and with Mzw-3p, II... D'a - Mr-3a, D2w = — Mw-3w, (2.) The vector a, with its tensor r, and the mass M, are given by the theory of the earth (or sun); and p, Dp, D2p are deduced from three (or more) near observations of the comet; operating then on III. with S. pDp, we arrive at the formula, Sp DpD2p T M Sp DpUa Z V... M 3 w3 which becomes by IV., when cleared of fractions and radicals, and divided by 2, an algebraical equation of the seventh degree, whereof one root is the sought distance* z, of the comet (or planet) from the earth. 421. As a sixth specimen, we shall indicate a method, suggested by quaternions, of developing and geometrically decomposing the disturbing force of the sun on the moon, or of a relatively superior on a relatively inferior planet. (1.) Let a, o be the geocentric vectors of moon and sun; r, s their geocentric distances (r= Ta, s = To); M the sum of the masses of earth and moon; and S the mass of the sun; then the differential equation of motion of the moon about the earth may be thus written (comp. 418, 419), if D be still the mark of derivation relatively to the time, and II... $a = $(a) = a ̄1Ta ̄1; so that pa is here a vector-function of a, but not a linear one. (2.) If we confine ourselves to the term Mpa, in the second member of k., we fall back on the equation 419, I., and so are conducted anew to the laws of undisturbed relative elliptic motion. (3.) If we denote the remainder of that second member by ŋ, then ŋ may be called the Vector of Disturbing Force; and we propose now to develope this vector, according to descending powers of T (σ: a), or according to ascending powers of the quotient rs, of the distances of moon and sun from the earth. (4.) The expression for that vector may be thus transformed: III.. Vector of Disturbing Force = n = D2a - Mpa = Ss ̄1σ ̄1 { 1 − (1 − ao ̄1) ̄1T (1 — ao ̄1)-1} * Compare the equation in the Mécanique Céleste (Tom. I., p. 241, new edition, Paris, 1843). Laplace's rule for determining, by inspection of a globe, which of the two bodies is the nearer to the sun, results at once from the formula V. CHAP. III.] DEVELOPMENT OF DISTURBING FORCE. 735 that is, if IV.. N = N1 + 72 + 73 + &c., = S V... n1 = Ss 1o ̄1 (fo ̄1a + fao1): · (a + 3oao-1) = N1,1+ N1,2; σ 233 VI. n2 = 3 Sr2 (aoa1+20+ 5oaoa ̄1o ̄1) = n2, 1+n2,2+n2, 3; &c. the general term of this development being easily assigned. (5.) We have thus a first group of two component and disturbing forces, which are of the same order as Sr2 as 84 Sr ; a second group of three such forces, of the same order 83 ; a third group of four forces, and so on. (6.) The first component of the first group has the following tensor and versor, MN', of the same first group, has an exactly triple intensity, MN'=3MN; and its direction is such that the angle NMN', between these two forces of the first group, is bisected by a line мs' from the moon, which is parallel to the sun's geocentric vector ES. (8.) If then we conceive a line EM' from the earth, having the same direction as the last force MN', this new line will meet the heavens in what may be called for the moment a fictitious moon D1, such that the arc DD1 of a great circle, connecting it with the true moon ) in the heavens, shall be bisected by the sun, as represented in Fig. 88. (9.) Proceeding to the second group (5.), we have by VI. for the first component of this group, a line from the earth, parallel to this new force, meets therefore the heavens in what may be called a first fictitious sun, 1, such that the arc of a great circle, OO1, connecting it with the true sun, is bisected by the moon D, as in the same Fig. 88. * Such a general term was in fact assigned and interpreted in a communication of June 14, 1847, to the Royal Irish Academy (Proceedings, Vol. III., p. 514); and in the Lectures, page 616. The development may also be obtained, although less easily, by Taylor's Series adapted to quaternions. Compare pp. 427, 428, 430, 431 of the present work; and see page 332, &c., for the interpretation of such symbols as σασ', ασα, (10.) The second component force, of the same second group, has an intensity exactly double that of the first (Tn2,2 = 2T72, 1); and in direction it is parallel to the sun's geocentric vector Es, so that a line drawn in its direction from the earth would meet the heavens in the place of the sun O. (11.) The third component of the present group has an intensity which is precisely five-fold that of the first component (Tn2,3 = 5Tn2, 1); and a line drawn in its direction from the earth meets the heavens in a second fictitious sun O2, such that the arc O O2, connecting these two fictitious suns, is bisected by the true sun (12.) There is no difficulty in extending this analysis, and this interpretation, to subsequent groups of component disturbing forces, which forces increase in number, and diminish in intensity, in passing from any one group to the next; their intensities, for each separate group, bearing numerical ratios to each other, and their direetions being connected by simple angular relations. (13.) For example, the third group consists (5.) of four small forces, n3,1 • • 13, ár Sr3 of which the intensities are represented by multiplied respectively by the four 1685' whole numbers, 5, 9, 15, and 35; and which have directions respectively parallel to lines drawn from the earth, towards a second fictitious moon D2, the true moon, the first fictitious moon Dı (8.), and a third fictitious moon D3; these three fictitious moons, like the two fictitious suns lately considered, being all situated in the momeRtary plane of the three bodies E, M, S and the three celestial arcs, D2D, DDI, DIDS, being each equal to double the arc O of apparent elongation of sun from moon in the heavens, as indicated in the above cited Fig. 88. (14.) An exactly similar method may be employed to develope or decompose the disturbing force of one planet on another, which is nearer than it to the sun; and it is important to observe that no supposition is here made, respecting any smallness of excentricities or inclinations. 422. As a seventh specimen of the physical application of quaternions, we shall investigate briefly the construction and some of the properties of Fresnel's Wave Surface, as deductions from his principles or hypotheses* respecting light. (1.) Let p be a Vector of Ray-Velocity, and μ the corresponding Vector of Wave-Slowness (or Index - Vector), for propagation of light from an origin o, within a biaxal crystal; so that I... == -1; II... Sμdp = 0; and therefore III... Spôμ = 0, The present writer desires to be understood as not expressing any opinion of his own, respecting these or any rival hypotheses. In the next Series (423), as an eighth specimen of application, he proposes to deduce, from a quite different set of physical principles respecting light, expressed however still in the language of the present Calculus, Mac Cullagh's Theorem of the Polar Plane; intending then, as a ninth and final specimen, to give briefly a quaternion transformation of a celebrated equation in partial differential coefficients, of the first order and second degree, which occurs in the theory of heat, and in that of the attraction of spheroids. CHAP. III.] CONSTRUCTION OF FRESNEL'S WAVE surface. 737 if do and du be any infinitesimal variations of the vectors p and με consistent with the scalar equations (supposed to be as yet unknown), of the Wave-Surface and its Reciprocal (with respect to the unit-sphere round o), namely the Surface of WaveSlowness, or (as it has been otherwise called) the Index*-Surface: the velocity of light in a vacuum being here represented by unity. (2.) The variation dp being next conceived to represent a small displacement, tangential to the wave, of a particle of ether in the crystal, it was supposed by Fresnel that such a displacement op gave rise to an elastic force, say dɛ, not generally in a direction exactly opposite to that displacement, but still a function thereof, which function is of the kind called by us (in the Sections III. ii. 6, and III. iii. 7) linear, vector, and self-conjugate; and which there will be a convenience (on account of its connexion with certain optical constants, a, b, c) in denoting here by p ́1ồp (instead of pop) so that we shall have the two converse formulæ, (3.) The ether being treated as incompressible, in the theory here considered, so that the normal component μ 'Sude of the elastic force may be neglected, or rather suppressed, there remains only the tangential component, as regulating the motion, tangential to the wave, of a disturbed and vibrating particle. (4.) If then it be admitted that, for the propagation of a rectilinear vibration, tangential to a wave of which the velocity is Tu1, the tangential force (3.) must be exactly opposite in direction to the displacement dp, and equal in quantity to that displacement multiplied by the square (Tμ-2) of the wave-velocity, we have, by V. and VI., the equation, VII... -1♪p — μ-1Sμdε =μ-2dp, or VIII... op (p1 — μ-2)-1μ ̃1Sμde; combining which with II., we obtain at once this Symbolical Form of the scalar equation of the Index Surface, * This brief and expressive name was proposed by the late Prof. Mac Cullagh (Trans. R. I. A., Vol. XVIII., Part I., page 38), for that reciprocal of the wave-surface which the present writer had previously called the Surface of Components of Wave-Slowness, and had employed for various purposes: for instance, to pass from the conical cusps to the circular ridges of the Wave, and so to establish a geometrical connexion between the theories of the two conical refractions, internal and external, to which his own methods had conducted him (Trans. R. I. A., Vol. XVII, Part I., pages 125-144). He afterwards found that the same Surface had been otherwise employed by M. Cauchy (Exercises de Mathématiques, 1830 p. 36), who did not seem however to have perceived its reciprocal relation to the Wave. while the direction of the vibration dp, for any given tangent plane to the wave, is determined generally by the formula VIII. (5.) That formula for the displacement, combined with the expression V. for the elastic force resulting, gives v being thus an auxiliary vector; and because the equation XI. of the index surface gives, XVI... Sμv = −1, while XVII. . . Vudɛ = 0, by XIII., it follows that the vector v, if drawn like p and μ from o, terminates on the tangent plane to the wave, and is parallel to the direction of the elastic force. (6.) The equations XIV. XVI. give, XVIII. . . μ2v2 — Supu = 1, whence XIX... v2Suèμ = Sμ&v=– Svčμ, because ¿Sμv = 0, by XVI., and ¿Sv¢v = 28(pv.cv), by the self-conjugate property of; comparing then XIX. with III., we see that † p (as being ↓ every dμ) has the direction of μ + v1, and therefore, by I. and XVI., that we may write, which last equation shows, by (5.), that the ray is perpendicular (on Fresnel's principles) to the elastic force de, produced by the displacement dp. (7.) The equations XX. and XXI. show by XIV. that XXIII... (pp)v=p1, whence XXIV... v = (p-2)-1p-1; we have therefore, by XXII., the following Symbolical Form (comp. (4.)) of the Equation of the Wave Surface, XXV... 0=Sp ̄1 (p − p ̃2) ̄1p ̄1; or, by transformations analogous to X. and XI., XXVI... 1 = Sp¢ ( − p ̃3) ̄1p ̄1; XXVII... 1 = Sp (p2 — p ̄1) ̄1p; and we see that we can return from each equation of the wave, to the corresponding equation of the index surface, by merely changing p to μ, and to -1; but this result will soon be seen to be included in one more general, which may be called the Rule of the Interchanges. (8.) The equation XXV. may also be thus written, XXVIII. . . Sp († − p ̃2)-1p = 0 ; but under this last form it coincides with the equation 412, XLI.; hence, by 412, (19.), the Wave Surface may be derived from the auxiliary or Generating Ellipsoid, XXIX... Spøp = 1, by the following Construction, which was in fact assigned by Fresnel* himself, but as the result of far more complex calculations:-Cut the ellipsoid (abc) by an arbitrary plane through its centre, and at that centre erect perpendiculars to that plane, which shall have the lengths of the semiaxes of the section; the locus of the extremities of the perpendiculars so erected will be the sought wave surface. * See Sir John F. W. Herschel's Treatise on Light, in the Encyclopædia Metropolitana, page 545, Art. 1017. ། |