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in the sense that it is cut by an arbitrary rectilinear transversal in two (real or imaginary) points, and in no more than two, let us assume two points L, M, or their vectors λ = OL, μ =ом, as given; and let us seek to determine the points P (real or imaginary), in which the indefinite right line LM intersects the locus II.; or rather the number of such intersections, which will be sufficient for the present purpose.
without proceeding to resolve which, we see already, by its mere degree, that the number sought is two; and therefore that the locus II. is, as above stated, a surface of the second order.
p is substituted for p; the
(3.) The equation II. remains unchanged, when surface has therefore a centre, and this centre is at the origin o of vectors. (4.) It has been seen that the equation of the surface may also be thus written:
it gives therefore, for the reciprocal of the radius vector from the centre, the expres
and this expression has a real value, which never vanishes,* whatever real value may be assigned to the versor Up, that is, whatever direction may be assigned to p: the surface is therefore closed, and finite.
(5.) Introducing two new constant and auxiliary vectors, determined by the two expressions,
and under these conditions, y is said to be the harmonic mean between the two former vectors, a and ß; and in like manner, & is the harmonic mean between a and -B; while 2a is the corresponding mean between y, 8; and 23 is so, between y and -8.
* It is to be remembered that we have excluded in (1.) the case where ß + a ; in which case it can be shown that the equation II. represents an elliptic cylinder.
CIRCULAR SECTIONS, CYCLIC PLANES.
(6.) Under the same conditions, for any arbitrary vector p, we have the transformations,
the equation IV. of the surface may therefore be thus written :
+KS + V
the geometrical meaning of which new forms will soon be seen.
(7.) The system of the two planes through the origin, which are respectively perpendicular to the new vectors y and d, is represented by the equation,
These two diametral planes therefore cut the surface in two circular sections, with T3 for their common radius; and the normals y and d, to the same two planes, may be called (comp. 196, (17.)) the cyclic normals of the surface; while the planes themselves may be called its cyclic planes.
(8.) Conversely, if we seek the intersection of the surface with the concentric sphere XIV., of which the radius is Tẞ, we are conducted to the equation XII. of the system of the two cyclic planes, and therefore to the two circular sections (7.); so that every radius vector of the surface, which is not drawn in one or other of these two planes, has a length either greater or less than the radius Tẞ of the sphere.
(9.) By all these marks, it is clear that the locus II., or 204, (14.), is (as above asserted) an Ellipsoid; its centre being at the origin (3.), and its mean semiazis being Tẞ; while Uẞ has, by 204, (15.), the direction of the axis of a circumscribed cylinder of revolution, of which cylinder the radius is Tß; and a is, by the last cited sub-article, perpendicular to the plane of the ellipse of contact.
(10.) Those who are familiar with modern geometry, and who have caught the notations of quaternions, will easily see that this ellipsoid II., or IV., is a deformation of what may be called the mean sphere XIV., and is homologous thereto; the infinitely distant point in the direction of ß being a centre of homology, and either of the two planes XI. or XII. being a plane of homology corresponding.
217. The recent form, X. or X'., of the quaternion equation of the ellipsoid, admits of being interpreted, in such a way as to conduct (comp. 216) to a simple construction of that surface; which we shall first investigate by calculation, and then illustrate by geometry.
(1.) Carrying on the Roman numerals from the sub-articles to 216, and observing that (by 190, &c.),
when and are two new constant vectors, and t is a new constant scalar, which we shall suppose to be positive, but of which the value may be chosen at pleasure.
(2.) The comparison of the forms X. and X'. shows that y and d may be interchanged, or that they enter symmetrically into the equation of the ellipsoid, although they may not at first seem to do so; it is therefore allowed to assume that
XVIII... Ty > To, and therefore that XVIII'... Ti > Tk ;
for the supposition Ty To would give, by VI.,
T(B+a)=T(B-a), and.. (by 186, (6.) &c.) BLα,
CONSTRUCTION OF THE ELLIPSOID.
so that the equation XVI. becomes,
XXIV. . . t = T. AE. T.DB.
(5.) The point B is external to the diacentric sphere (4.), by the assumption (2.); a real tangent (or rather cone of tangents) to this sphere can therefore be drawn from that point; and if we select the length of such a tangent as the value (1.) of the scalart, that is to say, if we make each member of the formula XXI. equal to unity, and denote by D' the second intersection of the right line BD with the sphere, as in Fig. 53, we shall have (by Euclid III.) the elementary relation,
XXV... t2 = T.DB.T.BD';
whence follows this Geometrical Equation of the Ellipsoid,
or in a somewhat more familiar notation,
XXVII... AE = BD;
where AE denotes the length of the line AE, and similarly for BD'.
(6.) The following very simple Rule of Construction (comp. the recent Fig. 53) results therefore from our quaternion analysis :
From a fixed point A, on the surface of a given sphere, draw any chord AD; let D' be the second point of intersection of the same spheric surface with the secant BD, drawn from a fixed external point B; and take a radius vector AE, equal in length to the line BD', and in direction either coincident with, or opposite to, the chord AD: the locus of the point E will be an ellipsoid, with a for its centre, and with B for a point of its surface.
(7.) Or thus:
If, of a plane but variable quadrilateral ABED', of which one side AB is given in length and in position, the two diagonals AE, BD' be equal to each other in length, and if their intersection D be always situated upon the surface of a given sphere, whereof the side AD' of the quadrilateral is a chord, then the opposite side BE is a chord of a given ellipsoid.
218. From either of the two foregoing statements, of the Rule of Construction for the Ellipsoid to which quaternions have conducted, many geometrical consequences can easily be inferred, a few of which may be mentioned here, with their proofs by calculation annexed: the present Calculus being, of course, still employed.
(1.) That the corner B, of what may be called the Generating Triangle ABC, is in fact a point of the generated surface, with the construction 217, (6.), may be
It is merely to fix the conceptions, that the point в is here supposed to be external (5.); the calculations and the construction would be almost the same, if we assumed B to be an internal point, or Ti <Tk, Ty < Td.
proved, by conceiving the variable chord AD of the given diacentric sphere to take the position AG; where G is the second intersection of the line AB with that spheric surface.
(2.) If D be conceived to approach to A (instead of G), and therefore D' to G (instead of A), the direction of AE (or of AD) then tends to become tangential to the sphere at A, while the length of AE (or of BD) tends, by the construction, to become equal to the length of BG; the surface has therefore a diametral and circular section, in a plane which touches the diacentric sphere at A, and with a radius = BG.
(3.) Conceive a circular section of the sphere through A, made by a plane perpen dicular to BC; if D move along this circle, D' will move along a parallel circle through G, and the length of BD', or that of AE, will again be equal to RG such then is the radius of a second diametral and circular section of the ellipsoid, made by the lately mentioned plane.
(4.) The construction gives us thus two cyclic planes through A; the perpendiculars to which planes, or the two cyclic normals (216, (7.)) of the ellipsoid, are seen to have the directions of the two sides, CA, CB, of the generating triangle ABC (1.).
(5.) Again, since the rectangle
BA. BG BD. BDBD. AE = double area of triangle ABE: sin BDE, we have the equation,
XXVIII... perpendicular distance of E from AB = BG. sin BDE;
the third side, AB, of the generating triangle (1.), is therefore the axis of revolution of a cylinder, which envelopes the ellipsoid, and of which the radius has the same length, BG, as the radius of each of the two diametral and circular sections.
(6.) For the points of contact of ellipsoid and cylinder, we have the geometrical relation,
XXIX... BDE = a right angle; or
XXIX. . . ADB = a right angle;
the point D is therefore situated on a second spheric surface, which has the line AB for a diameter, and intersects the diacentric sphere in a circle, whereof the plane passes through A, and cuts the enveloping cylinder in an ellipse of contact (comp. 204, (15.), and 216, (9.)), of that cylinder with the ellipsoid.
(7.) Let AC meet the diacentric sphere again in F, and let BF meet it again in F (as in Fig. 53); the common plane of the last-mentioned circle and ellipse (6.) can then be easily proved to cut perpendicularly the plane of the generating triangle ABC in the line AF'; so that the line F'в is normal to this plane of contact; and therefore also (by conjugate diameters, &c.) to the ellipsoid, at B.
(8.) These geometrical consequences of the construction (217), to which many others might be added, can all be shown to be consistent with, and confirmed by, the quaternion analysis from which that construction itself was derived. Thus, the two circular sections (2.) (3.) had presented themselves in 216, (7.); and their two cyclic normals (4.), or the sides CA, CB of the triangle, being (by 217, (4.)) the two vectors κ, 1, have (by 217, (1.) or (3.) ) the directions of the two former vectors y, d; which again agrees with 216, (7.).
(9.) Again, it will be found that the assumed relations between the three pairs of constant vectors, a, ẞ; y, d; and 1, 4, any one of which pairs is sufficient to deter