CHAP. III.] MAC CULLAGH'S POLAR PLANE.

759

(6.) Solving then for 7, by the rules of the present Calculus, this system of the three linear and scalar equations VI. XIII. XIV., we find for the incident vibration the following vector expression,*

XV... T=

Πρυτ
2Spv

or XV'... 27Spv=T'Spv' – v'Spr' ; '

and accordingly it may be verified by mere inspection, with the help of VII. and IX., that this vector value of satisfies the three scalar equations (5.). And when the incident vibration has been thus deduced from the refracted vibration r', the reflected vibration" is at once given by the formula XI., or by the expression,

XVI... 7" = T′-T;

(7.) The relation XV'. gives at once the equation of complanarity,

XVII... SV'TT' = 0, or the formula XVIII... μ'-p' ||| T, T' ;

if then a plane be anywhere so drawn, as to be parallel (4.) to the three vibrations 7, 7, 7", it will be parallel also to the line 'p', which connects two corresponding points, on the wave and index surface in the crystal: but this is one form of enunciation of Professor Mac Cullagh's Theorem of the Polar Plane, which theorem is thus deduced with great simplicity by quaternions, from the principles above supposed.

(8.) For example, if we suppose that or and oq, in Fig. 89, represent the refracted ray p', and the index vector μ' corresponding, and if we draw through the line PQ a plane perpendicular to the plane of the Figure, then the plane so drawn will contain (on the principles here considered) the refracted vibration r', and will be parallel to both the incident vibration r and the reflected vibration 7"; whence the directions of the two latter vibrations may be in general determined, as being also perpendicular respectively to the incident and reflected rays, p and p": and then the relative intensities (Tr2, Tr'2, T7") of the three lights may be deduced from the relative amplitudes (Tr, Tr', Tr") of the three vibrations, which may them elves be found from the three complanar directions, by a simple resolution of one line 'into two others, of which it is the vector sum, as if the vibrations were forces.

(9.) The equations II'. IV'. V. and XIII'. enable us to express the four vectors, . μ' (= p + v), 1 (= p − v ̄1Svp), p′′ (= p − 2v-1Svp), and p' (= p + 7− v'), in terms of the three vectors p, v, v', which are connected with each other by the relation, XIX... (= p-v1Svp), p′′ (=p-2v-1Svp), and p' (=p+v-v'), XIX. v2 + 2Svp = Sv' (p+v), because XIX'... Svp' S (v′ − v) p,

* The expressions XV. XVI. enable us to determine, not only the directions Ur, Ur" of the incident and reflected vibrations, but also their amplitudes Tr, Tr", or the intensities Tr2, Tr" of the incident and reflected lights, for any given or assumed amplitude Tr of the refracted vibration, or intensity Tr2 of the refracted light, after having determined the direction Ur' of the refracted vibration by means of the formula X.

as in XIII., or because 2-p2 = Su'v') by I. and XIII'.; and with which is connected (VII. and IX.), by the two equations,

T

"

XX. . . § (p + v) r' = 0, and XXI. . . Sv'r' = 0;

while and are connected with the same three vectors, and with r', by the relations VI. VIII. XI. XIII., which conduct, by elimination of ", to the following system (comp. (5.)) of three linear and scalar equations in 7,

XXII... Spr=0; 2SvpSvr = Sv′ (p + v) Svr′; 2SvpSr'-IT = Sv'e ;

and therefore to the vector expression,

2rSvp=Vpv'+', as in XV.

(10.) By these or other transfomations, there is no difficulty in deducing this new equation, in which w may be any vector,

XXIII... VvV { (p − w) + − (p' — w) 7' + (p′′ — w) T1"} 7′ = 0;

and conversely, when w is thus treated as arbitrary, the formula XXIII., with the relations (9.) between the vectors p, p′, p”, v, v', μ', but without any restriction (ezcept itself) on 7, T′, 7", is sufficient to give the two vector equations,

XI... 7 - 7' + 7"=0, and XXIV... pr - p'r' + p′′r" = xv 1 + y,

in which

-

XXV. . . x = Sv (pr — p′r′ + p′′r′′) = Svv ́7′, and XXI...y = S (pr — p'r'v + p′′r′′) ; and which conduct to the two scalar equations (among others),

and

XXVII... SK (pT − p'r' + p′′T') = 0, if XXVII'. . . S«v = 0,

XXVIII... Svp (Spr - Sp"r") = Svp'Sμ'r';

so that if we now suppose the equations VI. VIII. IX. to be given, the equation VII. will follow, by XXVIII.; while, as a case of XXVII., and with the signification IV. or IV'. of t, we have the equation,

XXIX... St (pr − p'r' + p′′r′′) = 0.

(11.) And thus (or otherwise) it may be shown, that the three scalar equations *VI. VIII. IX., combined with the one vector formula XXIII., which (on account of the arbitrary w) is equivalent to fire scalar equations, are sufficient to give the same direction of 7', and the same dependencies of 7 and "thereon, as those expressed by the equations X. XV. XVI.; and therefore (among other consequences), to the formulæ XII. and XVII.

(12.) But the equations VI. VIII. IX. contain what may be called the Principle of Rectangular Vibrations (or of vibrations rectangular to rays); and the formula XXIII. is easily interpreted (416.), as expressing what may be termed the Principle of the Resultant Couple: namely the theorem, that if the three vibrations (or displacements), 7, 7', 7", be regarded as three forces, RT, R ́T′, R ̈T", acting at the ends of the three rays, p, p', p', or OR, OR, OR" (drawn in the directions (1.) from the point of incidence o), then this other system of three forces, RT, — R'T′, R′′T” (conceived as applied to a solid body), is equivalent to a single couple, of which the plane is parallel (or the axis perpendicular) to the face of the crystal.

CHAP. III] PRINCIPLE OF EQUIVALENT MOMENTS.

761

(13.) It follows then, by (10.) and (11.), that from these two principles,* (I.) and (II.), we can infer all the following:

(III.) the Principle of Tangential Vibrations (or of vibrations tangential to the waves);

(IV.) the Principle of Equivalent Vibrations (4.);

(V.) the Principle of the Vis Viva, as expressed (in conjunction with that of the Constant Density of the Ether) by the equation XII.;

(VI.) the Principle (or Theorem) of the Polar Plane;

And (VII.) what may be called the Principle of Equivalent Moments,† namely

* The word “Principle" is here employed with the usual latitude, as representing either an hypothesis assumed, or a theorem deduced, but made a ground of subsequent deduction. The principle (I.) of rectangular vibrations coincides, for the case of an ordinary medium, with the principle (III.) of tangential vibrations; but, for an extraordinary medium, except for the case (not here considered) of ordinary rays in an uniaxal crystal, these two principles are distinct, although both were assumed by Mac Cullagh and Neumann. The present writer has already disclaimed (in the Note to page 736) any responsibility for the physical hypotheses; so that the results given above are offered merely as instances of mathematical deduction and generalization attained through the Calculus of Quaternions.

In a very clear and able Memoir, by Arthur Cayley, Esq. (now Professor Cayley), "On Professor Mac Cullagh's Theorem of the Polar Plane," which was read before the Royal Irish Academy on the 23rd of February, 1857, and has been printed in Vol. VI. of the Proceedings of that Academy (pages 481-491), this name "principle of equivalent moments," is given to a statement (p. 489), that “the‍ moment of R't' round the axis AH, is equal to the sum of the moments of Rt and Rt" round the same axis"; the line AH being (p. 487) the intersection of the plane of incidence with the plane of separation of the two media, that is, with the face of the crystal: while Rt, R't', R"t' are lines representing (p. 488) the three vibrations (incident, refracted, and reflected), at the ends of the three rays AR, AR′ AR", which are drawn from the point of incidence A, so as to lie, all three (p. 487), within the crystal. And in fact, if this statement be modified, either by changing the sign of the moment of R''t" (p. 491), or by drawing the reflected ray AR", like the line or" of the present investigation in the air (or in vacuo), instead of prolonging it backwards within the biaxal crystal, it agrees with the case XXIX. of the more general formula XXVII., which is itself included in what has been called above the Principle of the Resultant Couple. In venturing thus to point out, as the subject obliged him to do, what seemed to him to be a slight inadvertence in a Paper of such interest and value, the present writer hopes that he will not be supposed to be deficient in the admiration (long since publicly expressed by him), which is due to the vast attainments of a mathematician so eminent as Professor Cayley.

Since the preceding Series 423, including its Notes (so far), was copied and sent to the printers, the writer's attention has been drawn to a later Paper by Mac Cullagh (read December 9th, 1839, and published in Vol. XXI., Part I., of the Transactions of the Royal Irish Academy, pp. 17-50), entitled "An Essay towards a Dynamical Theory of crystalline Reflexion and Refraction;" in which there is given at p. 43) a theorem essentially equivalent to the above-stated "Principle of the

theorem that the Moment of the Refracted Vibration (R'T') is equal to the Sum of the Moments of the Incident and Reflected Vibrations (RT and R ̋T"), with respect to any line, which is on, or parallel to, the Face of the Crystal.

[It appears by the Table of Initial Pages (see p. lix.), that the Author had intended to complete the work by the addition of Seven Articles.]

Resultant Couple," but expressed so as to include the case where the vibrations are not uniradial, so that the double refraction of the crystal is allowed to manifest itself. Mac Cullagh speaks, in his enunciation of the theorem, of measuring each ray, in the direction of propagation: which agrees with, but of course anticipates, the direction of the reflected ray, adopted in the preceding investigation. The writer believes that subsequent experiments, by Jamin and others, are considered to diminish much the physical value of the theory above discussed.

39 PATERNOSTER ROW, E.C.
LONDON: January 1870.

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