418.] The optical interpretation of the common result of the last two sections is that the planes of polarization of the two rays whose wave-fronts are parallel, bisect the angles contained by planes passing through the normal to the wave-front and the vectors (optic axes) X', '. 419.] As in § 409, the normal velocity is given by v2 = Sw $w = 2 SX' to Sμ'-p'w2 = p2 = 13 §2. λ' μ'α [This transformation, effected by means of the value of in § 417, is left to the reader.] Hence, if v1, v2 be the velocities of the two waves whose normal is a, v―v2 = 2 T.VXaVμ'a oc sin a sin a. That is, the difference of the squares of the velocities of the two waves varies as the product of the sines of the angles between the normal to the wave-front and the optic axes (X', μ'). 420.] We have, obviously, Hence (T2 — S2).VX ́aVμ'a = T2V.VX'aVμ'a = S.2x'μ'a. v2 = p' + (T± S). Vλ'aVμ'a. The equation of the index surface, for which This will, of course, become the equation of the reciprocal of the index-surface, i.e. the wave-surface, if we put for the function & its reciprocal: i. e. if in the values of X', μ', p' we put 1 1 1 a с for a, b, c respectively. We have then, and indeed it might have been deduced even more simply as a transformation of § 409 (7), 1 = −pp2 F (T+S).VλpVμp, as another form of the equation of Fresnel's wave. If we employ the , « transformation of § 121, this may be written, as the student may easily prove, in the form (x2-12) 2 = S2 (i−k) p+ (TV 1p+TVκp)2. 421.] We may now, in furtherance of our object, which is to give varied examples of quaternions, not complete treatment of any one subject, proceed to deduce some of the properties of the wavesurface from the different forms of its equation which we have given. 422.] Fresnel's construction of the wave by points. From § 273 (4) we see at once that the lengths of the principal semidiameters of the central section of the ellipsoid by the plane Spp-1p = 1, S.a (p-1-p-2)-1a = 0. are determined by the equation Up, If these lengths be laid off along a, the central perpendicular to the cutting plane, their extremities lie on a surface for which a = and Tp has values determined by the equation. Of course the index-surface is derived from the reciprocal ellipsoid by the same construction. Spopp = 1 To find the nature of the surface near these points put where T is very small, and reject terms above the first order in Ta. The equation of the wave becomes, in the neighbourhood of the singular point, 2pSλw + S.@V.XVλμ = ±T.Vλ@Vλμ, which belongs to a cone of the second order. 424.] From the similarity of its equation to that of the wave, it is obvious that the index-surface also has four conical cusps. As an infinite number of tangent planes can be drawn at such a point, the reciprocal surface must be capable of being touched by a plane at an infinite number of points; so that the wave-surface has four tangent planes which touch it along ridges. To find their form, let us employ the last form of equation of the wave in § 420. If we put we have the equation of a cone of the second degree. It meets the wave at its intersections with the planes S (i −k) p = ± (x2 —‚2). .... (2) Now the wave-surface is touched by these planes, because we cannot have the quantity on the first side of this equation greater in absolute magnitude than that on the second, so long as p satisfies the equation of the wave. That the curves of contact are circles appears at once from (1) and (2), for they give in combination p2 = S (i + k) p2 ..... ..... (3) the equations of two spheres on which the curves in question are situated. The diameter of this circular ridge is TV. (1+x) U (−k) = 2TVLK 1 T(−x) = ¿ √ (a2 — b2) (12 — c2). [Simple as these processes are, the student will find on trial that the equation Sp (4-1 — p−2)−1p = 0, gives the results quite as simply. For we have only to examine the cases in which -p-2 has the value of one of the roots of the symbolical cubic in 4-1. In the present case Tp=6 is the only one which requires to be studied.] 425.] By § 413, we see that the auxiliary vector of the succeeding section, viz. r = (μ2 — p−1)−1μ = (p−1—p−2)−1p−1, is parallel to the direction of the force of restitution, p. Hence, as Hamilton has shewn, the equation of the wave, in the form Stp = 0, (4) of § 414, indicates that the direction of the force of restitution is perpendicular to the ray. Again, as for any one versor of a vector of the wave there are two values of the tensor, which are found from the equation S.Up (p-1-p-2)-1 Up = 0, we see by § 422 that the lines of vibration for a given plane front are parallel to the axes of any section of the ellipsoid made by a plane parallel to the front; or to the tangents to the lines of curvature at a point where the tangent plane is parallel to the wavefront. 426.] Again, a curve which is drawn on the wave-surface so as to touch at each point the corresponding line of vibration has Hence pdp || (p-1—p-2)-1 p. Sppdp = 0, or Spopp = C, so that such curves are the intersections of the wave with a series of ellipsoids concentric with it. 427.] For curves cutting at right angles the lines of vibration we have Hence -1 dp || Vpp-1(4-1— p−2)−1p || Tp (p—p2)−1p. Spdp = 0, or Tp = C, so that the curves in question lie on concentric spheres. They are also spherical conics, because where Τρ the equation of the wave becomes S.p (p1+C-2)-1p = 0, the equation of a cyclic cone, whose vertex is at the common centre of the sphere and the wave-surface, and which cuts them in their curve of intersection. (Quarterly Math. Journal, 1859.) 428.] As another example we take the case of the action of electric currents on one another or on magnets; and the mutual action of permanent magnets. A comparison between the processes we employ and those of Ampère (Théorie des Phénomènes Electrodynamiques, &c., many of which are well given by Murphy in his Electricity) will at once shew how much is gained in simplicity and directness by the use of quaternions. The same gain in simplicity will be noticed in the investigations of the mutual effects of permanent magnets, where the resultant forces and couples are at once introduced in their most natural and direct forms. 429.] Ampère's experimental laws may be stated as follows: I. Equal and opposite currents in the same conductor produce equal and opposite effects on other conductors: whence it follows. that an element of one current has no effect on an element of another which lies in the plane bisecting the former at right angles. II. The effect of a conductor bent or twisted in any manner is equivalent to that of a straight one, provided that the two are traversed by equal currents, and the former nearly coincides with the latter. III. No closed circuit can set in motion an element of a circular conductor about an axis through the centre of the circle and perpendicular to its plane. IV. In similar systems traversed by equal currents the forces are equal. : To these we add the assumption that the action between two elements of currents is in the straight line joining them and two others, viz. that the effect of any element of a current on another is directly as the product of the strengths of the currents, and of the lengths of the elements. 430.] Let there be two closed currents whose strengths are a and a; let a, a, be elements of these, a being the vector joining their middle points. Then the effect of a on a, must, when resolved along a1, be a complete differential with respect to a (1.e. with respect to the three independent variables involved in a), since the total resolved effect of the closed circuit of which a' is an element is zero by III. Also by I, II, the effect is a function of Ta, Saa', Saa1, and Sa'a1, since these are sufficient to resolve a. and a, into elements parallel and perpendicular to each other and to a. Hence the mutual effect aa1Uaf (Ta, Saa', Saa1, Sa'α1), is Also, that action and reaction may be equal in absolute magnitude, f must be symmetrical in Saa' and Saa1. Again, a' (as differential of a) can enter only to the first power, and must appear in each term of f. Hence f=ASa'a1+BSaa Saa1. But, by IV, this must be independent of the dimensions of the system. Hence A is of - 2 and B of -4 dimensions in Ta. There {ASaa1Sa'a,+BSaa'S2aa1} is a complete differential, with respect to a, if da = a'. Let |