With (7), (5) becomes so that, if we write we have Now, the two vectors have, as is easily seen, equal tensors; the first is parallel to the line drawn horizontally northwards from the point of suspension, the second horizontally eastwards. Let, therefore, ☎ = x (ai — a sin λ) + y Via, ....... which (x and y being very small) is consistent with (6). Va[(ai—a sin λ) + ÿVia—2ż w sin λVia—2ýw (a—ai sin λ) (11) +n2x(ai — a sin λ)+n2yVia] = 0, a (ai — a sin λ), (— ï — 2 ỳ w sinλ — n2x) a Via + (ÿj — 2 ¿ w sinλ+n2y)a (ai — a sin λ) = 0. This gives at once +n2x+2wy sin λ = 0, ) ÿ+n2y-2 wx sin λ = 0, } ...... (12) which are the equations usually obtained; and of which the solution is as follows : If we transform to a set of axes revolving in the horizontal plane at the point of suspension, the direction of motion being from the positive (northward) axis of x to the positive (eastward) axis of y, with angular velocity Q, so that x = έ cos Qt—ŋ sin Qt, -n کرے y = έ sin t+n cos St, S (13) and omit the terms in 2 and in w (a process justified by the results, see equation (15)), we have o, (§+n2¿) cos Qt — (ÿ+n2ŋ) sin Qt — 2ỷ (N—w sin λ) = 0, } (§+n2§) sin Qt+(ï+n2n) cos £t+2 ở (Q—w sin λ) : = 0. the usual equations of elliptic motion about a centre of force in the centre of the ellipse. (Proc. R. S. E., 1869.) 406.] To construct a reflecting surface from which rays, emitted from a point, shall after reflection diverge uniformly, but horizontally. Using the ordinary property of a reflecting surface, we easily obtain the equation 8. dp (B+aVap)&p P P = 0. By Hamilton's grand Theory of Systems of Rays, we at once write down the second form Tp-T (B+aVap) = constant. w The connection between these is easily shewn thus. Let and any two vectors whose tensors are equal, then τ be Hence, putting w=U(B+aVap) and rUp, we have from the first equation above But and S.dp[Up+U(B+aVap)] = 0. d (B+aVap) = aVadp-dp-a Sadp, so that we have finally S.a (B+aVap) = 0, which is the differential of the second equation above. A curious particular case is a parabolic cylinder, as may be easily seen geometrically. The general surface has a parabolic section in the plane of a, ß; and a hyperbolic section in the plane of ß, aß. It is easy to see that this is but a single case of a large class of integrable scalar functions, whose general type is is the equation of the surface of the reflected wave: the integral of the former being, by the help of the latter, at once obtained in the form TpT(o-p) constant *. 407.] We next take Fresnel's Theory of Double Refraction, but *Proc. R. S. E., 1870-71. merely for the purpose of shewing how quaternions simplify the processes required, and in no way to discuss the plausibility of the physical assumptions. Let to be the vector displacement of a portion of the ether, with the condition @2 = −1, the force of restitution, on Fresnel's assumption, is t(a2iSiw+b2 j$j+c2 kSkw) = tow, (1) using the notation of Chapter V. Here the function is obviously self-conjugate. a2, 62, c2 are optical constants depending on the crystalline medium, and on the colour of the light, and may be considered as given. Fresnel's second assumption is that the ether is incompressible, or that vibrations normal to a wave front are inadmissible. If, then, a be the unit normal to a plane wave in the crystal, we have of a2=-1, course (2) (3) or w-1Vwpw || a, S.awow = 0. (4) This equation (4) is the embodiment of Fresnel's second assumption, but it may evidently be read as meaning, the normal to the front, the direction of vibration, and that of the force of restitution are in one plane. 408.] Equations (3) and (4), if satisfied by, are also satisfied by a, so that the plane (3) intersects the cone (4) in two lines at right angles to each other. That is, for any given wave front there are two directions of vibration, and they are perpendicular to each other. 409.] The square of the normal velocity of propagation of a plane wave is proportional to the ratio of the resolved part of the force of restitution in the direction of vibration, to the amount of displacement, hence v2 = Swpw. Hence Fresnel's Wave-surface is the envelop of the plane Formidable as this problem appears, it is easy enough. From (3) In passing, we may remark that this equation gives the normal velocities of the two rays whose fronts are perpendicular to a. In Cartesian cöordinates it is the well-known equation [It will be a good exercise for the student to translate the last ten formulae into Cartesian coordinates. He will thus reproduce almost exactly the steps by which Archibald Smith* first arrived at a simple and symmetrical mode of effecting the elimination. Yet, as we shall presently see, the above process is far from being the shortest and easiest to which quaternions conduct us.] *Cambridge Phil. Trans., 1835. R 410.] The Cartesian form of the equation (7) is not the usual This last quaternion equation can also be put into either of the new forms T p = 0, 411.] By applying the results of §§ 171, 172 we may introduce a multitude of new forms. We must confine ourselves to the most simple; but the student may easily investigate others by a process precisely similar to that which follows. Writing the equation of the wave as and the equation of the wave in Cartesian cöordinates is, putting r2 = a2x2 + b2 y2+c2x2, 412.] By means of equation (1) of last section we may easily prove Plücker's Theorem. The Wave-Surface is its own reciprocal with respect to the ellipsoid whose equation is |