: or, with the present symbols, that

IX... Sp'r' = 0; whence X... 7' || Vμ'p',

the direction of the refracted vibration r' being thus in general determined, when those of the vectors p' and μ' are given.

(4.) To deduce from 'the two other vibrations, 7 and 7′′, Mac Cullagh assumes, (b), the Principle of Equivalent Vibrations, expressed here by the formula,

in virtue of which the three vibrations are parallel to one common plane, and the refracted vibration is the vector sum (or resultant) of the other two; (c), the Principle of the Vis Viva, by which the reflected and refracted lights are together equal to the incident light, which is conceived to have caused them; and (d), the Principle of Constant Density of the Ether, whereby the masses of ether, disturbed by the three lights, are simply proportional to their volumes: the two last hypothesest being here jointly expressed by the equation,

(5.) Eliminating p′′ and 7′′ from XII. by V. and XI., 72 goes off; and we find, with the help of I. and II'., the following linear equation in 7,

a second such equation is obtained by eliminating p" and r" by III. and XI. from VIII., and attending to I. VI. VII., namely,

XIV... 2Spv Sμ'r= (p2 - μ'2) Spr' = - Su'v'Spr';

and a third linear equation in r is given immediately by VI.

The equations VI. VII. VIII. hold good, for instance, on Fresnel's principles; but Fresnel's tangential vibration in the crystal has a direction perpendicular to that adopted by Mac Cullagh.

In the concluding Note (p. 74) to this Paper, Professor Mac Cullagh refers to an elaborate Memoir by Professor Neumann, published in 1837 (in the Berlin Transactions for 1835), as containing precisely the same system of hypothetical principles respecting Light. But there was evidently a complete mutual independence, in the researches of those two eminent men. Some remarks on this subject will be found in the Proceedings of the R. I. A., Vol. I., pp. 232, 374, and Vol. II., p. 96.