bodies (earth and comet) is the nearer to the sun, results at sight from ARTICLE 421.-On the Development of the Disturbing Force of the Sun on the Moon; or of one Planet on another, which is nearer (a). Let a, o be the geocentric vectors of moon and sun; r(= Ta), and (To), their geocentric distances; M the sum of the masses of earth and moon; S the mass of the sun; and D (as in recent Series) the mark of derivation with respect to the time: then the differential equation of the disturbed motion of the moon about the earth is, D2a = Moa+n, (Ls), if pa=4(a) = a ̄1Ta ́1, and n= Vector of Disturbing Force = S(po − p (o − a)); ⚫ denoting here a vector function, but not a linear one. (b). If we neglect ŋ, the equation (Ls) reduces itself to the form D'a Moa; which contains (comp. (04)) the laws of undisturbed (c). If we develope the disturbing vector ŋ, according to ascend- ing powers of the quotient r: s, of the distances of moon and sun from the earth, we obtain an infinite series of terms, each representing a finite group of partial disturbing forces, which may be thus denoted, these partial forces increasing in number, but diminishing in intensity, (d). For example, the two forces ni, 1, 1,2 of the first group are, rigorously, proportional to the numbers 1 and 3; the three forces n2, 1, 2, 2, N2,3 of the second group are as the numbers 1, 2, 5; and the four forces of the third group are proportional to 5, 9, 15, 35: while the separate intensities of the first forces, in these three first 734-736 .(e). All these partial forces are conceived to act at the moon; but (f). Denoting then the geocentric elongation OD of moon from sun Pages. - circle, are 20, 20, and -9, +30, 30, as illustrated by Fig. 88 (p. 735); it is found that the directions of the two forces of the first group are represented by the two radii of this unit-circle, which termi- nate in D and D1; those of the three forces of the second group, by the three radii to 1, O, and O2; and those of the four forces of the third group, by the radii to D2, D, D1, and D3; with facilities for ex- tending all these results (with the requisite modifications), to the fourth and subsequent groups, by the same quaternion analysis. (9). And it is important to observe, that no supposition is here made respecting any smallness of excentricities or inclinations (p. 736); so that all the formula apply, with the necessary changes of geocen- tric to heliocentric vectors, &c., to the perturbations of the motion of a comet about the sun, produced by the attraction of a planet, which is (a). If p and μ be two corresponding vectors, of ray-velocity and wave-slowness, or briefly Ray and Index, in a biaxal crystal, the velo- city of light in a vacuum being unity; and if dp and du be any infi- nitesimal variations of these two vectors, consistent with the equa- tions (supposed to be as yet unknown), of the Wave (or wave-surface), and its reciprocal, the Index-Surface (or surface of wave-slowness): we have then first the fundamental Equations of Reciprocity (comp. p. which are independent of any hypothesis respecting the vibrations of (b). If op be next regarded as a displacement (or vibration), tan- gential to the wave, and if dɛ denote the elastic force resulting, there the function being of that linear, vector, and self-conjugate kind, (e). The fundamental connexion, between the functional symbol , and the optical constants abc of the crystal, is expressed (p. 741, comp. the formula (W3) in p. xlvi) by the symbolic and cubic equa- (p+a ̄2) (p + b-2) (p + c-2) = 0 ; of which an extensive use is made in the present Series. (Ws) (d). The normal component, μ-1Sμde, of the elastic force dɛ, is in- 736-756 CONTENTS. 2p, for the propagation of a rectilinear vibration (p. 737); we ob- lv Pages. (XS) (Ys) which is a Symbolical Form of the scalar Equation of the Index-Sur- (e). The Wave-Surface, as being the reciprocal (a) of the index- or 0 = Sp-1(p-2)-1p-1; (A6) (f). In such transitions, from one of these reciprocal surfaces to the other, it is found convenient to introduce two auxiliary vectors, v and (= $v), namely the lines ou and ow of Fig. 89; both drawn from the common centre o of the two surfaces; but v terminating (p. 738) on the tangent plane to the wave, and being parallel to the direc- tion of the elastic force de; whereas w terminates (p. 739) on the tan- gent plane to the index-surface, and is parallel to the displacement dp. and generally (p. 739), the following Rule of the Interchanges holds good: In any formula involving p, p, v, w, and p, or some of them, it is permitted to exchange p with μ, v with w, and with p-1; pro- vided that we at the same time interchange dp with dɛ, but not gene- rally* du with op, when these variations, or any of them occur. * This apparent exception arises (pp. 739, 740) from the circumstance, that with others easily deduced, which may all be illustrated by the above- (i). Among such deductions, the following equations (p. 740) (Vvpv)2 + Svpv =0, (J6); (Vwp-1w)2 + Swp ̄1w=0; (K6) which show that the Locus of each of the two Auxiliary Points, u and Spøp = 1, (Le), and Spp 1p=1; p denoting, for each, an arbitrary semidiameter. (M6) (j). It is, however, a much more interesting use of these two (k). In fact, on comparing the symbolical form (A) of the equa- (1). A precisely similar construction applies, to the derivation of $ (m). The cubic (W3) in ø enables us easily to express (p. 741) the (N6) 0 = (p ̄1p)2 + (p2 + a2 + b2 + c2) Sp¢ ̈1p — a2b2c2 ; (n). If either Spp-p or p2 be treated as constant in (Ne), the in a (real or imaginary) curve of the fourth degree of which two quar- Pages. CONTENTS. tic curves, answering to all scalar values of the constants h and r, the (o). The new ellipsoid (0) is similar to the ellipsoid (M6), and for the squares of its scalar semiaxes, and is therefore confocal with (p). For any point P of the wave, or at the end of any ray p, the tangents to the two curves (n) have the directions of w and uw; so (2). But the vibration dp is easily proved to be parallel to w; (r). In general, the vibration dp has (on Fresnel's principles) the direction of the projection of the ray p on the tangent plane to the wave; and the elastic force dɛ has in like manner the direction of the projection of the index-vector μ on the tangent plane to the index- surface so that the ray is thus perpendicular to the elastic force lvii Pages. *For real curves of the second system (n), this new quadric (Re) is an hy- perboloid, with one sheet or with two, according as the constant r lies between a and b, or between b and c; and, of course, the conjugate hyperboloid (0) has two sheets or one, in the same two cases respectively. In a different theory of light (comp. the next Series, 423), these sphero- i |