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(2.) Employing still only quotients of vectors, but introducing two other pairs of vector-constants, y, & and ɩ, к, instead of the pair a, ẞ in the equation I., which were however connected with that pair and with each other by certain assigned relations, that equation was transformed successively to
and to a form which may be written thus (comp. 217, (5.)),
III. . . T 、 + K
and this last form was interpreted, so as to lead to a Rule of Construction* (217, (6.), (7.)), which was illustrated by a Diagram (Fig. 53), and from which many geometrical properties of that surface were deduced (218, 219) in a very simple manner, and were confirmed by calculation with quaternions: the equation and construction being also modified afterwards, by the introduction (220) of a new pair of vector-constants, i' and ', which were shown to admit of being substituted for and x, in the recent form III.
(3.) And although the Equation of Conjugation,
which connects the vectors A, μ of any two points L, M, whereof one is on the polar plane of the other, with respect to the ellipsoid I., was not assigned till near the end of the First Chapter of the present Book, yet it was there deduced by principles and processes of the Second Book alone: which thus were adequate, although not in the most practically convenient way, to the treatment of questions respecting tangent planes and normals to an ellipsoid, and similarly for other surfacest of the same
*This Construction of the Ellipsoid, by means of a Generating Triangle and a Diacentric Sphere (page 227), is believed to have been new, when it was deduced by the writer in 1846, and was in that year stated to the Royal Irish Academy (see its Proceedings, vol. iii. pp. 288, 289), as a result of the Method of Quaternions, which had been previously communicated by him to that Academy (in the year 1843).
+ The following are a few other references, on this subject, to the Second Book. Expressions for a Right Cone (or for a single sheet of such a cone) have been given
in pages 119, 179, 220, 221. In page 179 the equation Ss=1, has been asα P
signed, with a transformation in page 180, to represent generally a Cyclic Cone, or a cone of the second order, with its vertex at the origin; and to exhibit its cyclic planes, and subcontrary sections (pp. 181, 182). Right Cylinders have occurred in pages 193, 196, 197, 198, 199, 218. A case of an Elliptic Cylinder has been already mentioned (the case when 6a in I.); and a transformation of the equation III. of the Ellipsoid, by means of reciprocals and norms of vectors, was assigned in page 298. And several expressions (comp. 403), for a Sphere of which the ori
CHAP. III.] TRANSFORMATIONS OF THE EQUATION.
(4.) But in this Third Book we have been able to write the equation III. under the simpler form,
V... T(ip + px) = x2 — 12,
which has again admitted of numerous transformations; for instance, of all those which are obtained by equating (x2-12)2 to any one of the expressions 336, (5.), for the square of this last tensor in V., or for the norm of the quaternion ɩp + pr; cyclic formst of equation thus arising, which are easily converted into focal forms (359); while a rectangular transformation (373, XXX.) has subsequently been assigned, whereby the lengths (abc), and also the directions, of the three semiaxes of the surface, are expressed in terms of the two vector-constants, i, x: the results thus obtained by calculation being found to agree with those previously deduced, from the geometrical construction (2.) in the Second Book.
(5.) The equation V. has also been differentiated (336), and a normal vector v=pp has thus been deduced, such that, for the ellipsoid in question,
VI... Svdp = 0, and VII... Svp = 1;
a process which has since been extended (361), and appears to furnish one of the best general methods of treating surfaces‡ of the second order by quaternions: especially when combined with that theory of linear and vector functions (pp) of vectors, which was developed in the Sixth Section§ of the Second Chapter of the present Book.
gin was not the centre, occurred in pages 164, 179, 189, and perhaps elsewhere, without any employment of products of vectors.
* Mentioned by anticipation in the Note to page 233.
Compare the second Note to page 633. The vectors and are here the cyclic normals, and is one of the focal lines; the other being the line '-' of page 232.
The following are a few additional references to preceding parts of this Third Book, which has extended to a much greater length than was designed (page 302). In the First Chapter, the reader may consult pages 305, 306, 307, for some other forms of equation of the ellipsoid and the sphere. In the Second Chapter, pages 416, 417 contain some useful practice, above alluded to, in the differentiation and transformation of the equation r2=T(ip+pk). As regards the Sixth Section of that Chapter, which we are about to use (405), as one supposed to be familiar to the reader, it may be sufficient here to mention Arts. 357-362, and the Notes (or some of them) to pages 464, 466, 468, 474, 481, 484. In this Third Chapter, the subarticles (7.)-(21.) to 373 (pages 504, &c.) might be re-perused; and perhaps the investigations respecting cones and sphero-conics, in 394 and its sub-articles (pages 541, &c.), including remarks on an hyperbolic cylinder, and its asymptotic planes (in page 547). Finally, in a few longer and later series of sub-articles, to Arts. 397, &c., a certain degree of familiarity with some of the chief properties of surfaces of the second order has been assumed; as in pages 571, 588, 591, and generally in the recent investigations respecting the osculating twisted cubic (pages 591, 620, &c.), to a helix, or other curve in space.
§ It appears that this Section may be conveniently referred to, as III. ii. 6; and similarly in other cases.
405. Dismissing then, at least for the present, the special consideration of the ellipsoid, but still confining ourselves, for the moment, to Central Surfaces of the Second Order, and using freely the principles of this Third Book, but especially those of the Section (III. ii. 6) last referred to, we may denote any such central and non-* conical surface by the scalar equation (comp. 361),
I... fp = Spøp = 1;
the asymptotic cone (real or imaginary) being represented by the connected equation,
II. . . fp = Spøp = 0;
and the equation of conjugation, between the vectors p, p' of any two points P, P', which are conjugate relatively to this surface I. (comp. 362, and 404, (3.), see also 373, (20.)), being,
III. . . f(p, p')= ƒ (p', p) = Sp$p' = Sp'pp = 1;
while the differential equation of the surface is of the form (361), IV... 0=dfp=2Svdp, with with V... = $p;
this vector-function pp, which represents the normal to the surface, being at once linear and self-conjugate (361, (3.)); and the surface itself being the locus of all the points P which are conjugate to themselves, so that its equation I. may be thus written,
I'...f(p, p) = 1, because f(p, p)=fp, by 362, IV.
(1.) Such being the form of op, it has been seen that there are always three real and rectangular unit-lines, a1, a2, aз, and three real scalars, c1, ez, ez, such as to satisfy (comp. 357, III.) the three vector equations,
VI... a1 = c1α1, pa2 = c2A2,
whence also these three scalar equations are satisfied,
фаз = -сзаз;
VII. fa1 = c1, fa2 = c2, fa3 = C3;
and therefore (comp. 362, VII.),
VIII...ƒ(ca1) =ƒ(c2 ̄1a2)=ƒ(c3 1a3) = 1.
(2.) It follows then that the three (real or imaginary) rectangular lines,
IX... ẞ1 = c1a1, B2 = c2a2, B3=c31a3,
are the three (real or imaginary) vector semiaxes of the surface I.; and that the three (positive or negative) scalars, c1, c2, c3, namely the three roots of the scalar and cubie equation* M = 0 (comp. 357, (1.)), are the (always real) inverse squares of the three (real or imaginary) scalur semiaxes, of the same central surface of the second order.
*It is unnecessary here to write Mo= 0, as in page 462, &c., because the function is here supposed to be self-conjugate; its constants being also real.
GENERAL CENTRAL SURFACE.
(3.) For the reality of that surface I., it is necessary and sufficient that one at least of the three scalars c1, c2, c3 should be positive; if all be such, the surface is an ellipsoid; if two, but not the third, it is a single-sheeted hyperboloid; and if only one, it is a double-sheeted hyperboloid: those scalars being here supposed to be each finite, and different from zero.
(4.) We have already seen (357, (2.)) how to obtain the rectangular transformation,
which may now, by IX., be thus written,
XI... fp = (Sẞ1-1p)2 + (Sẞ2 ̄1p)2 + (Sẞ3 ̄1p)2;
but it is to be remembered that, by (2.) and (3.), one or even two of these three vectors ẞiẞ2ß3 may become imaginary, without the surface ceasing to be real. (5.) We had also the cyclic transformation (357, II. II.'),
XII. . . fp = gp2 + §\pμp = p2 (g — Sλμ) + 2SXpSμp,
in which the scalar g and the vector λ, μ are real, and the latter have the directions of the two (real) cyclic normals; in fact it is obvious on inspection, that the surface is cut in circles, by planes perpendicular to these two last lines.
(6.) It has been proved that the four real scalars, c¡c2cзg, and the five real vectors, aιa2a ̧λμ, are connected by the relations† (357, XX. and XXI.),
XIII... c1=-g- Tλμ,
= U (XTμ – μTX),
at least if the three roots cic2c3 of the cubic M=0 be arranged in algebraically ascending order (357, IX.), so that c1 <C2<C3.
(7.) It may happen (comp. (3.)), that one of these three roots vanishes; and in that case (comp. (2.)), one of the three semiaxes becomes infinite, and the surface I. becomes a cylinder.
(8.) Thus, in particular, if c1 = 0, or g=- TAμ, so that the two other roots are both positive, the equation takes (by XII., comp. 357, XXII.) a form which may be thus written,
XV... (SAμp)2+ (SλpTμ + SμpTλ)2 = Tλμ — Sλμ > 0;
and it represents an elliptic cylinder.
and represents an hyperbolic cylinder; the root c1 being in this case negative, while the remaining root c3 is positive.
* Compare the Note to page 468; see also the proof by quaternions, in 373, (16.), &c., of the known theorem, that any two subcontrary circular sections are homospherical, with the equation (373, XLIV.) of their common sphere, which is found to have its centre in the diametral plane of the two cyclic normals λ, μ.
+ These relations and a few others mentioned are so useful that, although they occurred in an earlier part of the work, it seems convenient to restate them here.
(10.) But if we suppose that c3 = 0, or g=Tλμ, so that c1 and c2 are both negative, the equation may (by 357, XXIII.) be reduced to the form,
XVII... (Sλμp)2 + (SλpTμ − SμpTX)2 = − Tλμ – Sλμ < 0 ;
it represents therefore, in this case, nothing real, although it may be said to be, in the same case, the equation of an imaginary* elliptic cylinder.
(11.) It is scarcely worth while to remark, that we have here supposed each of the two vectors A and μ to be not only real but actual (Art. 1); for if either of them were to vanish, the equation of the surface would take by XII. the form,
XVIII... p2 = g1, or XVIII'. . . Tp = (−g) ̄*,
and would represent a real or imaginary sphere, according as the scalar constant g was negative or positive: λ and μ have also distinct directions, except in the case of surfaces of revolution.
(12.) In general, it results from the relations (6.), that the plane of the two (real) cyclic normals, λ, μ, is perpendicular to the (real) direction of that (real or imaginary) semiaxis, of which, when considered as a scalar (2.), the inverse square c2 is algebraically intermediate between the inverse squares ci, c3 of the other two; or that the two diametral and cyclic planes (Sλp = 0, Sμp = 0) intersect in that real line (Vλμ) which has the direction of the real unit-vector a2 (1.), corresponding to the mean root c2 of the cubic equation M=0: all which agrees with known results, respecting the circular sections of the (real) ellipsoid, and of the two hyperboloids.
406. Some additional light may be thrown on the theory of the central surface 405, I., by the consideration of its asymptotic cone 405, II.; of which cone, by 405, XII., the equation may be thus written,
I. • fp=gp2 + S\pμp = p2 (g − Sλμ) + 2SλpSμp = 0; and which is real or imaginary, according as we have the inequality,
II... g2 <3μ3, or III... g>\2μ2;
that is, by 405, (6.), according as the product cc, of the extreme roots of the cubic M=0 is negative or positive; or finally, according as the surface fp = 1 is a (real) hyperboloid, or an ellipsoid (real or imaginary†).
*In the Section (III. ii. 6) above referred to, many symbolical results have been established, respecting imaginary cyclic normals, or focal lines, &c., on which it is unnecessary to return. But it may be remarked that as, when the scalar function fo admits of changing sign, for a change of direction of the real vector p, so as to be positive for some such directions, and negative for others, although ƒ(−p)=ƒ(+p), the two equations, fp =+1, fp = 1, represent then two real and conjugate hyperboloids, of different species: so, when the function fo is either essentially positive, or else essentially negative, for real values of p, the equations fp = 1 and fp=-1 may then be said to represent two conjugate ellipsoids, one real, and the other imaginary. + Compare the Note immediately preceding; also the second Note to page 474.